Percolation on complex networks: Theory and application
Ming Li, Run-Ran Liu, Linyuan Lü, Mao-Bin Hu, Shuqi Xu, Yi-Cheng Zhang
aa r X i v : . [ phy s i c s . s o c - ph ] J a n Percolation on complex networks: Theory and application
Ming Li a , Run-Ran Liu b , Linyuan L¨u c,b,d, ∗ , Mao-Bin Hu a , Shuqi Xu c , Yi-Cheng Zhang e a Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei, 230026, P. R. China b Alibaba Research Center for Complexity Sciences, Hangzhou Normal University, Hangzhou, 310036, P. R. China c Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, P. R. China d Beijing Computational Science Research Center, Beijing, 100193, P. R. China e Department of Physics, University of Fribourg, Fribourg CH-1700, Switzerland
Abstract
In the last two decades, network science has blossomed and influenced various fields, such as statistical physics, com-puter science, biology and sociology, from the perspective of the heterogeneous interaction patterns of componentscomposing the complex systems. As a paradigm for random and semi-random connectivity, percolation model playsa key role in the development of network science and its applications. On the one hand, the concepts and analyticalmethods, such as the emergence of the giant cluster, the finite-size scaling, and the mean-field method, which areintimately related to the percolation theory, are employed to quantify and solve some core problems of networks. Onthe other hand, the insights into the percolation theory also facilitate the understanding of networked systems, such asrobustness, epidemic spreading, vital node identification, and community detection. Meanwhile, network science alsobrings some new issues to the percolation theory itself, such as percolation of strong heterogeneous systems, topolog-ical transition of networks beyond pairwise interactions, and emergence of a giant cluster with mutual connections.So far, the percolation theory has already percolated into the researches of structure analysis and dynamic modelingin network science. Understanding the percolation theory should help the study of many fields in network science,including the still opening questions in the frontiers of networks, such as networks beyond pairwise interactions, tem-poral networks, and network of networks. The intention of this paper is to o ff er an overview of these applications, aswell as the basic theory of percolation transition on network systems. Keywords:
Percolation, Complex Network, Network Structure, Network Dynamics, Phase Transition and CriticalPhenomena ∗ Corresponding author
Email address: [email protected] (Linyuan L¨u )
Preprint submitted to Physics Reports January 29, 2021 ontents1 Introduction 2 / multiplex networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Model and phase transition characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Algorithms for reducing time complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.3 Variants and related models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Explosive percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Achlioptas process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Phase transition characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Percolation transition during the growth of networks . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Growing random network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Variants and related models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ffi c and transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3.1 Percolation in urban tra ffi c networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3.2 Percolation in connected vehicle networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.3 Percolation in urban tra ffi c planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.4 Percolation in post-disaster tra ffi c networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.4 Evolutionary game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 . Introduction In contrast to many other modern research fields, the network problem is often easy to define by abstracting fromeveryday life [1, 2]. For examples, how many people an epidemic can infect in a social contact network, whethera communication network can maintain its function after an intentional attack, which node has the largest impact ina social network, and so on [3–6]. The key point of these problems with network involved can be summarized as acluster forming process within a chosen fraction of nodes, those might be infected people, preserved nodes after anintentional attack, or individuals with the same opinion. In principle, these processes are easy to define, however, notso easy to solve.Fortunately, in statistical physics, a profound theoretical system, called percolation theory, just touches this prob-lem, i.e. , the behaviors of a networked system when some of nodes or links are not available [7]. Indeed, whenthe network science was just a new rising topic, the percolation theory has already been widely used for explainingempirical results, and solving models [2]. Now, after more than twenty years’ development of network science, thepercolation theory, including conceptions, analytical methods, and algorithms, can be found almost in all the researchfields of network science.It is known that the classical percolation in statistical physics only considers regular lattices, therefore, with theseapplications to complex networks, the percolation theory itself also has been enriched and developed. In recent hotareas of network science, such as higher-order networks [8, 9] and networks beyond pairwise interactions [10], modelsand methods of percolation have still been widely touched. This is obviously because the connection property mustalways be a key point to understand network structure and dynamics.Due to importance of the percolation theory in the study of complex networks, almost all the review articlesabout networks have the relevant sections to introduce distinguishable developments and applications of percolationtheory on complex networks, however, a systematic comparison and summary specifically from the perspective ofpercolation is still absent. This review article aims to fill this gap, and comb the scattered discussions on the networkpercolation and its applications, which can facilitate a wide area of sciences, ranging from physics and computerscience to biology and sociology, as well as various branches of probability theory in mathematics.
Percolation now usually refers to a class of models that describe geometric features of random media. In statisticalphysics, percolation theory is often accompanied by scaling law, fractal, self-organization criticality, and renormaliza-tion, which are all of immense importance theoretically in many diverse fields of physics [7]. Therefore, percolationhas long served as a basic ideal model for demonstrating phase transition and critical phenomena. However, quiteapart from the role percolation theory now plays, it originates from an honest applied problem in the study of gelationin the 1940s [11–13]. To be a mathematical subject, it first starts from Broadbent and Hammersley’s paper in 1957[14], in which its name, and the geometrical and probabilistic concepts were introduced.The study of percolation then becomes popularized in the physics community, and many of the open problemshave been proposed and solved [15–21]. Now, the percolation theory has also been found to be of a broad range ofapplications to diverse problems. The applications in network science that this article focuses on are one of the typicalrepresentatives. This is also benefited from the development of computational technology, as the simulation plays acrucial role in the study of percolation [22].In the following, we will firstly give a brief introduction to the percolation model, as well as some physicalconcepts and quantities involved.
Imagine a large porous stone immersed in water. Does the water come into the core of the stone? If so, theremust be some paths formed by the pores running through the stone. However, as a whole, the connection of any twoadjacent pores is probabilistic, supposing the probability is p . The problem thus reduces to whether there exist suchpaths for a given probability p . This actually is a typical example of percolation problems. The connection of twoadjacent pores, in the terminology of percolation, is called occupying the corresponding bond between the two pores,and hence p is the occupied probability. Obviously, if p is large enough, the core of the porous stone can be wetted.In that case, the connected pores are able to form a cluster that penetrates the stone. Accordingly, percolation theoryis mainly concerned with the existence of such a cluster and its structure with respect to the occupied probability p .2he above process is just a special case of the percolation process. The same question can also be proposed formany other systems constituted by random medium. Another typical example is the forest fire model. Suppose aburning tree can only ignite the trees in the adjacent sites, the destructiveness of the forest fire depends on the densityof the trees, i.e. , the probability p of finding a tree on a site. This is also obviously a percolation process – when theadjacent trees form a cluster that can penetrate the entire forest, the fire could raze almost the entire forest; otherwise,the fire is constrained in a small area. Furthermore, similar regulations can also be applied to model the spreading ofepidemics among individuals, where infected individuals can infect their neighbors probabilistically.Although the soaking of porous stone, the forest fire and the spreading of epidemics seemingly belong to separatedfields, the three are now converging in an intriguing manner. That is, is there a cluster of connected sites through thesystem? As is quite typical, it is actually easier to examine such an infinite cluster in an infinite system than just largeones. With the increasing of the probability p , there must be a critical p c , called percolation threshold or critical point,below which such a cluster cannot be found. For a more detailed understanding of this criticality, the percolationmodel needs to be defined explicitly. That is what we are going to talk about. To facilitate presentation, we consider a two-dimensional lattice here as shown in Fig.1 (a). Mathematically, eachbond is occupied with probability p or unoccupied with probability 1 − p . Then, the occupied bonds connect thesites into clusters. This model is called bond percolation, which can be used to model the process of the soaking ofporous stone and the spreading of epidemics. For the forest fire model, one often uses a slightly di ff erent percolationmodel, the so-called site percolation model. For this model, we occupy each site with probability p rather than bondsas shown in Fig.1 (b). In general, the bond percolation is considered less general than the site percolation due to thefact that the bond percolation can be reformulated as a site percolation on a di ff erent lattice, but not vice versa.The percolation theory mainly focuses on the emergence of the infinite cluster with the increasing of probability p . To characterize this phenomenon, one often adopts the size of the giant cluster, which is defined as S = lim N →∞ s N . (1)Here, N is the size of the system (site number), and s is the number of sites in the largest cluster. As shown inFig.1, with the increasing of p , there must be a critical point p c , above which a non-zero S can be found. This figuresout the percolation transition of the system with respect to the control parameter p , and S is the corresponding orderparameter.In addition, there is another commonly used order parameter called wrapping probability, which is defined asthe probability that a cluster wraps around the periodic boundary conditions on a regular lattice. For large systems,this probability is equal to the probability that the system percolates. This parameter is usually used to estimate theposition of the percolation threshold, since it is a size-independent parameter at the critical point. Specifically, clusterwrapping can be defined in a number of di ff erent ways, such as wrapping around one direction, wrapping around onedirection but not the other, and wrapping around both directions. This probability, however, cannot be well definedin network systems without spatial constraints, so that the size of the giant cluster is the preferred order parameter innetwork science.To describe the features of finite clusters, the distribution of cluster sizes p s = m s / P s m s is also used, where m s isthe number of the clusters with size s . Sometimes, the normalized cluster number n s = m s / N is also used to featurethe cluster size distribution. It is obvious that p s = n s N / P s m s = n s h s i , where h s i = N / P s m s = P s sp s is the averagecluster size.It must be pointed out that the average size h s i is not the mean cluster size χ commonly used in percolation theory,which is defined as χ = P s s n s . It is not hard to know that sn s is the probability that a randomly chosen site belongsto a cluster with size s . Thus, the mean cluster size χ actually is the average size of the cluster that a randomly chosensite belongs to. Together with the characteristic length ξ and the characteristic size s ξ , above which the clusters areexponentially scarce, these statistics are often used to describe and characterize the percolation transition [7]. Thecritical behaviors of these parameters will be briefly introduced in the next section.Besides, there exist several other variants of percolation. For example, one could drop the assumption of inde-pendence of occupation, so that the occupation of a site or a bond could depend on the state of other sites or bonds.This type of percolation is often called dependent percolation, which is widely used in network science to model the3 = 0.4 p = 0.585 p = 0.605p = 0.51p = 0.49p = 0.3 (b)(a) p = 0.3 Figure 1: (Color online) Schematic diagrams of the classical percolation on square lattice, where di ff erent colors denote di ff erent clusters. The sizeof the system used here is N = L × L = ×
80. The values of p labeled in the figures are the corresponding site / bond occupied probabilities. (a)Bond percolation. For ease of identification, the sites and the unoccupied bonds are not shown here. For p = .
51, a giant cluster exists indicatedby yellow. (b) Site percolation. For ease of identification, the bonds and the unoccupied sites are not shown here. For p = . The prerequisite for the study of critical phenomena is determining the percolation threshold p c . Figure 1 indi-cates that the site percolation and the bond percolation have di ff erent critical points. In fact, the threshold of bondpercolation on square lattice can be solved exactly by the duality transformation or real-space renormalization [7, 24–27], that is p c = /
2. However, there is no exact solution for site percolation on square lattice. Through MonteCarlo simulation, a good estimate of the critical point can be obtained, such as p c = . p c = . Table 1: Selected percolation thresholds for various networks. Here, z is the coordination number of the Bethe lattice, m is the minimum degree ofSF network, ζ ( s , a ) is the Hurwitz zeta function, G ( x ) is the generating function of the excess-degree distribution, and h k n i is the n th moment ofthe degree distribution.Network Site Bondsquare lattice 0 . . / / π/
18) [30]honeycomb lattice 0 . . . − π/
18) [30]simple cubic 0 . . . . . . . . d = . . . . . . . . . . . d = . . . . . . d = . . . . . . d = . . . . . . / z / z ER network 1 / h k i [47, 48] 1 / h k i [47, 48]SF network ζ ( λ − , m ) / [ ζ ( λ − , m ) − ζ ( λ − , m )] [48] ζ ( λ − , m ) / [ ζ ( λ − , m ) − ζ ( λ − , m )] [48]tree-like network 1 / G ′ (1) = h k i / [ h k i − h k i ] [48] 1 / G ′ (1) = h k i / [ h k i − h k i ] [48] In the subcritical regime ( p < p c ), all clusters are finite and the size distribution has a tail which decays exponen-tially; in the supercritical regime ( p > p c ), an infinite cluster can be found in the system, and the size distribution ofother finite clusters has a tail which decays slower than exponential. Near the critical point, some asymptotic behav-iors can be found, referred to as critical phenomena. In the percolation theory, these behaviors are characterized bythe critical exponents [7], S ∝ ( p − p c ) β , (2) χ ∝ | p − p c | − γ , (3) ξ ∝ | p − p c | − ν , (4) s ξ ∝ | p − p c | − /σ , (5) p s ∝ s − τ . (6)with relations β = τ − σ , (7) γ = − τσ . (8)5ere, β , γ , ν , σ , and τ are the so-called critical exponents, which determine the universal class of the percolationtransition. Besides, at the critical point the characteristic length ξ and size s ξ also have a relation s ξ ∝ ξ d f . (9)The exponent d f is often called fractal dimension [7, 49], characterizing the structure of the infinite cluster at thecritical point. Assuming the dimension of the system is d , there is another relation between critical exponents, calledhyperscaling, d f = d − βν . (10)Note that this relation is also universal, and independent of the topological structure of the system. The phase transitiontheory points out that there is an upper critical dimension d c (for percolation d c = d ≤ d c .It is also worth mentioning that the geometric structure of high-dimensional percolation clusters cannot be fullyaccounted for by the counterparts of random networks, though both of them have the mean-field nature [52].In addition, the critical behavior below d c is di ff erent from the mean-field approximation which is valid away fromthe phase transition. For these systems, the renormalization group theory has made remarkable predictions about thebehavior of the percolation near and at the threshold [53, 54]. In the renormalization group theory, there is also alower critical dimension (for percolation d l = p c is determined by the local structure of the sys-tem, whereas the behavior of clusters that is observed near p c is independent of the local structure (lattice type andpercolation type). In this sense, percolation is believed to be a substrate-dependent but model-independent process,and therefore, the critical exponents of the transition are only determined by the geometry of the system, and identicalfor the bond and site percolation. The critical exponents are thus more natural to be considered than the threshold p c ,and there is no need to deal with the site and bond percolation, individually.To end this subsection, it must be pointed out that the critical exponent β is distinguishable for the site and bondpercolation on scaling-free (SF) networks [55]. This is a special case derived from the vanished percolation threshold,which has no e ff ect on the other properties discussed above. This paper mainly focuses on the percolation theory on networks and its applications. So, in this section we willprovide an overview of some simple and general properties and models of networks.
In network science, the elements of a system and the connections between them are no longer known as site andbond. Instead, they are often called node and link, or vertex and edge, respectively. The number of links a node has iscalled degree, labeled k .To exactly represent the connection pattern of a network, one often uses a N × N matrix A , called adjacent matrix,where N is the number of nodes in the network. In this way, the element a i j of A is one when there is a link from node i to node j , and zero when there is no link. For undirected networks, it must have a i j = a ji , thus A is a symmetricmatrix. For weighted networks, the element a i j can further be any non-zero value to represent the corresponding linkweight. Instead of studying a special network with a given adjacent matrix, the network science is more on discovering thecommon nature of a class of networks, i.e. , a network ensemble. For that, networks are often featured by the degreedistribution p k , which provides the probability that a randomly selected node in the network has degree k .A typical network ensemble is the one with Poisson degree distribution p k = e −h k i h k i k k ! , (11)6here hi means the average over all the nodes. This just is the ensemble of the known Erd˝os–R´enyi (ER) randomnetworks / graphs [47], and can be simply realized by randomly connecting a given number of links among a set ofnodes, or connecting each pairs of nodes with a given probability. Quite obviously, the generation of ER networks isactually a percolation process. Only when the system percolates ( h k i > h k i < N , at h k i =
1, the largest cluster has a size of order N / , and at h k i >
1, there exists a single giant cluster of size of order N and all other clusters have a size of order log N . Sincesuch systems have no spatial constraint, the critical phenomena in this ensemble just recover the mean-field solution.Besides, distinguishing from the degree distribution, there is another kind of commonly used networks, whichrepresents a deeper organizing principle of real networks called the SF property [2]. In mathematical terms, the SFproperty translates into a power-law function of the form p k ∝ k − λ , k ≥ m . (12)For a real network, there must also be an upper bound of degree, called degree cuto ff K .The main di ff erence between an ER and a SF network comes in the tail of the degree distribution. For ER networks,most nodes have comparable degrees and hence the degree cuto ff is in the order of the average degree. In contrast,nodes with very large degrees are expected in SF networks, called hubs of the networks. Indeed, the degrees of hubsgrow with the network size, and thus can grow quite large. This indicates that SF networks have strong heterogeneity,and hence the percolation transition bears a strong dependence upon the degree distribution. Beyond this, there aremany other measurements to further subdivide the network ensembles, such as clustering, correlation, and community.In recent years, temporal networks, multiplex networks, and high-order networks have also received a lot of attention.Because of space limitation and the focus of this paper, the details will not be included here and brief discussions willbe given in this article when necessary. Further details on these can also be found in Refs.[1–3, 5, 6]. After a review of the fundamental properties of complex networks, we explore the mathematical modeling ofnetwork ensembles, and mainly focus on the configuration model and the hidden parameter model [1, 2]. In principle,the two models can generate any network ensembles with a meaningful degree distribution. In fact, these two modelsare usually only used to realize SF property, since there exist easy-implemented models for Poisson degree distribution(ER networks) and small-world (SW) networks [1, 2]. Note that the SF network also has an easy-implementedmodel, Barab´asi–Albert (BA) model [56], nevertheless, the generated SF network has its own structural features andlimitations rather than a network with programmable and tunable SF properties.From random graph theory [47], there are two ways to generate a network with Poisson degree distribution. Oneis randomly connecting nodes until the excepted number of links is met, and the other is connecting each pair of nodeswith a given probability. For a given average degree, networks implemented by the former method must be a subsetof those by the later one. In spite of this, the two methods will become equivalent in the thermodynamic limitation( N → ∞ ).The SW network grows out of a regular network, such as the triangle lattice and the square lattice, by randomlyrewiring a small fraction of links [57–60]. Since there is no spatial constraint for the rewired link, the average distanceof nodes becomes much smaller than that of the underlying regular network. This is why it is called small-world. Inturn, the regular backbone of the SW network usually leads to a high clustering, namely, the neighbors of a node arealso connected. This is another characteristic of the SW network. For this type of networks, the degree distribution isoften not the concern, and one often uses the backbone network and the rewiring probability to define a SW networkensemble.The configuration model can help us build a network with a pre-defined degree distribution [1, 2, 61, 62]. Thealgorithm consists of two steps. First, assigning degrees to each node drawn randomly from the pre-defined degreedistribution, represented as stubs. Then, two stubs are selected randomly and connected. Repeating this procedureuntil all stubs are paired up. If there is nothing in this procedure to forbid self- and multi-connections, the obtainednetwork is probably not a simple graph as usually studied in network science. While rejecting such connections couldadd much more overhead to the program’s execution time.Indeed, one of the e ff ective methods to generate random networks by the configuration model without self- andmulti-connections can be as follows. First, connecting the stubs in any fast and easy-to-implement ways (the details7 bc d a bbcc dda R R Figure 2: Schematic diagram of the rewiring procedure. Here, the rewiring means that swapping the ends of two links. There are two ways torewire, labeled R and R respectively in the figure. Note that, to ensure the ergodicity and the detailed balance, the two ways must be chosenrandomly. are dependent on the specific degree distribution), and recording the self- and multi-connections. Then, rewiring afraction of links of the obtained network, and the self- and multi-connections must be included. Note that the rewiringleading to self- or multi-connections is forbidden.A schematic diagram of the rewiring procedure is shown in Fig.2. For the degree distribution with the maximumdegree K larger than √ N , this method could be much more e ff ective than the original one, which may not evencomplete the network. This is because there must be some degree correlation in the network with K > √ N [63, 64],randomly connecting stubs with self- and multi-connection forbidden is a very ine ffi cient procedure. Of course, thee ffi ciency of the algorithm also depends on the implementing way and the detailed property of the network, thus thisjust is a general discussion and not always the case.By the way, the rewiring operation as shown in Fig.2 is often called degree preserving randomization, whichcan randomize the connections of a network without changing the degree distribution. With this operation, one caneliminate other properties from the network, for example, clustering, and assure that a certain network feature ispredicted by its degree distribution alone. If the network is large enough and all the degrees are much smaller than thenetwork size N , the degree preserving randomization could turn the network to be locally tree-like.Note that the correlation derived from the degree distribution cannot be undone by this randomization procedure.A typical example is a network with K > √ N . Consequently, when we study networks with hubs, i.e. , SF networks,the configuration model and the degree preserving randomization must be carefully dealt with. In turn, with someconstraints, this rewiring operation can also “randomize” the network into the one with a given high-order property[65, 66].There is another model, hidden parameter model, to generate networks with a pre-defined degree distribution[67–71]. In this model, each node i is assigned a hidden parameter η i , chosen from a pre-defined distribution p η .Then, we connect each node pair with probability p ( η i , η j ) = η i η j / N h η i . The expected number of links is thus L = P i , j η i η j / N h η i . For SF networks, we can set η i ∝ i − α , and the obtained network has the degree distribution p k ∝ k − (1 + /α ) .The SF networks generated by the configuration model and the hidden parameter model also have their ownproperties, thus are not exactly equivalent, at least for small systems. First of all, the boundaries of degrees must begiven for the configuration model to normalize the degree distribution. However, the boundaries of degrees are notneeded for the hidden parameter model, in which a degree cuto ff occurs naturally at N / ( λ − , thus known as the naturalcuto ff of the SF network [72–76]. This cuto ff can be also found in the configuration model, if the upper boundary M used to normalize the degree distribution is larger than N / ( λ − . Then, the hidden parameter model generates no self-8nd multi-connections, since only a single connection would occur between each node pair. Another is that the degreedistribution of the SF network generated by the hidden parameter model has negative deviations for small degrees,while the one generated by the configuration model almost perfectly matches the pre-defined degree distribution. Inaddition, setting the total number of links is not allowed in the configuration model, since it is fixed by the pre-defineddegree distribution. Rather, the hidden parameter model allows controlling the link number, accurately. Finally, whatneed to be pointed out is that both the networks generated by the configuration model and by the hidden parametermodel are only a possible realization of the pre-defined degree distribution. To check the property induced by a certaindegree distribution, the ensemble average must be done.In recent years many works have also been devoted to the so-called multiplex networks (or multilayer networks,interdependent networks) [8, 9]. In this network ensemble, links are divided into di ff erent layers to represent di ff erenttypes of connections between nodes. To generate such a network ensemble, one can first generate several networksusing the methods mentioned above, and then bundle nodes from di ff erent networks together to form the layeredconnections. It can also be interpreted as inserting some inter-connections between nodes in di ff erent networks (lay-ers), which represent the dependence relations between them. The bundling of nodes (or inter-connection) could beone-to-one, one-to-many, or with some other rules, that depending on the inter-correlation between layers. Of course,there are many other network models with di ff erent properties, one can find them in special books and reviews aboutcomplex networks. This paper contains six sections. In the next section, we start by reviewing the basic properties of the percolationprocess on networks, including the model definition, the analytical methods, and the corresponding discussions. Sec-tion 3 reviews some typical percolation models on networks, mainly focusing on the phase transition and the criticalphenomena. The percolation model is by nature featuring the cluster forming process in random media. Therefore,the percolation theory including conceptions, theoretical methods, and algorithms, is widely used in network structureanalysis. This is the main content of Sec.4. Moreover, the percolation process is often used to model and analyzenetwork dynamics, which will be reviewed in Sec.5. Finally, we summarize and prospect to this paper in Sec.6.In addition, the main notations used in this paper are listed in Tab.2 for readability.
2. Classical percolation on networks
In statistical physics, the percolation model is often displayed on regular lattices. However, the regular topologicalstructure is not always the case in reality. In this section, we will review the basic properties of the classical percola-tion on heterogeneous network systems, mainly focusing on analytic methods, critical phenomena, and Monte Carloalgorithms of the classical percolation. It should be pointed out that there are a lot of theoretical studies of percolationtransition on networks from the perspective of random graphs, one can refer to Refs.[47, 77–82] and the relevantreferences to learn more. Here we mainly focus on the network percolation from the perspective of statistical physicsand network science.
As a lattice model, the percolation model can be easily extended to networked systems, namely, nodes or links arerandomly designated either occupied (with probability p ) or unoccupied (with probability 1 − p ). For convenience, innetwork science one usually removes each node or link with probability 1 − p to realize the percolation model. Theparameters and the corresponding problems can be defined on the preserved network as that on regular lattices. As aresult, there is no need to repeat the arguments. Here we will concentrate on the relationships between the classicalpercolation model and some typical issues of network science, such as network robustness, and epidemics spreadingon network.For site percolation, the network science often translates the unoccupied nodes as failed / removed ones. In thisway, the percolation process is just the performance of the network under random nodal failures. If the giant clusterremains after the percolation process, the resulted network can be thought of as a whole, and still functional. Thusthe percolation threshold p c can be used to evaluate the robustness of the network [72, 83–86]. A large threshold p c indicates that only a small amount of failed nodes can disconnect the network, so its robustness is poor, and vice9ersa. Furthermore, if the unoccupied nodes are chosen with preference, this model can further explain the robustnessof networks under intentional attacks. A remarkable knowledge of this model is the finding of the strong robustnessof SF networks under random failures, and extremely fragile for intentional attacks [85]. A specific discussion on thisproblem can be found in Sec.4.In network science, bond percolation often refers to the spreading process [23]. The connection between thespreading of epidemic and percolation is in fact one of the original motivations for the percolation model itself [7,16, 17, 25]. For this process, the occupation of a link means a successful infection between the two connected nodes.That is, the occupied probability reflects the infectiousness of the epidemic. However, di ff erent from the mappingbetween site percolation and models for network robustness, the link occupied probability is not simply equivalent tothe infectious probability, but an integrated probability of transmission of the disease between two individuals [87].In this way, the emergence of the giant cluster corresponds to the outbreak of an epidemic, so that the percolationthreshold can also be used to evaluate the infectiousness of the epidemic. An epidemic with smaller percolationthreshold p c has a stronger infectious ability. Specific discussion on this problem can be found in Sec.5.The basic mechanisms of the network percolation and their extensions can be further applied in various fields ofnetwork science from structure analysis to dynamics modeling, which this article explores next. Although the percolation model is easy to define, the exact solutions are absent for most systems. If the networktopology is restricted to be tree-like, like that of Bethe lattice, there is a mean-field method based on the branchingprocess for solving the percolation model. Here we introduce the framework of this method, including the solutionsfor the size of the giant cluster and the distribution of finite clusters [1, 48, 87].
The percolation process is tantamount to dilute links and thus changes the degree distribution p k . So, for con-venience and generality, the occupation of nodes or links will not be considered in the following discussion. Afterobtaining the result, we can revise it by considering the diluting e ff ect of the percolation on the degree distribution p k .When a network percolates, there must be an infinite cluster, in which the branching process is endless. Specif-ically, when arriving at a node by following a link, the node must have some other links (at least one) to ensure thecontinuity of branching. In terms of this, assuming a link belongs (connects) to the giant cluster with probability R ,we have a self-consistent equation R = X k = q k h − (1 − R ) k − i = − G (1 − R ) , (13)where q k = p k k / h k i is the probability that the node reached by following a link has degree k , known as the excess-degree distribution, and G ( x ) = P k q k x k − is the corresponding generating function. The term 1 − (1 − R ) k − in thesquare brackets just means that at least one excess link is needed to keep branching.After obtaining R from Eq.(13), the order parameter S , i.e. , the probability that a randomly chosen node belongsto the giant cluster (the infinite cluster), can be expressed as S = X k = p k h − (1 − R ) k i = − G (1 − R ) , (14)where G ( x ) = P k p k x k is the generating function of the degree distribution. The right hand side of this equationmeans that at least one of the neighbors must belong to the giant cluster. From the perspective of branching, thisequation can also be interpreted as the start point of the branching process, while Eq.(13) is the intermediate process.In the infinite cluster any nodes can be the start point, so S is an average probability, as well as the parameter R .Furthermore, for the percolation with occupied probability p , we only need to replace the generating functionsin Eqs.(13) and (14) with those of the diluted network. Next, let us calculate the degree distribution of the dilutednetwork. Regardless of the type of the percolation (site percolation or bond percolation), the diluting ratio of links is10 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 1.0 bond percolation site percolation S p (a) ER network RR network p (b) Figure 3: Percolation on ER networks and random regular (RR) networks. The scatters are the simulation results on networks with size N = ,the corresponding lines are the theoretical prediction of Eqs.(17)-(19). (a) ER networks with average degree h k i =
4. For this case the generatingfunctions can be simplified as G ( x ) = G ( x ) = e −h k i (1 − x ) . (b) RR networks with all the nodes having identical degree k =
4. For this case thegenerating functions can be simplified as G ( x ) = x k and G ( x ) = x k − . p , i.e. , each link is preserved with probability p . So the generating functions g ( x ) and g ( x ) of the diluted networkscan be written as g ( x ) = ∞ X k ′ = p k ′ x k ′ = ∞ X k ′ = ∞ X k = k ′ p k kk ′ ! p k ′ (1 − p ) k − k ′ x k ′ = G (1 − p + px ) , (15) g ( x ) = ∞ X k ′ = p k ′ k ′ p h k i x k ′ − = ∞ X k ′ = ∞ X k = k ′ p k kk ′ ! p k ′ (1 − p ) k − k ′ k ′ p h k i x k ′ − = G (1 − p + px ) . (16)Then, inserting them into Eqs.(13) and (14), we have [48, 87] R = − g (1 − R ) = − G (1 − pR ) , (17) S = − g (1 − R ) = − G (1 − pR ) . (18)Before continuing to consider the details of these two equations, we must point out that the order parameter S here isthe size respected to the diluted network. For bond percolation, the dilution is only for links, so the diluted networkhas the same size as the original network and S is also the size respecting to the original network. However, for sitepercolation the diluted network consists of only occupied nodes (fraction p respected to the original network). As aresult, Eq.(18) should be revised for site percolation, that is S = p (cid:2) − G (1 − pR ) (cid:3) . (19)In general, one can solve Eq.(17) first, and then insert R into Eq.(18) or (19) to get the order parameter S for thebond percolation and the site percolation, respectively. From Eqs.(17)-(19), we can also find that the two types of11ercolation models give the same probability R , but di ff erent giant clusters. For mathematical tractability, sometimesone just uses R to represent the meaning of pR for site percolation in Eqs.(17) and (19), that is R = p [1 − G (1 − R )] , (20) S = p [1 − G (1 − R )] . (21)These two equations are fully equivalent to Eqs.(17) and (19).Although the mean-field approximation is used in the above discussion, i.e. , all the nodes have the same probability S and all the links have the same probability R , Eqs.(17)-(21) can give an exact solution for the percolation on tree-likenetworks, see Fig.3 as an example. A comparison between the mean-field prediction and the numerical simulation onsome real-world networks can also be found in Ref.[88].Following the idea of branching, the microstructure of the giant cluster in random networks [89], such as degreedistribution, and degree correlations, as well as that of temporal networks [90], can also be obtained. For networkswith loops, di ff erent links could lead to the same nodes in the branching process. In other words, the branchingprocesses starting from di ff erent links are no longer independent of each other, thus all the equations (13)-(21) are notvalid. That is why the above method is only applicable to the tree-like structure. Besides the giant cluster, the percolation theory also concerns the behaviors of finite clusters, such as mean clustersize, and cluster size distribution. Next, along with the idea of the branching process used above, we will further showhow to obtain the information of finite clusters in percolation model.As discussed in Ref.[48], it is more convenient to study the distribution π s = n s s = p s s / h s i rather than n s or p s ,which means the probability distribution of the size of the cluster that a randomly chosen node belongs to. One canfurther define ρ s as the size distribution of the cluster at the end of a link. Accordingly, the generating functions havethe forms H ( x ) = P s = π s x s and H ( x ) = P s = ρ s x s . For convenience, the occupation of links and nodes is not takeninto account in the following. When necessary, one can use the revised generating functions Eqs.(15) and (16) to getthe corresponding results.Note that a zero size has no physical senses, so π =
0, while ρ can be non-zero corresponding a node with degreeone. Since H (1) and H ( x ) only contain the finite clusters, and the giant cluster is excluded (if there is one), we have H (1) = P s π s = − S and H (1) = P s ρ s = − R , where S and R are defined as that for Eqs.(13) and (14). Belowthe critical point, R = S =
0, so H (1) = H (1) = π s and ρ s are dependent on both the degree / excess-degreedistribution and the branching size at the end of a link. Thus, Fig.4 can be translated as the equations H ( x ) = x X k = p k [ H ( x )] k = xG [ H ( x )] , (22) H ( x ) = x X k = q k [ H ( x )] k − = xG [ H ( x )] . (23)Note that there is one more factor x on the right hand sides of the two equations for the contribution of the root node.Using the relations H (1) = − S and H (1) = − R , these two equations just recover Eqs.(13) and (14) when x = H ( x ), and then use π s = d s − dx s − H ( x ) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = (24)to find the cluster size distribution. However, usually no closed form can be found for H ( x ). Instead, we can substituteEqs.(22) and (23) into Eq.(24), then do a change of variables via Cauchy formula, and finally get π s = h k i ( s − d s − dx s − [ G ( x )] s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = . (25)12 + + + + ···= + + + + ··· (a)(b) Figure 4: Schematic diagrams of the self-consistent equations for the cluster size distribution. The line, circle and square represent link, node andcluster, respectively. (a) The possible cases of the branchings at the end of a link, corresponding to Eq.(23). (b) The possible cases of the branchingsstarting from a node, corresponding to Eq.(22). Such schematic diagrams are originally shown in Ref.[48].
With this, the cluster size can be obtained directly from G ( x ), therefore avoiding solve Eqs.(22) and (23) [91–93]. As π s ∝ sn s ∝ sp s , the result must have the form π s ∝ s − τ , where τ is the Fisher exponent.Furthermore, the mean cluster size χ can also be obtained from Eqs.(22) and (23). From the definition, the meancluster size χ can be represented as χ = P s s n s P s sn s = P s s π s P s π s = H ′ (1) H (1) = + G ′ (1) H ′ (1) H (1) . (26)In the last equation the di ff erential of Eq.(22) is used. We can further employ Eq.(23) to replace the term H ′ (1), andyield χ = H (1) " + G ′ (1)1 − G ′ (1) . (27)It is obvious that χ is divergent when G ′ (1) = . (28)This is the Molloy–Reed criterion h k ( k − i = G ′ (1) >
1) thegiant cluster exists.In addition, the discussion of finite clusters based on generating functions H ( x ) and H ( x ) can also work withoutthe help of generating functions G ( x ) and G ( x ), so it is not limited to uncorrelated tree-like networks, see Sec.3.4for an example of the percolation on growing networks. The Potts model is a generalization of the Ising model with more than two components [94], and related to anumber of other outstanding problems in lattice systems [24], including percolation model. Although the Potts modelis also unsolved for an arbitrary network, the connection between the Potts model and the percolation model hasmade it possible to explore the network percolation from the known information on the Potts model. This section willprovide the mapping between Potts model and percolation model, as well as some discussions.13 .3.1. Fortuin-Kasteleyn cluster representation
It is known that the q → q -state Potts model without external field corresponds to the percolationproblem, which is thus applied to the study of percolation on networks [16, 24, 95, 96]. To consider the percolationon a network G , we introduce a q -state Potts model with the Hamiltonian H = − J X h i , j i δ α i ,α j − H X i δ α i ,α , (29)where α i = , , , . . . , q is the spin state of node i , and the sums P h i , j i and P i are over all the links and nodes of thenetwork, respectively. δ x , y is the Kronecker delta function, i.e. , δ x , y = , x , y and x = y , respectively. Here, theferromagnetic interaction is used J >
0, and the magnetic field H > α H . Then, the partitionfunction can be written as Z = X { α } e − β H = X { α } e K P h i , j i δ α i ,α j e L P i δ α i ,α H = X { α } Y h i , j i e K δ α i ,α j Y i e L δ α i ,α H = X { α } Y h i , j i h + ( e K − δ α i ,α j i Y i h + ( e L − δ α i ,α H i , (30)where K = β J , L = β H , and the sum P { α } is over all the spin configurations of the system. We can further expand theproducts and use the subnetworks of G to represent the terms in the expansion, which is often called Fortuin–Kasteleyncluster representation [97].Next, we give a brief derivation for the Fortuin–Kasteleyn cluster representation. For arguments a i , i ∈ I , theproduct Q i ∈ I (1 + a i ) is equivalent to the sum of all the possible polynomials formed by a i , i ∈ I , so that Q i ∈ I (1 + a i ) = P I ′ ⊆ I Q i ∈ I ′ a i , where I ′ is a subset of I , and the sum P I ′ is over all the possible cases. By this formula, the product Q h i , j i in Eq.(30) can be represented by the sum of the subnetworks G ′ of G , that is Z = X { α } X G ′ ⊆G Y h i , j i∈G ′ ( e K − δ α i ,α j Y i h + ( e L − δ α i ,α H i = X G ′ ⊆G ( e K − l ( G ′ ) X { α } Y h i , j i∈G ′ δ α i ,α j Y i h + ( e L − δ α i ,α H i , (31)where l ( G ′ ) is the number of links in subnetwork G ′ . Note that the term Q ( i , j ) ∈G ′ δ α i ,α j gives nonzero value only whenthe nodes in the same clusters are in the same states (any of the q states α = , , . . . , q ). This can further simplify thepartition function to be Z = X G ′ ⊆G ( e K − l ( G ′ ) X { α c } Y c h + ( e L − δ α c ,α H i s c = X G ′ ⊆G ( e K − l ( G ′ ) Y c (cid:16) e Ls c + q − (cid:17) . (32)Here, the product Q c is over all the clusters formed by the links in subnetwork G ′ , and s c is the number of nodes incluster c . The sum over spin { α } in Eq.(30) has now been replaced by a sum over subnetwork {G ′ } , this is just theFortuin–Kasteleyn cluster representation of the partition function of Potts model.Let e K − = p / (1 − p ), the partition function Eq.(32) can be rescaled as Z ≡ (1 − p ) l ( G ) Z = X G ′ ⊆G p l ( G ′ ) (1 − p ) l ( G ) − l ( G ′ ) Y c (cid:16) e Ls c + q − (cid:17) = X G ′ ⊆G P G ′ Y c (cid:16) e Ls c + q − (cid:17) , (33)14here l ( G ) is the total number of links in G . If p is interpreted as the occupied probability of links, the term P G ′ = p l ( G ′ ) (1 − p ) l ( G ) − l ( G ′ ) is just the probability of a percolation configuration with l ( G ′ ) occupied links. Thus, Eq.(33)bridges the Potts model and the percolation model.From Eq.(33), we can write the free energy per node as f ( K , L , q ) = lim N →∞ ln Z N . (34)For convenience, we further define h ( K , L ) ≡ ∂∂ q f ( K , L , q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = = P G ′ ⊆G P G ′ P c e − Ls c N P G ′ ⊆G P G ′ = DP c e − Ls c E N . (35)For L > S = − h P c s c i N = + ∂∂ L h ( K , L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = , (36)and the mean cluster size is χ = ∂ ∂ L h ( K , L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = . (37)By the above two equations, the percolation property can then be extracted from the Potts model.The product Q c in Eq.(33) is over all the clusters. It thus can be rewritten in another form of Z = X G ′ ⊆G P G ′ Y s (cid:16) e Ls + q − (cid:17) m s , (38)where the product Q s is over the size of the clusters, and m s is the number of clusters with size s . By this formula,function h ( K , L ) can also be expressed as h ( K , L ) = DP s m s e − Ls E N = X s n s e − Ls . (39)Here, n s = h m s / N i is the cluster size distribution. This is just the generating function h ( K , x ) = P s n s x s with x = e − L ,which can be used to study the cluster size distribution.Specifically, once the Hamiltonian (Eq.(29)) of a network is known, all the percolation parameters can be obtainedby the above equations. However, for most network models, it is di ffi cult to simplify the sum over links in Eq.(29). Asolvable case is the hidden parameter model, for which the pre-defined connection probability is given for each pair ofnodes. Thus, the sum over links can be translated as a weighed (connection probability) sum over all the node pairs.With this technique, the Potts model formulation has been used in solving the percolation on SF networks [98, 99],on correlated hypergraphs [100], and in generalized canonical random network ensembles [101]. Through the so-called recurrence relation, the mean-field equations for tree-like networks can be recovered by thePotts model formulation [102]. For this purpose, the partition function can be represented as an integration of a root,15abeled node 0, and the branchings from it, that is Z = q X α = e L δ α ,α H Y i = z i ( α ) = e L Y i = z i ( α H ) + ( q − Y i = z i ( α ) . (40)Here, z i ( α ) is the partition function of the branching from node i , and the product Q i runs over all the root’s nearestneighbors (nodes in the first shell of node 0). Due to the symmetry, α in the second term of Eq.(40) can be any stateexcluding a H . Obviously, this formulation is only for the tree-like networks, if not, the product Q i must include somedouble counting.Di ff erent from Eq.(30), the partition function equation (40) represents the e ff ective state of the root node. In thisway, the free energy per nodes is ln Z , then we have h ( K , L ) = ∂∂ q ln Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = = e − L Q i = z i ( α ) Q i = z i ( α H ) . (41)To study the percolation model, we assume here that z i ( α ) describes the state for q = L =
0, thus it is irrelevantto the di ff erentials ∂/∂ q and ∂/∂ L . For convenience, we further rescale z i ( α ) by z i ( α H ), i.e. , x i ≡ z i ( α ) / z i ( α H ). Then,by Eq.(36), we can find the order parameter of the percolation transition S = + ∂∂ L h ( K , L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = = − Y i = x i . (42)From this equation, the physical meaning of x i becomes clear, that is the probability that the branching from node i isnot the giant cluster.To obtain the order parameter, we need to further find x i ( α ). Comparing Eqs.(40) and (30), we have z i ( α ) = X { α } e K P h j , k i δ α j ,α k + L P j δ α H ,α j + K δ α ,α i , (43)where the sums P j and P h j , k i run over the nodes and the links in the branching from node i , respectively. Note that theinteraction between node i and the root K δ α ,α i is also included in this partition function. Furthermore, it is not hardto find that z i ( α ) satisfies the following recurrence relation z i ( α ) = q X α i = e K δ α ,α i + L δ α H ,α i Y m z m ( α i ) . (44)Here, the product Q m is over all the neighbors of node i except the root (nodes in the second shell of node 0), and z m ( α i ) is the partition function for the subnetwork branching from node m .For infinite system, the recursive relation Eq.(44) holds for any two adjacent shells. If α , α H , it can be rewrittenas z i ( α ) = e K Y m z m ( α ) + e L Y m z m ( α H ) + ( q − Y m z m ( α ) . (45)While for α = α H , Eq.(44) reduces to z i ( α H ) = e K + L Y m z m ( α H ) + ( q − Y m z m ( α ) . (46)16hen, we can find the recursive equation for x i , x i ( α ) = z i ( α ) z i ( α H ) = e K Q m z m ( α ) + e L Q m z m ( α H ) + ( q − Q m z m ( α ) e K + L Q m z m ( α H ) + ( q − Q m z m ( α ) = e K Q m x m ( α ) + e L + ( q − Q m x m ( α ) e K + L + ( q − Q m x m ( α ) . (47)For q = L =
0, it reduces to x i = e − K + (1 − e − K ) Y m x m = − p + p Y m x m , (48)where p = − e − K .From the view of mean-field theory, the probabilities x i for di ff erent nodes can all be replaced by an averageprobability x . Then, Eqs.(42) and (48) become S = − X k p k x k = − G ( x ) , (49) x = − p + p X k p k k h k i x k − = − p + pG ( x ) . (50)These are just the mean-field equations (17) and (18) found in Sec.2.2 with x = − pR . By this equation, we bridge thePotts model formulation of the percolation model with that of the mean-field method based on the branching process[102]. Note that the tree-like structure is also required here. In recent years, there has been a growing interest in the analysis of the percolation problem on real networks.These systems all have finite sizes and are featured by the adjacency matrix rather than the degree distribution. Asa consequence, the above mean-field method does not work in this case. Instead, the most common method toestimate the percolation threshold on these networks is the so-called message passing method [103–113]. Note thatthe percolation threshold (transition) is theoretically defined in the thermodynamic limit ( N → ∞ ), here we assumethat the system is large enough to observe a percolation transition. Next, we will go to the details of this method.Di ff erent from the above mean-field method, the message passing method allows each node i has its own proba-bility s i to express whether it is a part of the giant cluster. From this perspective, the size of the giant cluster can bewritten as S = P i s i N . (51)Similar to Eqs.(20) and (21), we can also express s i as s i = p − Y j ∈N i (1 − r i → j ) , (52) r i → j = p − Y k ∈N j , k , i (1 − r j → k ) , (53)17here r i → j is the probability that the link i → j leads to the giant cluster, and N i is the set of neighbors of node i .Furthermore, by evaluating the logarithm of the above two equations, we can turn the products into sums,ln − s i p ! = X j ∈N i ln(1 − r i → j ) , (54)ln − r i → j p ! = X k ∈N j , k , i ln (cid:16) − r j → k (cid:17) = X k A k j ln (cid:16) − r j → k (cid:17) − A ji ln (cid:16) − r j → i (cid:17) . (55)Here A is the adjacency matrix of the network, i.e. , A i j = i → j , otherwise A i j = w i → j = ln(1 − r i → j / p ) and v i → j = ln(1 − r i → j ), Eq.(55) becomes equivalent to the vectorial equation w = Mv , (56)where M is a 2 L × L matrix ( L is the number of links). From Eq.(55), we can find that only when j = k and i , l , theentry M i → j , k → l is non-zero. In other words, the two links must be head to tail, and excluding the case of backtracking.This matrix is known as the Hashimoto or non-backtracking matrix of graphs [114, 115]. Mathematically, M i → j , k → l = δ j , k (1 − δ i , l ), where δ x , y is the Kronecker delta function δ x , y = x = y , and δ x , y =
0, otherwise.When p → p c , all r i trend to zero. Then, expanding Eq.(55) around the critical point, we obtain an eigenvalueequation of matrix M , 1 p c r = Mr . (57)According to the Perron-Frobenius theorem, only the largest eigenvalue λ max of matrix M can give a meaningfuleigenvector r (with all elements non-negative). Therefore, it gives a lower bound for the site percolation threshold onan infinite graph p c = /λ max [103]. Considering the e ff ects of triangles, this framework can also be used to establisha tighter lower bound of the bond percolation threshold on clustered networks [116].The above discussion is for site percolation, it can be easily extended to bond percolation, that is s i = − Y j ∈N i (1 − pr i → j ) , (58) r i → j = − Y k ∈N j , k , i (1 − pr j → k ) . (59)Further discussion of the two equations can be done like that of the site percolation.In addition, as was done by the mean-field method, the message passing method can also be adopted to study thecluster size distribution. One can refer to Refs.[103–113] for details. Due to the heterogeneous structure, the classical percolation on networks demonstrates many interesting phenom-ena. In this subsection we will review the findings on tree-like network ensembles, for which the percolation problemcan be solved exactly by the methods reviewed in the above subsections.
To discuss the critical phenomena, we first need to determine the critical point. For tree-like networks, it is easyto know from Eqs.(17)-(21) that only a non-zero R can lead to a non-zero S . Thus the percolation threshold p c can befound by analyzing the non-trivial solution of Eq.(17), and the site and bond percolations have the same critical point.For this purpose, we can construct a function w ( R ) = R − + G (1 − pR ) . (60)18 c w ( R ) R (0,0) non-trivial solution p = p c p > p c p < p c R p Figure 5: (Color online) Schematic of the solution of Eq.(17). Only when p exceeds p c , function w ( R ) has a non-trivial crossing point with R -axis.The critical point corresponds to the tangency of the function w ( R ) and R -axis, which gives a vanished R . The inset shows the non-trivial solution R vs. p . The solution of Eq.(17) corresponds to the crossing point of w ( R ) and R -axis. Since w ( R ) is continuous with w (0) = w (1) >
0, we can draw a qualitative curve of w ( R ) as shown in Fig.5. It is easy to find that the critical point p c corresponds to the tangency of w ( R ) and R -axis with R c =
0, so we have dw ( R ) dR = − p c G ′ (1) = . (61)Thus, the percolation threshold can be written as p c = G ′ (1) . (62)This is a general form for any tree-like networks. Using the expression of the generating function G ( x ), it can berewritten as p c = h k ih k i − h k i . (63)For p =
1, this equation reduces to Eq.(28), i.e. , the Molloy–Reed criterion h k ( k − i = p k = e −h k i h k i k / k !, so that G ( x ) = G ( x ) = e h k i ( x − , (64) h k i = h k i + h k i . (65)This allows the percolation threshold has a simple form p c = / h k i . Besides, due to the Poisson degree distribution, ERnetworks can have a closed solution of percolation threshold for many network percolation models. For comparison,we list the percolation thresholds of ER networks for di ff erent models in Tab.3.A more interesting case is the SF network with 2 < λ <
3, for which h k i is divergent, indicating a vanishedpercolation threshold. Employing the Hurwitz zeta function ζ ( x , a ) = P n = a n − x , the percolation threshold of SF19etworks can be represented as p c = P ∞ k = m k − λ P ∞ k = m k − λ − P ∞ k = m k − λ = ζ ( λ − , m ) ζ ( λ − , m ) − ζ ( λ − , m ) . (66)One can find that for m =
1, this formula gives a percolation threshold p c larger than 1 when λ > . . . . ,indicating there is no percolation transition. This is due to the absence of the spanning cluster in such SF networks[78]. The typical value λ ≈ . ζ ( λ − = ζ ( λ − ζ ( x ) is the Riemann zeta function.This problem can be overcome by simply setting m ≥
2. Furthermore, the Hurwitz zeta function ζ ( x , a ) is divergentfor x ≤
1. Thus a vanished percolation threshold can be found for SF networks with λ ≤ Varied percolation thresholds in di ff erent networks are excepted, since it is not universal, whereas things getinteresting in that the heterogeneous structures could produce a di ff erent universal property of percolation transition, i.e. , the critical exponent. Indeed, we can expand the analytical results (see Sec.2.2) around the critical point to findthe critical exponents. It is also worth pointing out that the critical exponents derived from the results obtained inSec.2.2 are still the sense of mean-field solutions, although they might be di ff erent from the regular mean-field nature.For a few cases that have a closed form of the generating functions G ( x ) and G ( x ), such as ER networks andRR networks, they give the critical exponents above the critical dimension (the mean-field critical exponents), sincethese systems have no spatial constraint. Mathematically, for these networks, the degree distribution only a ff ects thecoe ffi cients of the expansion series around the percolation threshold, then yields the same leading order. For thecalculation of these cases, one can refer to Refs.[48, 87, 91, 92] for details.A special case is SF networks, which have strong heterogeneity. Although there is also no spatial constraint, itshows a λ -dependent critical behavior ( λ is the exponent of the degree distribution p k = ck − λ ). Even for a vanishedpercolation threshold (2 < λ < G ( x ) and G ( x ) have no closed form for SF networks, they can be representedby the Lerch’s transcendent Φ ( x , s , a ) = P ∞ n = x n ( n + a ) − s , G ( x ) = X k = m ck − λ x k = x m Φ ( x , λ, m ) ζ ( λ, m ) = cx m Φ ( x , λ, m ) , (67) G ( x ) = P k = m k − λ x k − P k = m k − λ = x m − Φ ( x , λ − , m ) ζ ( λ − , m ) = ζ ( λ − , m ) ∂∂ x (cid:2) x m Φ ( x , λ, m ) (cid:3) = c h k i ∂∂ x (cid:2) x m Φ ( x , λ, m ) (cid:3) , (68)20here h k n i = c P k = m k n − λ = c ζ ( λ − n , m ). To find the critical behaviors of the percolation on SF networks, we needfurther to know the series expansion of G ( x ) and G ( x ) at the critical point p c , which corresponds to x →
1, seeSec.2.5.1. However, Φ ( x , s , a ) has a singularity at x =
1, which is dependent on s . This is the mathematical origin ofthe specific critical behaviors of SF networks.From Ref.[130], for | ln x | < π and a , , − , − , · · · , the Lerch’s transcendent Φ ( x , s , a ) can be expanded as Φ ( x , s , a ) = x − a Γ (1 − s )( − ln z ) s − + X n = ζ ( s − n , a ) n ! (ln x ) n , s , , , , · · · . (69)Let x ≡ − ǫ → − , we also have − ln(1 − ǫ ) = ǫ + ǫ + ǫ + · · · . (70)Then, we can further write Φ ( x , s , a ) as a series of ǫ , Φ ( x , s , a ) = x − a Γ (1 − s ) X n = ǫ n n ! s − + X n = ( − ǫ ) n n ! n X l = S ( n , l ) ζ ( s − l , a ) , s , , , , · · · , (71)where S ( n , l ) is the signed Stirling numbers of the first kind. With this equation, we can find the leading terms ofthe generating functions G ( x ) and G ( x ) for x → i.e. , the asymptotic series to the percolation threshold. Then,substituting into the corresponding equations in Sec.2.2, the λ -dependent critical exponents can be obtained, whichare summarized in Tab.4. We can find that the case of 3 < λ < < λ < λ ), the regular mean-field exponents as that found in ER networks are excepted.Cohen et al . pointed out that the regular mean-field results can be found when λ > λ c = ff erent criticalexponent β for 2 < λ < S ∝ p (1 − R ).For λ >
3, the system has a non-trivial p c , and the leading term reduces to S ∝ − R , which gives the same criticalexponent β as that of the bond percolation. However, for 2 < λ <
3, it can be rewritten as S ∝ ( p − p c )(1 − R ), where p c =
0. This leads to β = / (3 − λ ) + = (4 − λ ) / (3 − λ ).It is also worth noting that with the Potts model formulation, Lee et al . shows τ = λ for 2 < λ < et al . pointed out that due to the vanished threshold for 2 < λ <
3, the exponent τ = (2 λ − / ( λ −
2) is calculated at a small but fixed occupied probability [131].With the framework shown above, many other features about the percolation transition on SF networks were alsostudied, such as the fractal dimensions of percolating networks [132], the giant cluster in the large-network limit[133], branching trees [134], statistical ensemble [135–137], width of percolation transition [138], the upper criticaldimension [139], and the cluster forming in SF networks with exponent less than two [140] or one [141].Although the above discussions are concentrated on the degree distributions of networks, we must point outthat the spatial constraint still plays a key role in the nature of percolation transition. By embedding networks in aphysical dimension, the network percolation can also exhibit a range of phase behaviors, as well as reconstructing theuniversality class of physical dimensions, depending on the dominant structure properties, such as spatial constraint,small-world, fractal, and hierarchy [142–146]. In Refs.[58, 60, 147, 148], Newman and the coauthors also developed amethod to study the percolation transition on SW networks, suggesting that the SW network resembles more a randomnetwork in infinite dimension.
Real world networks are commonly clustered and correlated rather than a tree-like random structure [2]. Someworks also contributed to the percolation of clustered and correlated networks. Note that the clustering and thecorrelation discussed here are local, i.e. , they are the properties of adjacent nodes.21 .6.1. Networks with low clustering
As a starting point, we first review the method proposed by Berchenko et al. to find the approximate solution of thepercolation model for networks with low clustering [149]. Network science often employs the clustering coe ffi cient C to feature how well connected the neighborhood of a node is [1, 2, 57, 150]. If the neighbors are fully connected, theclustering coe ffi cient is 1, while a value close to 0 means that there are hardly any connections in the neighborhood.Indeed, as the definition of the clustering coe ffi cient, it also represents the connection probability of a node’s twoneighbors. In the branching process one can exclude the backward links from the excess degrees, see Fig.6 (a).Therefore, the e ff ective excess-degree k ′ of a node with degree k can be expressed as k ′ = k − k ′ ! C k − − k ′ (1 − C ) k ′ . (72)For C =
0, this reduces to the tree-like structure k ′ = k −
1. This equation has no physical meaning for C =
1, since itis only an approximate treatment for low clustering. With Eq.(72), the generating function of the excess degrees canbe revised as G c ( x ) = X k = p k k h k i X k ′ = k − k ′ ! C k − − k ′ (1 − C ) k ′ x k ′ = G ( C + x − Cx ) , (73)where G ( x ) is that with no thought of clusterings. With this revised generating function, we can solve the percolationmodel on networks with low clustering, which gives the critical point [149] p c = − C ) G ′ (1) . (74)This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold. This mainlybecause for a fixed number of links, the clustering structure reinforces the core of the network with the price ofdiluting the global connections [151–154]. However, the clustering cannot restore a finite percolation threshold forSF networks with 2 < λ < G ′ (1) only depends on the degree distribution in thisapproximation.It is important to notice that Eq.(74) works only for networks with low clusterings. Strong clustering could inducethe core-periphery structure [156, 157], in which the core and periphery might percolate at di ff erent critical points[158–160], and the above approximate treatment is not applicable. In this perspective, the clustered networks areoften characterized as a robust system [161–163]. For high clustering, the triangles formed by adjacent nodes could share links, which cannot be directly reflectedby the clustering coe ffi cient, Eq.(72) thus overcounts the excess links. Of course, one can further use the polynomialsof the clustering coe ffi cient to estimate these high-order clustering structures [65]. However, to completely eliminatethe overcounting, the history of the branching process has to be considered, which is beyond the compass of themean-field method. In addition, this problem in a sense relates to the minimum spanning tree of networks [164–166],which is also a problem without a unified solution.If the triangles or other non-tree-like motifs do not share links with each other, there is another framework to solvethe corresponding percolation problem [152, 153, 167, 168]. Beyond the degree distribution, this approach requiresto know the distribution of all other motifs that a node has, see Fig.6 (b). Taking the triangle structure as an example,when the number of triangles that a node has is known, one can treat the triangle as a special link in the branchingprocess. Then, the network can percolate by means of not only single links but also triangles. Under this premise, theclustered networks can also be regarded as tree-like, and the processing method provided above can thus be used. Inprinciple, this method can be applied to networks with any motifs. However, it is a tedious work to classify diverseclustered structures and get the corresponding distribution. In addition, this approach can be well summarized asmapping the clustered network into a tree-like hyper-network [169].22 (a) (b) Figure 6: (Color online) Schematic of the branching process in clustered networks. (a) The branching process starts from node 1 takes the breadth-first strategy, i.e. , searching all the neighbors of node 1 firstly. At the end of link l indicated by the arrow, node 2 has three excess links, oneof which ( l ) leads to a branching (node 3) that has already been reached as a neighbor of node 1. Therefore, only links l and l can lead tonew branchings, i.e. , there are only two e ff ective excess degrees. In general, a local backward link, such as l , corresponds to the formation of aclustering, thus occurs with probability C on average. (b) Considering the non-tree-like motifs as special links / nodes indicated by di ff erent colors,the network can be seen as a tree-like one. (a) (b) Figure 7: (Color online) Schematic of the percolation configuration of a SW network. (a) A SW network consisted of an underlying two-dimensional square lattice and several shortcuts indicated by red. (b) A possible configuration after removing some links from the SW networkshown in (a). Since the shortcuts are sparse, it is nearly impossible to form loop structures with shortcuts near or below the percolation threshold.The loop structures only exist in the clusters extracted from the underlying lattice.
The general thought in dealing with this problem, namely, recognizing a certain connection structure as an indi-vidual (link or node), is also used in the so-called clique percolation [170]. In this percolation problem, the k -clique,a complete subnetwork with k nodes, is treated as a node, and two k -cliques sharing l nodes are considered to beconnected. This percolation model is usually used as an algorithm for the detection of communities with overlap[171]. For percolation transition, the main finding is the seemingly discontinuous transition, which essentially is acontinuous one [124]. The details will be discussed in Sec.4.3.Another application of this framework is the solution of the percolation model on SW networks [147, 148], whichtypically show a highly clustering e ff ect. In Refs.[147, 148] the authors treated the finite percolation clusters extractedfrom the underlying lattice of the SW network as e ff ective nodes, then the system is just a set of e ff ective nodesconnected by the shortcuts, which are links added between randomly selected pairs of nodes in the underlying lattice[57, 59]. As the setting of SW networks, the shortcuts are sparse, so near or below the percolation threshold the re-duced network demonstrates a tree-like structure composed of e ff ective nodes and shortcuts, see Fig.7. Consequently,the formulation of the self-consistent equation as Eq.(23) can be used to solve the model, too. Based on this treatment,they found that the shortcuts not only modify the percolation threshold but also the universality class.Since the clusterings are naturally excluded in the non-backtracking matrix, with some techniques the message23assing method can also be used to find the percolation threshold of clustered networks, see the references listed inSec.2.4 for details. Along with clusterings, real networks generally show a structure with degree correlation rather than random con-nections. For such networks, the branching process cannot be simply described by the excess-degree distribution.Instead, a joint degree-degree distribution is used to figure out the degree correlation of the two nodes at two ends ofa link [172, 173].Theoretical results show that a finite amount of random mixing of the connections in SF networks with 2 <λ < ffi cient to give a divergent h k i , and thus leads to a vanished percolation threshold [172]. Moreover, theassortative correlation makes networks more robust, while the disassortative correlation makes networks fragile evenwith a divergent second moment of degree distribution. However, in the spatially constrained ER networks, degreecorrelations favor or do not favor percolation depending on the connectivity rules [174].The Monte Carlo simulations on the exponential random network show that the disassortative correlation has noe ff ect on the critical phenomena so that the percolation transition on disassortative networks belongs to the sameuniversality class as on uncorrelated networks [175]. While assortative correlation is relevant, percolation transitionshows continuously varying critical exponents [175]. Recently, Mizutaka et al. proposed a maximally disassortativenetwork model, which realizes a maximally negative degree-degree correlation [176]. Both the analytical and simula-tion results suggest this maximally disassortative network can also give new critical exponents for SF networks with2 < γ < et al. further summarized three conditions [173]: (i) The largest eigenvalue λ of the branching matrix isfinite if h k i is finite, or λ → ∞ if h k i → ∞ ; (ii) The second largest eigenvalues of the branching matrix is finite;(iii) The sequence of entries of the eigenvector associated with the largest eigenvalue converges to a nonzero value.When these conditions are fulfilled, the critical exponents are completely determined by the asymptotic behavior ofthe degree distribution at large degrees, thus the percolation transition on a correlated network belongs to the sameuniversality class as the percolation on an uncorrelated network. If at least one of the three conditions is not fulfilled,the critical exponent becomes model-dependent and hence non-universal [177–179]. Furthermore, by the techniquethat dividing nodes into di ff erent types with hyper-links, the site and bond percolations on clustered and correlatednetworks can also be solved [108, 180, 181]. The percolation transition can be also defined on directed networks. The di ff erence is that the giant cluster ofdirected networks can be further specified as [48]: (i) the giant strongly connected cluster (GSCC), in which eachnode is reachable from others; (ii) the giant in-cluster (GIC), from which GSCC are reachable but those are notreachable from GSCC; (iii) the giant out-cluster (GOC), from which GSCC are not reachable but those are reachablefrom GSCC; (iv) the giant weakly connected cluster (GWCC), in which each pair of nodes are reachable withoutregard to the direction of links. A schematic of these giant clusters is shown in Fig.8.The mean-field method reviewed in Sec.2.2 can also be extended to directed networks [182–189]. For this, wedefine R I and R O as the probabilities that a directed link leads to GSCC and comes from GSCC, respectively. Thus,similar to Eq.(13), we have two self-consistent equations, R I = − X i j p i j i h k i (cid:16) − R I (cid:17) j = − G I (1 , − R I ) , (75) R O = − X i j p i j j h k i (cid:16) − R O (cid:17) i = − G O (1 − R O , , (76)24 SCCGIC GOCtendrilstube
Figure 8: (Color online) Schematic of the giant cluster of directed networks. GSCC is a set of nodes, in which each node is reachable from allothers. The set of nodes, from which GSCC are reachable but those are not reachable from GSCC, form the GIC. In turn, GOC is the set of nodesthat GSCC are not reachable but those are reachable from GSCC. Besides, GIN can also lead to some nodes which do not belong to GSCC, if thesenodes can lead to GOC then called tube, otherwise called tendril. Moreover, the nodes lead to GOC but not belong to GSCC and GIC are alsocalled tendril. These giant clusters and the attached smaller clusters, including tubes and tendrils but not just, form the GWCC, which is just thegiant cluster by ignoring the direction of links. In addition to these giant clusters, there are some small clusters that are disconnected to these giantclusters, though not depicted here. where p i j is the probability that a node has in-degree i and out-degree j , and G I ( x , y ) = X i j p i j i h k i x i − y j , (77) G O ( x , y ) = X i j p i j j h k i x i y j − (78)are the generating functions of the excess-degree distribution following and against the link direction, respectively.Note that a directed network has an identical average in-degree and average out-degree h k i = P i j p i j i = P i j p i j j .When Eqs.(75) and (76) only have the trivial solution R I = R O =
0, there is thus no GSCC in the system.Otherwise, the giant cluster can be expressed by R I and R O according to their definitions, S GIC = − G (1 , − R I ) , (79) S GOC = − G (1 − R O , , (80) S GSCC = − G (1 − R O , − G (1 , − R I ) + G (1 − R O , − R I ) , (81)where G ( x , y ) = X i j p i j x i y j (82)is the generating function of the degree distribution. Similar as that for undirected networks, let R I ( R O ) → ∂ G I (1 , y ) ∂ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = = , (83) ∂ G O ( x , ∂ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = = . (84)25he two equations are equivalent, that is X i j p i j (2 i j − i − j ) = . (85)This is the condition that the GSCC exists in a directed network. Considering the dilution of the initial node or linkremoval as that for undirected networks (see Sec.2.2), one can also find the percolation threshold from Eq.(85), p c = h k ih i j i . (86)where h i j i = P i j p i j i j . If there is no correlation between in-degrees and out-degrees, i.e. , h i j i = h k i , Eq.(86) reducesto p c = / h k i . This indicates that in contrast to undirected networks, a non-zero percolation threshold can exist indirected SF networks with λ I > λ O > λ I and λ O are the in-degree and out-degree distributionexponents, respectively.From the asymptotic expansion of Eqs.(75)-(81), the critical exponents can also be obtained [183]. In general,the GIC and GOC can give di ff erent critical exponents β , which are dependent on the in-degree and out-degreedistributions. As the definition, GSCC is the intersection of GIC and GOC. Therefore, it behaves as the smaller oneof GIC and GOC. For directed SF networks, the critical exponents can be also written in the form of undirected SFnetworks, see Tab.4, but with an e ff ective λ ∗ , which is dependent on the existence of correlations and on the degreedistribution exponents λ I and λ O [183].The above discussion can also be generalized to the cases with the bidirectional links and the degree correlations[184, 190]. The result shows that the percolation threshold can be generally expressed as a function of the maximumeigenvalue of the connectivity matrices. In particular, for networks with no degree correlations, bidirectional linksact as a catalyst for percolation, favoring the emergence of the GSCC, and for SF networks, only an infinitesimalfraction of bidirectional links is needed. The interface links, i.e. , the ones joining GIC / GOC and GSCC, can also beinvestigated analytically in this framework [185]. In addition, the GWCC of a directed network can be obtained bymapping it into an undirected network. It should be noted that the GWCC could emerge at a smaller p c than that ofGSCC, which is dependent on the degree correlations of the in-degrees and the out-degrees. This indicates that thedirected network may have the GWCC and, simultaneously, may not have the GSCC. For most network systems, the percolation model cannot be solved exactly. In consequence, the simulation algo-rithm becomes very important to estimate the critical point and to study the critical phenomena. However, due to thediversity of network connections, the algorithms for classical percolation on regular lattices cannot be transplantedinto network percolation directly. One needs to store the connection information of the network, while this is notnecessary for that on regular lattices [191]. Hence, the simulation of the network percolation requires more memorythan of the classical percolation on physical dimensions.In general, we can first construct the percolation configuration following the percolation rule, then use the classicalalgorithm of graph searching to identify the connections of the configuration, including the breadth-first searching andthe depth-first searching. For a given configuration with M links, these searching algorithms take time O ( M ) to findall the clusters. This is the minimal overhead for scanning an unknown structure. If the data structure of the networkdoes not allow to directly scan the neighbors of nodes, such as the adjacency matrix, the time complexity will behigher. This two-step algorithm, i.e. , constructing the configuration first and then identifying the clusters, does nothave a specific requirement for percolation rules and network structures, and thus is universally applicable.To be more e ff ective, the percolation algorithm should be a creative blend of configuration constructing and clusteridentifying, not a process with two standalone steps. In other words, while constructing the percolation configuration,we must simultaneously update the cluster information. The detailed techniques are dependent on the percolationmodel and the network model. Here, we only introduce the algorithm proposed by Newman and Zi ff for the classicalpercolation on networks [28, 192], which is applicable to any given networks. Other algorithms will be reviewedwhen necessary.The Newman-Zi ff algorithm employs an exquisite data structure [28, 192], with which the nodes / links can be oc-cupied one by one, and the cluster size information can be updated concurrently, which improves greatly the e ffi ciencyfor checking the percolation properties of a system under a sequence of occupied probability p . In this algorithm, each26luster is stored as a separate tree with a single root node. Each node is allocated a pointer either to the root of thecluster or to another node in the cluster, such that by following a pointer chain we can travel from any node to the rootof the cluster. The root nodes can be identified by the fact that they have a null pointer. To be more e ffi cient, we canalso use the pointer of the root to store the size of the cluster.Taking bond percolation as an example, this percolation algorithm can be summarized as follows [28, 192]:1. Initially each node is its own root, and contains a record of its own size, which is 1.2. Links of the network are occupied in random order. When a link is occupied, two nodes are joined together.Follow the pointer chains from the two nodes separately until we reach the root nodes of the clusters to whichthey belong.(a) If the two roots are the same node, we do nothing further.(b) If the two roots are di ff erent, we examine the cluster sizes stored in them, and add a pointer from the rootof the smaller cluster to the root of the larger, thereby making the smaller tree a subtree of the larger one.If the two have the same size, we may choose whichever tree we like to be the subtree of the other. Wealso update the size of the larger cluster by adding the size of the smaller one to it.Step 2 is repeated until the expected number of occupied links is reached. The cluster size can be obtained fromthe pointer of the root, thus it allows us to evaluate the observable quantities of interest. In addition, to improve thee ffi ciency of the algorithm, we can compress the pointer chain as much as possible. For site percolation, the algorithmis similar, see Refs.[28, 192] for details.
3. Network-specific percolation models
In the previous section, we reviewed the studies of the classical percolation model on networked systems. As aframework to evaluate the system performance, the emergence of the giant cluster with respect to some control param-eters is widely used in network science [1–3, 5, 6, 8, 86]. However, the cluster forming rule is diversiform rather thana probabilistic occupation as that in the classical percolation. They usually contain some recursive / iterative processesor with a broad sense of occupation in the cluster forming. Thus, these derivative models are often categorized intoa family called dependent percolation. Typical examples include k -core percolation, clique percolation, core percola-tion, explosive percolation, percolation on interdependent / multiplex networks, etc . In this section we will give a briefintroduction of the theoretical findings of these models.In Tab.5, we list the key rules and the types of the percolation transitions for di ff erent models. -core percolation is one of the typical representatives of the dependent percolation on networks[193, 211–213].For a given network k -core is the subnetwork in which nodes have at least k links connecting with other nodes in thissubnetwork. So, as a percolation model, it mainly studies the emergence of the giant k -core when a fraction 1 − p nodesare removed, i.e. , occupying each node with probability p [193]. The case k = k = / biconnectivity, which is a mean-ingful concept of network robustness [117]. For k ≥ k -core percolation transition becomes discontinuous. In thissubsection we will review the theoretical researches of k -core percolation, as well as some related percolation models. k -core percolation and its variants are often used as algorithms to provide the structure information of networks [214],some of which will be included in Sec.4. We can iteratively remove nodes with degrees smaller than k to obtain the k -core of a network. Note that evenwithout the initial node removal, k -core may be absent in networks, dependent on the network topology. An extremecase is the tree-like network, for which there cannot exist any finite k -cores with k ≥
2. This is not di ffi cult to beunderstood from the pruning process of k -core percolation. In any finite and tree-like networks, there must be somenodes with degree k =
1, located at the periphery of the network. The removal of these nodes must introduce somenew periphery nodes with degree k =
1. In this way the pruning process will destroy the whole network. However,this does nothing to the theoretical analysis of k -core percolation transition on tree-like networks, since an infinite27 > p c p = p c p < p c non-trivial solution R w ( R ) (0,0) p c R p Figure 9: (Color online) Schematic of the solution of Eq.(90). Only when p exceeds p c , function w ( R ) has non-trivial crossing points with R -axis.The critical point corresponds to the tangency of the function w ( R ) and R -axis, which gives a non-zero R . The inset shows the non-trivial solution R for di ff erent p . spanning tree does not have periphery nodes, and all the degrees are larger than or equal to 2. This also means thatthere are no finite k -core clusters ( k ≥
2) in tree-like networks, thus the k -core, if exists, is just the giant cluster, andcan be used to feature the percolation transition.As the model setting, if a node is in the k -core, it must have at least k links connecting to other nodes in the k -core.By this, similar to Eqs.(17) and (19), we can write the self-consistent equations for k -core percolation on tree-likenetworks as [193], S k = p ∞ X n = k ∞ X d = n p d dn ! R nk (1 − R k ) d − n (87) = p − k − X n = R nk n ! d n G ( x ) dx n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = − R k , (88) R k = p ∞ X n = k − ∞ X d = n + p d d h k i d − n ! R nk (1 − R k ) d − − n (89) = p − k − X n = R nk n ! d n G ( x ) dx n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = − R k , (90)where S k is the size of the k -core, and R k is the probability that a node reached by following a randomly chosen linkbelongs to the k -core. Here, to avoid confusion we use p d to represent the degree distribution. The sum P n is overthe nodes with n preserved neighbors, and the sum P d is for the degree distribution. Note that the sum P n in Eq.(89)begins with k −
1, since the link used to reach the node has already provided a link for the k -core. For both the twocases k = k =
2, Eq.(90) reduces to that of the classical percolation equation (20), so 1-core percolation and2-core percolation have the same threshold equation (62). For k > k -core percolation becomesdistinguishable for k .To get the critical point, one can also define a function w ( R k ) as that for classical percolation, see Sec.2.5.1. Thecritical point thus corresponds to the tangency of w ( R k ) and R k -axis. However, since function w ( R k ) has more than28ne extreme point for k ≥
3, the tangency point gives a non-zero R k , labeled R k , c , which indicates that the percolationtransition is discontinuous, see the schematic shown in Fig.9.Although there is no closed form for the solution of Eqs.(88) and (90), the asymptotic expansion shows that thescaling behaviors can also be observed near the critical point, that is S k − S k , c ∝ ( p − p c ) β . (91)In consequence, the emergence of a k -core is a unique hybrid phase transition with a jump emergence of the k -coreas a first-order phase transition but also with a critical singularity as the second-order phase transition. This resultis first found in Bethe lattice with β = , , / k = , ≥ k -core percolation, and we will give the details later. It is worth noting that this result indicates that 2-core percolationbelongs to a di ff erent universality class from that of 1-core percolation (the classical percolation), although they havethe same percolation threshold [216]. This finding is also confirmed by the Monte Carlo simulation on ER networks[217].For SF networks that bear a strong heterogeneous degree distribution, a ratio β k = /β k = = λ − < λ < λ > λ is the exponent of the degree distribution of SF networks. This suggests thatthe k -core percolation could reconstitute the normal mean-field nature like that on ER networks when λ >
3, whilefor the classical percolation the criterion is λ > ff erent ways for k ≥ k -core percolation can onlysurvive in high dimensions [220–223].More generally, Dorogovtsev et al. showed that if the second moment of the degree distribution of a network isfinite, k -core percolation has a hybrid nature, and there is no principal di ff erence between di ff erent tree-like networks[193, 224, 225]. A dramatic di ff erence takes place if the second moment of the degree distribution diverges, i.e. , theSF networks with exponent 2 < λ <
3, for which an infinite order transition is observed. Besides, k -core percolationwas also analyzed for clustered networks [226, 227] and for biased initial node removal [228]. Moreover, k -corepercolation is also studied as models of the jamming transition [229, 230], granular gas [231], evolution [232], andnervous system [233]. Furthermore, the inducing mechanism has been found that can bridge the classical percolationand k -core percolation for a general k , which always causes a discontinuous percolation transition [234].In network science, the pruning process of k -core percolation has also attracted a lot of attention [235–239], whichis used to analyze network structures [211, 240–245]. It often refers to k -core / k -shell decomposition [214], and thenodes belong to k -core, but not ( k + k -shell. Focusing on the pruning process, one can adjust the criterion to observe the emergence of other core structuresof networks. Meanwhile, the corresponding percolation transitions can be also defined. Although some of them arenot initially motivated by k -core percolation, here we review these studies roughly attributed to variants and relatedmodels of the k -core percolation. Heterogeneous k-core percolation.
In the pruning process of k -core percolation, if di ff erent thresholds are allowedfor di ff erent nodes, the model becomes a sophisticated one known as heterogeneous k -core percolation [246–250].Specifically, after the initial node removal, some of nodes need k a subtrees to survive from the pruning process, someof nodes need k b subtrees, and so forth.A representative example of the heterogeneous k -core is the binary mixture k = ( k a , k b ) that nodes have a thresholdof either k a or k b distributed randomly through the network with probabilities f and 1 − f , respectively. With theframework figured by Eqs.(88) and (90), one can easily express the size of the heterogeneous k -core as a linearcombination of those for the two thresholds [246–250], each of which takes the form as the right hand side of Eq.(88)or Eq.(90).The transition nature of the heterogeneous k -core percolation obviously depends on the probabilities assigned todi ff erent thresholds, as well as the values of thresholds. Although the cases k = k = k -core percolation, the percolation transitions of k = (1 ,
3) and k = (2 ,
3) are much di ff erent. For k = (2 ,
101 percolating phase f p c non-percolating phase (a) p c non-percolating phase (b) Figure 10: (Color online) Schematic of two di ff erent phase diagrams of the heterogeneous k -core percolation. (a) The two lines indicate thecontinuous transition (blue solid line) and the hybrid transition (green dashed line) matching at a tricritical point (red scatter). (b) The line for thehybrid transition (green dashed line) ends at a point (orange scatter) that is out of the line for the continuous transition (blue solid line). Note thatthese two figures are conceptual, and do not represent the exact values. a crossover of the continuous and the hybrid percolation transitions [247], i.e. , in the phase diagram the lines for thetwo types of phase transitions match at a tricritical point, see Fig.10 (a). In contrast, the case k = (1 ,
3) demonstratesa complex phase diagram: the line for the hybrid transition ends at a point that is out of the line of the continuoustransition, see Fig.10 (b). This means that a double percolation transition can be observed in the system for someappropriate probabilities f [248], namely, first showing the continuous transition, and later the discontinuous hybridtransition. In general, the tricritical point never occurs for the case k = (1 , k ), whereas in the case k = (2 , k ) it ispresented only for k = k = (2 ,
3) can also map onto a model ofglasses on the Bethe lattice [251], which suggests the same universality class.More generally, the ternary mixture [252] and the generalized case k = (2 , , , . . . ) [250] were also investigatedto show new types of critical phenomena. Through the analytic calculations and the numerical simulations on ERnetworks, Chae et al. concluded that the heterogeneous k -core percolation can be featured by the series of continuoustransitions with order parameter exponents β = / n , n = , , , . . . , discontinuous hybrid transitions with β = / /
4, and three kinds of multiple transitions [250].
Cascading failure model.
A special case of the heterogeneous k -core percolation is the one with threshold α k fornodes with degree k , where α ∈ [0 ,
1] [194, 195, 253–260]. In other words, if a node i can be preserved from thepruning process, k i / k i , ≥ α must be satisfied for any stages, where k i and k i , are the current and the initial degree ofnode i , respectively. This node removal process is usually used to model the spreading of failures on networks, henceits name cascading failure model, sometimes also referred to as threshold model.The simulation results suggest the existence of the crossover of the continuous percolation transition and thediscontinuous percolation transition [195]. However, more detailed checking of the critical phenomena as that ofheterogeneous k -core percolation is still lacking. More often, the emergence of the giant cluster in this model isstudied as a criterion for network robustness rather than a characterization for phase transition. We will review theseworks in the later sections.In addition, there are two caveats to this model. First, many works on this cascading failure model focus on theclusters formed by failed nodes, not those of preserved nodes [194, 253, 256–259, 261]. Second, there is another30 core Figure 11: (Color online) Removal categories of nodes in the core percolation. Red nodes are non-removable, i.e. , they belong to the core indicatedby shaded background. Green nodes are removable: nodes 1 and 2 are α removable; nodes 3 and 5 are β removable; node 4 is γ removable. model in network science called cascading failure model which focuses on the overload failures, see Ref.[262] andsubsequent references for details. Bootstrap percolation.
Along with the k -core percolation, there is an important and well-known model, called boot-strap percolation. In the early literature the bootstrap percolation just refers to the model of k -core percolation in-troduced above, such as Refs.[215, 246, 263]. Now, the bootstrap percolation is generally referring to an activationprocess, which starts with a fraction f of active seeds (occupied nodes) and other nodes (unoccupied) will be activated(occupied) when they have at least k activated neighbors. A theoretical analysis for this model like Eqs.(88) and (90)can be found in Refs.[196, 264, 265]. This model and its variants are often used to describe the spreading dynamicsin network science. A typical example is the cascading failure model reviewed previously.Although the bootstrap percolation is an activation process beginning from a sparsely activated network, while the k -core percolation is a pruning process beginning from the whole network, a corresponding relation between them canalso be established [248]. For comparison, we construct a heterogeneous k -core percolation with k = (1 , k ). Threshold1 is randomly assigned to a fraction f of nodes corresponding to the seeds in bootstrap percolation, and other nodeshave threshold k . By this mapping, the heterogeneous k -core percolation k = (1 , k ) can demonstrate very similarcritical phenomena as the bootstrap percolation, i.e. , a double percolation transition [196, 248]. However, this modelcannot completely recover the bootstrap percolation [248]. This is because all the clusters in the bootstrap percolationmust grow from the seeds, while for heterogeneous k -core percolation nodes can also survive from the pruning processby forming a compact cluster. Moreover, Di Muro et al. also pointed out that the heterogeneous bootstrap percolationis the complement of the heterogeneous k-core percolation for complex networks with any degree distribution in thethermodynamic limit, as long as the thresholds of the nodes in both processes complement each other [264].Due to the correlation of bootstrap percolation and k -core percolation, many works for k -core percolation alsocontributed to bootstrap percolation. For systems in physical dimensions and the Bethe lattice, one can also referto Refs.[266–272]. In network science, bootstrap-like processes are often used to model the cascading dynamics[1, 3, 6, 23, 86, 273], such as the spreading of information, opinion, and disease. Instead of the critical phenomena,these researches are focusing on the dynamical characteristics and their relations with the problems in real life. For thisreason, there is no need to lump them into a clear category (bootstrap percolation or heterogeneous k -core percolation).Some of these studies will be reviewed in the later sections. Core percolation.
The core of a network is defined as the subnetwork after a greedy leaf removal (GLR) procedure.Here, the leaf of a network refers to the node of degree 1. The GLR procedure is just iteratively removing nodesof degree 1 along with its neighbor. Note that the nodes with degree 1 only have one neighbor. Apparently, theGLR procedure is more destructive than the pruning process of 2-core percolation, and finite clusters cannot exist intree-like networks. As the k -core percolation, the core percolation model is also used to identify the core structure ofnetworks, which could dominate the dynamical properties of complex networks [274].Considering the branching process, the infinite cluster (core) of a tree-like network can also be obtained exactly[118]. For that, nodes are divided into four categories, see Fig.11: (i) α removable: nodes that can become isolatedwithout directly removing themselves, i.e. , all neighbors are β removable; (ii) β removable: nodes that can becomea neighbor of a leaf, i.e. , at least one neighbor is α removable; (iii) γ removable: nodes that can become leaves but31re neither α nor β removable; (iv) non-removable: nodes that cannot be removed and hence belong to the core. Therule for γ removable nodes can be ignored because it is not useful to determine the size of the core. Denoting α (or β ) as the probability that a random neighbor of a random node in a network is α removable (or β removable), twoself-consistent equations can be derived, α = G ( β ) , (92) β = − G (1 − α ) , (93)where G ( x ) = P k p k kx k − / h k i is the generating function of the excess-degree distribution. From the two equations, itcan be found that α satisfies α = G [1 − G ( α )]. With this, we can find the solutions of α and β . For tree-like networks,if the core exists, it must have infinite size (a giant cluster). Hence, the size of the giant cluster, i.e. , the normalizedcore size, can be expressed as S core = ∞ X k = p k k X s = ks ! β k − s (1 − β − α ) s = G (1 − α ) − G ( β ) − c (1 − β − α ) α, (94)where G ( x ) = P k p k x k is the generating function of degree distribution. This approach can also be generalized fordirected networks with given in- and out-degree distributions [118].The theoretical results show that, for undirected networks, if the core percolation occurs, then it is always con-tinuous, while for directed networks, it becomes discontinuous with a hyperscaling β = / ff erent [118]. For ER networks the percolation threshold can be solved theoretically h k i c = . . . . [275], and the core of purely SF networks never percolates for any degree exponents larger than2. While for the SF networks generated by the static model with p k ∝ k − γ only for large k [276], the core developswhen the average degree is larger than a threshold value, which is actually similar to ER networks.The leaf removal procedure of the core percolation can also be generalized as that nodes of degree smaller than k ,together with their nearest neighbors and all incident links are progressively removed from a random network [277].With this pruning process, the percolation transition can also be well defined, which can be seen as either a generalizedcore percolation or a generalized k -core percolation. Similar to the ordinary k -core percolation, this model also showsa discontinuous phase transition for k ≥ Greedy articulation points removal.
The articulation points of a network are the nodes whose removal disconnects thenetwork, such as nodes 3, 4, 5 and 6 in Fig.11. Those nodes play an important role in keeping the connectivity of real-world networks, such as infrastructure networks, protein interaction networks and terrorist communication networks.Tian et al. proposed a greedy articulation point removal (GAPR) process to study the organizational principles ofcomplex networks [119].In each iterative step of GAPR, all the articulation points are removed from the network and their sub-clustersdisconnected from the giant cluster are also removed. After that new articulation points emerge. This removal processis repeated until there is no articulation point left in the network. Note that at each step all the articulation points areremoved simultaneously and the size of the giant cluster in the final network is studied as a key quantity.It is obvious that if the network is sparse, the giant cluster will finally be destroyed with the proceeding of GAPR.For a dense network, a giant cluster, in which all the nodes connect to each other with at least two paths, can beobtained. Note that this cluster is not equivalent to the 2-core, or the biconnected cluster (BCC) [117, 216, 218, 278,279], or the one obtained by GLR. A comparative sketch of the 2-core, the BBC, the cluster preserved after the GLR,and the one after the GAPR is shown in Fig.12.Next, we briefly cover the relations between these clusters. For convenience, let S − core , S BBC , S GLR , and S GAPR be the subnetworks after applying the pruning rules of 2-core, BBC, GLR, and GAPR, respectively. For 2-core, oneonly needs to recursively remove the nodes with degree 1, thus some tree-like structures might exist in 2-core. Thenodes in a BCC must connect each other with more than one independent paths. This further requires removing thenodes and links that do not belong to any loops from the 2-core, therefore, S BBC ⊆ S − core . For GLR, when we removethe nodes with degree 1, their root nodes will also be removed, thus we also have S GLR ⊆ S − core .32 a) 2-core(b) BCC(c) GLR(d) GAPR Figure 12: (Color online) A comparative sketch of the 2-core, the biconnected cluster (BCC), the cluster obtained by the greedy leaf removal(GLR), and the one obtained by the greedy articulation points removal (GAPR). For a given network (shown left), the four figures on the rightshow the corresponding subnetworks (indicated by red) under the four pruning rules, respectively. (a) 2-core. The cluster is obtained by iterativelyremoving nodes with degree 1. (b) BCC. The nodes in a BCC must connect each other with more than one independent paths, which can be foundby removing all the nodes and links that do not belong to any loop structure. Here, two separated BBCs are obtained. (c) GLR. The cluster isobtained by iteratively removing nodes of degree 1 along with their direct neighbors. (d) GAPR. The cluster is obtained by iteratively removingthe articulation points, which are the nodes that can bridge two clusters. After these pruning / connection processes, there generally exist more thanone clusters. The corresponding percolation theory considers the emergence of such a giant cluster. S BBC and S GLR is network dependent. The case shown in Fig.12 gives an example that S BBC ⊇ S GLR . Considering a network configuration that two triangles are connected by a degree-2 node, we can alsohave S BBC ⊆ S GLR . But both S BBC and S GLR are larger than S GAPR . This is because all the nodes and links that donot belong to S BBC and S GLR are certainly not in S GAPR , whereas the articulation points excluded from S GAPR mightcontribute to S BBC or S GLR , see Fig.12. In summary, for a given network, we have the relation S GAPR ⊆ S BBC ( S GLR ) ⊆ S − core .An additional consequence of the above relation is that the emergence of the giant cluster after GAPR needs adenser initial configuration than that after GLR. For ER networks, it has been found that the critical point of thepercolation induced by GAPR is h k i c = . . . . [119], which is obviously larger than the one induced by GLR(core percolation) [118], see Tab.3 for the percolation thresholds of some models on ER networks. This percolationis also proved to be a hybrid percolation transition with critical exponent β = /
2. Note that the core percolation isalways continuous for undirected networks. Besides, if one only does a finite iteration of the GAPR, the percolationtransition can also be observed, and will have the same nature of the ordinary percolation. This finding has the samemanner of the so-called history-dependent percolation on multiplex networks [199].There is also a variation of GAPR: iteratively remove the most destructive articulation point that will cause themost nodes disconnected from the giant cluster of the current network. This is called articulation point targeted attackstrategy [119]. Given a limited “budget” (that is, the number of nodes to be removed), this strategy is very e ffi cient inreducing the giant cluster, compared with strategies based on other node centrality measures, such as degree [84, 85]and collective influence [280].The statistical properties of articulation points, i.e. , the probability that a random node in a network is an artic-ulation point, has been also derived in Ref.[281]. It has been found that high-degree nodes are more likely to bearticulation points than low-degree nodes. Articulation point is related to another concept of “bridge”, which is a linkwhose removal disconnects the network and increases the number of connected clusters [282]. Both articulation pointand bridge can be used to qualify the importance of a bridge structure in damaging a network and provide useful ideasto destroy a network e ffi ciently. Color-avoiding percolation.
In the color-avoiding percolation, the nodes in the percolating cluster also need morethan one path that connect with each other. The details are as follows. First, each link is assigned a color chosen from C = { c , c , . . . , c n } . If there exists a path between two nodes after the removal of all the links of color c ∈ C , the twonodes are called c -avoiding connected. Then, if two nodes are c -avoiding connected for all the color c , we say thatthe two nodes are color-avoiding connected. Thus, the color-avoiding percolation studies the behaviors of the giantcluster, in which all the nodes are color-avoiding connected. The color avoiding can also be generalized as the casewhere arbitrary sets of colors are avoided. Furthermore, colors can also be assigned on nodes, that is the setting whenthe model was first proposed [283].Di ff erent from the previous percolation models, the color-avoiding connected clusters cannot be simply acquiredfrom a single pruning process. Instead, one can find c -avoiding connected clusters for all the color c ∈ C , then theintersections of these clusters are just the color-avoiding connected clusters, see Fig.13 for a schematic of a three-color case. Due to this intersection process, the connectivity of a color-avoiding cluster could access through somenodes / links outside the color-avoiding cluster.In Ref.[284], it is suggested that the complexity of finding the color-avoiding cluster highly depends on the exactdefinition, using a strong version the problem is NP-hard, while using a weaker notion makes it possible to find thecomponents in polynomial time. In spite of this, there are some special structures that surely do not contribute to thecolor-avoiding connected clusters. The first is the bridge link, whose removal disconnects the network. The second isthe nodes that all their links have identical colors. A typical example for these two structures is the nodes of degree 1and their links, i.e. , the leaves of the network.Besides, the size of C , i.e. , the number of colors n , also has significant restrictions on the structural features ofthe color-avoiding connected clusters. First, a small n generally results in a large frequency for each color, thus theemergence of color-avoiding clusters for small n requires a dense connection. Second, the size of the color-avoidingconnected cluster has a lower bound, which increases with the color number n . All these suggest that the color-avoiding connected cluster must have a relatively dense and complex structure, containing many loops.If there is only one color n =
1, the model reduces to a classical percolation with respect to the fraction ofcolored links / nodes. For n ≥
2, the critical behaviors have a rich, multifaceted nature, depending on the network34 a) (b)(c)(d)
Figure 13: (Color online) A schematic for color-avoiding percolation. Here, three colors, namely, red, yellow, and blue, are uniformly distributedamong all the links, see (a). If there exists a path between two nodes after the removal of all the blue / red / yellow links, the two nodes are calledblue / red / yellow-avoiding connected. Then, if two nodes meet this connection criterion for all the three colors, we say that the two nodes arecolor-avoiding connected. A cluster, whose nodes are all color-avoiding connected, is thus the color-avoiding connected cluster. (b)-(d) show theclusters after the removal of blue, red, and yellow links, respectively. The intersections of these clusters correspond to the color-avoiding connectedclusters indicated by green shades in (a). Note that the color-avoiding connections of the nodes in a color-avoiding cluster could access throughsome links outside the color-avoiding cluster. β is dependent on the number of colors, i.e. , β = n . This is identical for both node and link coloring. Varying the color frequencies, fractal exponents can also beobserved. However, topological constraints of SF networks are so strong that the corresponding critical behaviors areindependent of the number of the colors (only dependent on the exponent λ ), but still dependent on the existence ofthe colors and therefore di ff erent from standard percolation. Due to the vanished percolation threshold, the breakingof the universality class of the site percolation and the bond percolation can also be observed for 2 < λ <
3. InRef.[285] a generic analytic theory that describes how structure and sizes of all connected components in the networkare a ff ected by simple and color-dependent bond percolation was also established. / multiplex networks The hybrid percolation transition is not unique to the k -core percolation [20, 21, 286–291]. In the last decade a ma-jor concern in network science, the interdependent networks or multilayer networks [8, 9], also involves a dependentpercolation, for which both abrupt change and critical exponents can be found at the critical point [120, 121, 292].For convenience, the following discussions are given by means of interdependent networks [120], see Fig.14 for anexample of the interdependent networks.This model considers the iterative percolation process between di ff erent network layers. If a node does not belongto the giant cluster of one layer, its dependent nodes in other layers are also no longer eligible to be consideredin the percolation of their layers, and vice versa. Obviously, one must check the percolation process of each layerinteractively to obtain a steady giant cluster, referred to as the mutual giant cluster. Thus, this model checks theemergence of the mutual giant cluster after an initial node removal. To provide a direct and precise description for this percolation model, the dependence link is used to connectdependent nodes in di ff erent network layers, see Fig.14. Specifically, a dependence link between node i and node j means that if node i is removed, node j will also be removed, and vice versa. The node removal process caused bydependence links is often called dependence process. Correspondingly, all the finite percolation clusters in each layerwill also be removed from the system, which is called percolation process. Thus, after an initial node removal (fraction1 − p ), the removal of nodes caused by the percolation process and the dependence process will occur alternately untilno more nodes can be removed, see Fig.15. The remained nodes (if any) form the mutual giant cluster, in which nodesare reachable through each layer. Note that if we do not remove the finite clusters from the system, the finite mutualclusters can also be well defined in the steady state [199, 293].The above pruning process to find the mutual giant cluster was first proposed in Ref.[120], in which the dependencelinks are assigned randomly between two network layers, labeled A and B (see Fig.14). Assuming both the twonetwork layers have tree-like structures, we can express the size of the mutual giant cluster as S = p h − G A (1 − R A ) i h − G B (1 − R B ) i . (95)Here, R A ( R B ) represents the probability that a randomly chosen link in network layer A ( B ) leads to the mutual giantcluster. By analogy with that of classical percolation equations (15) and (16), we can know that the right hand sideof Eq.(95) just describes the final state of the pruning process, i.e. , all the preserved nodes must be reachable in bothlayers. Similarly, R A and R B must satisfy R A = p h − G A (1 − R A ) i h − G B (1 − R B ) i , (96) R B = p h − G A (1 − R A ) i h − G B (1 − R B ) i . (97)The above three equations (95)-(97) were first used to analyze the percolation on the networks with dependence linksby Son et al. [294, 295]. Considering the iterative process of the pruning process, this model can also be solved. Thatis the analytical method when this model was proposed [120, 121], which is equivalent to the method listed above[296, 297].From Eqs.(96) and (97), we can also find the critical point of the system as we do for the classical percolation.The solution of R shows a similar behavior as k -core percolation illustrated in Fig.9, indicating a discontinuous36 B Figure 14: (Color online) Schematic of the interdependent / multiplex network. Here, the system has two network layers A and B indicated by dif-ferent colors, both of which can have their own topology, and dependence links indicated by dashed lines are used to represent the interdependencebetween nodes from network layers A and B . More generally, there can be more than two layers, and the interdependence relation needs not to berestricted to one-to-one. (a) (b) (c) (d) (e) Figure 15: (Color online) Schematic of the cascading process on interdependent networks. Here, we consider an interdependent network withtwo layers, represented by the two columns of circles with the corresponding solid lines, respectively. The dashed lines are the dependence linksbetween the two network layers. With an initial node removal (indicated by red in (a)), the percolation process ((b) and (d)) and the dependenceprocess ((c) and (e)) alternately pruning nodes from the system (indicated by gray) until no more nodes can be removed. p c is significantly larger than that of classical percolation. For example, two interdependentER networks give p c ≈ . / h k i , while p c = / h k i for single ER networks. This also means that h k i ≈ . < λ <
3, the mutual giant cluster can emerge at a non-vanished p c . Further study have showed that thisdiscontinuous transition is also a hybrid transition with exponent β = / k -corepercolation with k ≥ et al. also pointed out that the number of iterations in the cascading process of interdependent networksdiverges when p → p c [121], and thus can be used to identify the transition point in numerical simulations. This ismainly because there is a plateau in the collapse of the system, and the plateau regime increases with the system size.Zhou et al. found that during the collapse there is also a random branching process at criticality, i.e. , a continuouspercolation transition. This simultaneously continuous phase transition is just the origin of the observed long plateauregime in the cascading failures and its dependence on system size [298]. This result coincides with the one foundin Ref.[199], in which the percolation transition is defined in each iteration, called generation. Theoretical analysisindicates that for any finite generation, the system demonstrates a continuous percolation transition. Monte Carlosimulations on ER networks further suggest that all these continuous transitions belong to the same universality class.Specifically, SF networks with exponent 2 < γ < i.e. , it has a nature of hypertransition. Hu et al. further pointed that in two-dimensional systems the infinite generation still presents a continuoustransition, but with a di ff erent universality class [200].By decreasing the coupled strength q (the fraction of nodes that have dependence links), this discontinuous per-colation transition can also reduce to the continuous transition [121]. An exception is the coupled lattices, for whichthe networks abruptly collapse for any finite q > r , the percolation transition will be continuous for r < r max ≈ r [300]. For r < r max , the percolation threshold increaseslinearly with r from 0 .
593 at r = .
738 for r = r max , and then gradually decreases to 0 . r = ∞ . This is mainly because the spatial embedding induces a similar structure in di ff erent network layers, andthus breaks the cascading picture of interdependence [294]. This model has also been generalized to high dimensions[301]. The simulation results show that the value r max decreases with dimension, and for lattices with dimension largerthan or equal to 6, the continuous percolation transition disappears. This suggests that a high degree of freedom (highdimension) decreases the structural similarity of the spatially embedded networks under random node or link dilution,thus reconstructs the cascading picture.It should be noted that the model, based on which the authors claimed that percolation transitions are not alwayssharpened by making networks interdependent [294], is a bond percolation model on the interdependent square latticewith typical length r =
0, and not the one discussed above. Thus their findings do not contradict that of Ref.[300],which considers the site percolation with di ff erent typical lengths. Grassberger also studied the bond percolationmodel with typical length r = d =
4, below which the percolation transition is always continuous, and at least for d ,
3– not in the universality class of ordinary percolation. This suggests that although the similar structure induced byspatial embedding breaks the cascading picture of interdependence, it indeed changes the nature of the branchingprocess in the continuous percolation transition. The bond percolation has also been studied on multiplex networks,and reported a multiple percolation phase transition [302]. Here, the multiple percolation transition refers to morethan one transition in a percolation process, also is reported in other related percolation models [303–308].In Ref.[293], an e ffi cient and implementable algorithm has also been proposed, based on which some other criticalbehaviors of the percolation on interdependent networks have been rechecked, numerically. The result suggests thatfor the model of Ref.[300] there exists such a r max above which the transition seems to be discontinuous, but a strongevidence that this is related to very large finite-size corrections was also found. Moreover, this work also providedsome evidences that the cluster statistics of independent ER networks can be exactly described by mean-field theory,while the cascade process cannot. For interdependent lattices with dimension d = k -core percolation[309], bootstrap percolation [310], and group percolation [311].38 .2.2. Algorithms for reducing time complexity For facilitating the studies on non-tree-like networks, some e ffi cient algorithms have been developed to find themutual giant cluster. Tracing the largest cluster during the whole pruning process, the algorithm proposed by Schnei-der et al. can achieve time complexity O ( N log N ) [312]. Nevertheless, this processing is too loose to study the criticalphenomena, since an initially large cluster could be taken over by a cluster that had started smaller after the pruningprocess, especially near the critical point. Apart from the critical phenomena or limited to tree-like networks, thisalgorithm might be a good choice to avoid brute-force searching.Although the finite mutual clusters are ignored in the original definition of this percolation model [120, 121], allmutual clusters must be taken into count for providing a more reliable simulation. A viable way to define such clusterscan be found in Refs.[199, 293], in which the percolation transition can be defined in each generation. Based on afully dynamic graph algorithm [279], Hwang et al. proposed an e ffi cient algorithm designed to proceed as links aredeleted, whose time complexity is approximately O ( N . ) for random networks [313]. With this algorithm, large scalesimulations for ER and 2-dimensional interdependent networks have been carried out, which verifies the exponentsfor the order parameter and its fluctuations numerically, as well as the finite avalanches [314]. Stippinger et al. latergeneralized this algorithm to e ffi ciently simulate interdependent networks with healing [315].Besides, Grassberger proposed an easily implemented algorithm to find all the mutual clusters by mapping theproblem onto a solid-on-solid model [293]. This method is simply done by alternately performing Leath-like processeson di ff erent network layers [316], each is referred to as a “wave” from the starting node. Waves are confined to thenodes that have the same “height” as the starting node. All the nodes’ heights are initially set as 0, and after thespreading of a wave, the heights of the involved nodes increase by 1. Do the searching process for all the possiblestarting nodes, a landscape of node heights can be obtained, from which the mutual giant cluster and the distributionof mutual cluster sizes can be found. With some techniques this algorithm can also simulate the percolation process oninterdependent networks, e ff ectively. In Ref.[315], it is also pointed out that compared to the algorithm of Ref.[293]their algorithm is generally intended for fast updates at the cost of higher memory use. With the growing trend of multiplex networks, this percolation model has also been employed frequently to figureout the robustness of multiplex networks [8, 9, 296, 317, 318], which can be generally classified into three categories,single networks with dependence links, networks of networks, and multiplex networks with general and special depen-dence. Most of these studies demonstrate a hybrid transition as shown above, and a crossover between the continuousand discontinuous percolation transition can also be found in some of them. Besides, the correlation between di ff erentnetwork layers can also be represented by interconnections, in which the percolation transition can be also observed[319].More importantly, these works show the robustness of multiplex networks in various environments, and are mean-ingful for understanding the structures and the robustness of multiplex networks. More results will be reviewed laterwhen we turn our thoughts to the applications of percolation model. Of course, one can find more information in thespecial papers and books on multiplex networks [8, 9, 296, 317, 318]. The explosive percolation was proposed by Achlioptas et al. in 2009 [201], hence this type of percolation processis named as Achlioptas process. In general, Achlioptas process is a competitive process that can be realized by manyrules, and the key is suppressing the emergence of a giant cluster. There is a specialized review article for this issue[206], here we only give a brief introduction of the main findings.
As we know, ER networks can be generated by randomly inserting links into a set of nodes one by one. TheAchlioptas process is just a variation of this process [201]. Assuming there are initially N nodes and no links, at eachtime step, m potential links are arbitrarily chosen, each of which is supposed to bridge two distinct clusters with size s and s , respectively. (The two nodes can belong to the same cluster with no impact on the following process).Then, according to a function f ( s , s ), the potential link with the smallest f ( s , s ) is inserted eventually, and otherpotential links are discarded, see Fig.16 (b) as an example. Repeating this process until the expected number of links39 a) (b) Figure 16: (Color online) (a) Illustrations for the classical percolation and the explosive percolation on ER networks. Here the explosive percolationis manufactured by the best-of-2 rule. (b) Schematic of the best-of-2 rule. At each time step, 2 potential links indicated by dashed lines are chosenrandomly from all the possible ones. According to the sum / product rule, here the potential link l will be inserted, and link l is discarded. is met. Such a selection mechanism is often called the best-of- m rule or the min-cluster- m rule [202], and the function f ( s , s ) represents the competitive rule.In order to suppress the emergence of the giant cluster, the competitive rule f ( s , s ) must be of positive correlationwith s and s . Two commonly used ones are the product rule f ( s , s ) = s s and the sum rule f ( s , s ) = s + s . For m =
1, this model reduces to the ER network model, which percolates when N / i.e. , h k i = m ≥
2, the emergence of a giant cluster will be suppressed, resulting in an explosive percolation. A comparisonof the two cases is shown in Fig.16 (a). One can find that the explosive percolation shows a sharper transition with alarger threshold.Since Ref.[201], many other generalized Achlioptas processes have been proposed and investigated. A typicalexample is the l -node ( l -vertex) model. In this model l nodes are chosen randomly at each step, and two of them areconnected according to a selection criteria, for example, the size di ff erence [320]. The original Achlioptas processjust corresponds to l =
4, and the case l = ff erent selection criteria is used in many studies [204, 321, 322].Besides, there are some other models can realize the Achlioptas process, such as BFW model [323–330], dCDGMmodel [203, 331–337], Hamiltonian model [205], di ff usion-limited cluster aggregation model [338, 339], spanningcluster-avoiding model [340, 341], and hybrid model [342–344]. One can find a contrast analysis of these models inRef.[345]. Most of these models can be generalized to any networks by constraining the potential links in a givenconfiguration. One can refer to Ref.[206] for more information. For a deep understanding of the explosive percolation, one can consider a complete network with size N [202, 321].In each step t , all the unconnected node pairs provide potential links, i.e. , m = N ( N − / − t +
1. In this case, up to N / N / N / N /
4. As an analogy, in step N −
1, the remaining two clustersof size N / m will relax the suppression for the emergence of the giant cluster, thus the jump of the largest clusteris blurred. Although a sharp transition might also be observed as shown in Fig.16, Riordan et al. pointed out thatany rule based on picking a fixed number of random nodes gives a continuous transition [324]. They also suggesteda lower end of the range: Whenever m → ∞ as N → ∞ , the Achlioptas process exhibits explosive percolation. The40ndings of da Costa et al. show that the Achlioptas process is irreversible, which also indirectly suggests a continuoustransition [203]. However, these findings do not mean that the Achlioptas process behaves exactly like a continuouspercolation. The discontinuities can also be observed in the Achlioptas process [320]. The double humped distributionof the sizes of the largest cluster and the hysteresis, which are both recognized as the sign of discontinuous transition,can be also found in finite systems [347, 348]. These works also reported a finite-size behavior with non-analyticscaling functions, and demonstrated that the explosive percolation transitions are indeed continuous but with someunusual scaling properties [203, 325, 345, 347, 349–352]. Riordan et al. also pointed out that the fluctuation of theorder parameter in explosive percolation does not disappear in the thermodynamic limit [322]. In contrast to the network models with a fixed size, such as the configuration model and ER network model, thereis a family of network models with growing size featuring the growing property of real networks [3, 4, 353]. A typicalexample is the famous BA networks [56]. However, there is always only one single cluster in the growing process,thus the system always percolates. To observe a percolation transition in the growth of a network, it must be allowedthat new nodes enter the system without connecting to the existing nodes. In this way, the network could containisolated nodes and finite clusters.Instead of looking at the network configuration, one can solve the growing network model by checking the evolu-tion process via the master equation. The key point is featuring the increment and decrement of the parameters in onetime step, which might vary from model to model [125, 354, 355].
A simple rule to achieve a percolation transition in network growing is the one proposed in Ref.[125], calledgrowing random network. This model begins with no node, and at each time step a new node is inserted into thesystem, then two nodes are chosen randomly from all the existing nodes and joined by a link with probability p .At time step t , there are N ( t ) = t nodes in the system. Let p k ( t ) be the degree distribution at time step t , then a rateequation for the evolution of the degree distribution p k ( t ) can be established, N ( t + p k ( t + − N ( t ) p k ( t ) = pp k − ( t ) − pp k ( t ) . (98)The two terms on the right hand side of this equation correspond to the expected increment and decrement of nodeswith degree k in one time step, respectively. For t → ∞ , p k ( t + = p k ( t ) ≡ p k , and Eq.(98) reduces to a recursionformula p k / p k − = p / (1 + p ). Note that Eq.(98) holds only for k >
0, and for k = N ( t + p ( t + − N ( t ) p ( t ) = − pp ( t ) , (99)which gives p = / (1 + p ) for t → ∞ . Consequently, the degree distribution has a closed form for t → ∞ , p k = + p p + p ! k . (100)Obviously, the degree distribution p k decreases with the increase of k . In fact, the power-law degree distribution canalso be produced in this model by introducing preferential attachment [355, 356].From Eq.(100), the generating functions are also available, those are G ( x ) = / (1 + p − px ) and G ( x ) = / (1 + p − px ) . Substituting G ( x ) into the Molloy-Reed criterion Eq.(28), we can find the percolation threshold p c = /
4. This result is for an uncorrelated tree-like network with degree distribution Eq.(100), whereas the growingrandom network has degree correlation. Specifically, earlier nodes are sure to form a core, in which there is a higheraverage degree. This indicates that the giant cluster forms more readily than in a network whose links are uniformlydistributed.To find the percolation threshold of the growing random network, we need to examine the cluster size distribution[125]. Let n s ( t ) be the normalized cluster number at time step t , i.e. , the ratio of the number of clusters with size s tothe total number of nodes, then a rate equation for the evolution of n s ( t ) can be expressed as N ( t + n ( t + − N ( t ) n ( t ) = − pn ( t ) , (101) N ( t + n s ( t + − N ( t ) n s ( t ) = p s − X i = in i ( t )( s − i ) n s − i ( t ) − psn s ( t ) , s ≥ . (102)41ote that sn s can be interpreted as the size of the cluster that a randomly chosen node belongs to. Therefore, p P s − i = in i ( t )( s − i ) n s − i ( t ) is the increment of clusters with size s ≥ s = psn s ( t ) is the decrement of clusters with size s inone time step by combining two clusters that at least one of them has size s .For t → ∞ , n s ( t + = n s ( t ) ≡ n s , then Eqs.(101) and (102) can be simplified to n = − pn , (103) n s = p s − X i = i ( s − i ) n i n s − i − psn s , s ≥ . (104)Although the two equations cannot give a closed form of n s , we can find a recursive function for the generatingfunction H ( x ) = P s π s x s = P s sn s x s by multiplying both sides of Eqs.(103) and (104) by sx s and then summing over s . The result reads H ( x ) = X s = sn s x s = x − pn x + p X s = s − X i = i ( s − i ) n i n s − i sx s − p X s = s n s x s = x + px ddx X s = sn s x s − px ddx X s = sn s x s = x + pxH ( x ) H ′ ( x ) − pxH ′ ( x ) , (105)where P n a n x n P n b n x n = P n P m a m b n − m x n is used. Now, we have H ′ ( x ) = H ( x ) − x px [ H ( x ) − . (106)Note that the mean cluster size χ = H ′ (1) / H (1) ∝ H ′ (1), then the critical behavior of χ can be derived fromEq.(106). In the supercritical phase, the giant cluster exists, thus H (1) <
1, Eq.(106) gives the mean cluster size χ = H ′ (1) / H (1) = / pH (1). For the subcritical phase, all the clusters have finite sizes, i.e. , H (1) =
1, then H ′ (1)can be solved by letting x → H ′ (1) = − p − p p . (107)For a physical meaning, p ≤ / p c = /
8. This thresholdis smaller than that for the configuration network with the same degree distribution, confirming that the growingmechanism facilitates the forming of the giant cluster, theoretically [125].As just described, the critical behavior of χ can be summarized as χ ∝ H ′ (1) = − √ − p p , p ≤ , p , p > . (108)This means that the mean cluster size jumps discontinuously at the percolation threshold, whereas diverges for thestandard percolation.Moreover, compared with the static network, the existence of the core makes the size of the giant cluster increasesmore slowly in the supercritical phase. The simulation results imply that the size of the giant cluster obeys S ∝ e α ( p − p c ) − β with β = /
2, and suggest an infinite order phase transition [125, 357–359], like the Berezinskii-Kosterlitz-Thouless phase transition in the condensed matter [360, 361]. In addition, such a phase transition was also found onthe substrate formed by XY spin configurations [362]. Some simulation verification for these findings and the fractalgeometry can be found in Refs.[363, 364]. In a growing network with a renormalization group treatment, Dorogovtsevalso found that the critical behavior of percolation on growing networks di ff ers from that in uncorrelated networks[365]. 42 .4.2. Variants and related models Similar results can also be found in a general model with preferential attachment [355], i.e. , the link between nodes i and j is inserted with a probability proportional to ( k i + a )( k j + a ), where k i ( k j ) is the degree of node i ( j ), and a is apositive constant which plays the role of additional attractiveness of node for new links [366]. The power-law form ofthe cluster size distribution in standard percolation has a decay for large sizes both above and below the percolationthreshold. However, in the growing network the power-law form can be observed in the entire phase without the giantcluster. This indicates that the system is in the critical state for the entire phase.Instead of connecting nodes randomly, other typical mechanisms for generating networks can also be incorporatedinto the growing network model, such as BA network model [367] and the explosive percolation model [308, 333,368, 369]. In these models, when a new node enters the system, with probability p links are inserted following thegiven rule, otherwise, do nothing. When the rule of BA network model is applied, the percolation transition is still ofinfinite order [367]. This mainly because the rule of BA network model does not break the picture of the forming ofthe giant cluster, which is the crucial factor for the nature of infinite order transition [355].Conversely, the Achlioptas process induces a suppression e ff ect against the growth of the large clusters. Thus,when the Achlioptas process is used, the percolation of growing network can revert to a nature of the second ordertransition with scaling behaviors depending on the specific rules [333, 369]. It should be pointed out that if there arenot enough potential nodes / links in the Achlioptas process, the growth mechanism will still dominate the nature ofthe percolation transition, hence an infinite order transition [368]; rather, if the Achlioptas process is done with globalinformation, the growth of large clusters can be completely suppressed, so that the continuous percolation transitionwill change to a discontinuous one [308, 370].Besides, the infinite order transition can occur even in the hierarchical networks with small-world e ff ects [142,144, 371, 372]. For the growing network without percolating state [373], the preferentially growing network [374],and the growing directed network [375, 376], the power-law distribution of cluster sizes can be also observed. Thepercolation transition of growing networks has also been studied by diluting links or nodes in a generated network[377]. The dense structures were also found to undergo phase transitions in the growth of networks[378, 379].
4. Applications to network structural analysis
The previous section mainly reviews the typical percolation models on complex networks, focusing on the phasetransition and the critical phenomena. In fact, the percolation theory, including conceptions, theoretical methods,and algorithms, has a broader range of applications in the study of network structures and dynamics. This is mainlybecause the percolation model is by nature for featuring the cluster forming process in random media, while networkscience is exploring the properties of various topological structures and the correlation with dynamics. In this sectionwe will review the applications to network structural analysis. We should point out that some of these problems arenot firstly motivated by percolation, however, a good interpretation can be given from the perspective of percolation.
Hierarchical structure is one of the important characteristics of real networks [1–3, 5, 6], which can be generallyidentified by either a branching process or a pruning process. For tree-like networks, the analytical method for networkpercolation is often employed to find theoretical results, while for non-tree-like networks percolation process can alsobe used as an algorithm to identify hierarchical structures.
As shown in Fig.17, the hierarchical structure of a tree-like network can be figured out by a spanning tree. For alarge enough network with tree-like structures, any node can be seen as the root of the spanning tree. Then the directneighbors of this node are the first shell of the network, and the neighbors’ neighbors belong to the second shell, andso on. This is just the branching process we discussed in Sec.2.2. Thus, the generating function can be also employedto study this shell structure [48].Since the root node is arbitrarily chosen, the sub-branchings from it can be described by the generating function G ( x ). Here, an x refers to an outgoing link from the root node. Following these outgoing links, we can find the nodesin the first shell of the network. As the explanation for Fig.4, the generating function G ( x ) can be used to represent43 st shell2nd shell3rd shell Figure 17: (color online) Schematic for the hierarchical structure of a tree-like network. The number of nodes in the ( m + m -th shell. x also refers to the outgoing link from this node.Therefore, replacing x in G ( x ) with G ( x ), we will obtain the sub-branchings from the nodes in the first shell G (1) ( x ) = G ( G ( x )). Similarly, the sub-branchings from the second shell can be represented by G (2) ( x ) = G ( G ( G ( x ))). Ingeneral, the sub-branchings coming from the m -th shell can be written as G ( m ) ( x ) = G ( G ( G ( ... G ( x ) ... )) | {z } m generations ) = G ( m − ( G ( x )) , m > . (109)With this iteration relation, Shao et al. found that the distribution of node numbers in these shells follows a powerlaw for a broad class of complex networks [380]. This branching process can also be used to realize the configurationmodel [381], i.e. , connecting nodes with given degrees shell by shell.Note that loops are forbidden here, so there must be a one-to-one correspondence between the parameter x and thesub-branching. Consequently, the exponent for the power of x in Eq.(109) is just the total number of sub-branchingsfrom the m -th shell. In other words, the derivative of Eq.(109) with respect to x at x = m + h k m + i = dG ( m ) ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = = dG ( m − ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = G ′ (1) = h k m i G ′ (1) , m ≥ , (110)where h k m i is the average number of nodes in the m -th shell, and h k i = h k i is just the average degree of the network.From Eq.(110), we can also find h k m + ih k m i = G ′ (1) , m ≥ . (111)This indicates that the growth rate of the node number with shells is a constant G ′ (1). For an infinite network, theshell must be endless, so the node number cannot decrease with the increasing of shells, i.e. , G ′ (1) = h k i − h k ih k i ≥ . (112)This is the existence condition of the giant cluster in a tree-like network, namely, percolation threshold, see Sec.2.2.For a finite network, the number of shells l must be finite. Next, we will show how to estimate l . Using Eq.(111),the number of nodes in the m -th shell can be written as h k m i = h k i h k ih k i ! m − = h k i h k i − h k ih k i ! m − , m ≥ . (113)Summing over all the shells, we obtain the network size N = + l X m = h k m i . (114)This yields l = ln h ( N − (cid:16) h k i − h k i (cid:17) + h k i i − h k i ln (cid:0) h k i − h k i (cid:1) − ln h k i . (115)45 -core 3-shell 2-shell 1-shell Figure 18: (color online) Schematic for the k -shell structure of networks. In a k -core, nodes are connected to one another by at least k paths. Thenodes that belong to the k -core but not belong to the ( k + k -shell of the network. Real networks usually have N ≫ h k i and h k i ≫ h k i , thus l can be approximately represented as l = ln N − ln h k i ln h k i − ln h k i + . (116)This result can be seen as an approximation of the average distance or the diameter of real networks [382, 383], whichindicates that l ∝ ln N . This is just the small-world e ff ect of random networks.The above derivations depend on a finite h k i (when N → ∞ ). However, SF networks with 2 < λ < h k i ∝ R N / ( λ − k k − λ dk ∝ N (3 − λ ) / ( λ − , where k is the lower boundary of degrees and N / ( λ − is thenatural cuto ff of degrees. Substituting h k i ∝ N (3 − λ ) / ( λ − into Eq.(116), we can find a size-independent l , indicatingthat SF networks are ultrasmall.Actually, the diameter l of SF networks increases with the system size in the form of ln ln N [70, 384, 385]. To findthis, the degree distribution of the nodes in each shell must be checked, separately. In general, an outgoing link is morelikely to lead to a node with large degree, which can be seen from the expression of the excess-degree distribution q k ∝ p k k . For finite networks, this means that earlier shells will bag almost all the nodes with large degrees, so thatthe small degree nodes are located in the periphery of the hierarchical structure. This e ff ect is more notable whennetworks have strong heterogeneity like SF networks. This mainly indicates that the degree cuto ff of nodes in eachshell decreases with the growth of shells. With this, the form l ∝ ln ln N can be found in [70, 381, 384–386]. As pointed above, the hierarchical structure shown in Fig.17 has an arbitrary root. In network science, there isanother hierarchical structure with deterministic shells, i.e. , k -shell structure [193]. For a given network, the k -shellrefers to the nodes that belong to the k -core but not belong to the ( k + k -core size S k for any k , the size of k -shell isthus S k − S k + . In general, the pruning process of k -core percolation is just used as an algorithm to identify the k -shell46tructure, called k -core / k -shell decomposition. The related discussions are far away from the percolation transition.So we will not go into details on this topic here, and one can refer to the specialized review on this topic for details[214]. In network science, percolation process has already become a paradigm for studying the network robustness.Specifically, one often removes a fraction 1 − p of nodes from a given network, then checks the existence of the giantcluster. If the giant cluster remains, the resulted network is still considered functional. Thus the percolation threshold p c can be used to evaluate the robustness of the network. A large threshold p c indicates that only a small amountof failed nodes can disconnect the network, so its robustness is poor, and vice versa. With di ff erent node removalmethods, the percolation model can further explain the robustness of networks under various failure mechanisms,some of which are also used to model network dynamics. Here, we review the main findings of the network robustnessbased on the percolation model from two aspects, single networks and multiplex networks. As mentioned above, the percolation threshold of SF networks is much smaller than that of ER networks with thesame link density, especially for SF networks with 2 < λ < ffi cients or assortative coe ffi cients, individually.If the hub nodes are preferred to be removed initially, i.e. , intentional attack, the percolation model can also beused to study the robustness of networks under targeted attack. A simple implementation approach is to remove afraction 1 − p of the nodes with the highest degrees [72], which reports that SF networks, known to be resilient torandom removal, are sensitive to intentional attack. More generally, the intentional attacks are commonly realized byremoving nodes with probability proportional to k α [391], where k refers to degree. When α >
0, nodes with largerdegrees are more vulnerable under the intentional attacks, while for α <
0, nodes with larger degrees are defendedand so have a lower probability to fail. The case α = α >
0, SF networks become extremely fragile [85], and the percolation threshold will be larger than that of ERnetworks with the same link density [85] or even equal to 1 [393]. At criticality, the topology of the network dependson the details of the removal strategy, implying that di ff erent strategies may lead to di ff erent kinds of percolationtransitions [391, 392]. An optimal attack is also directly related to optimal graph partitioning and immunization ofepidemics [394]. Di ff erent structures can also emerge from these pruning processes [395–397]. Moreover, by mappingthe intentional attack to a normal percolation problem, this robustness model can also be solved [398]. Further research47lso shows that an onion-like structure could be a nearly optimal structure against simultaneous random and targetedhigh degree node attacks [399–401].There is also a special percolation model for the case α → −∞ , called degree-ordered percolation [402], wherenodes are occupied in degree-descending order (or alternatively, removed in degree-ascending order). The mean-fieldresults on SF networks suggest that the critical exponents depend on the heterogeneity of the network, and do notbelong to the universality class of the standard percolation [403].Besides, the localized attack, meaning that an attack on a node can destroy the node together with its neighborswithin a range, has also been introduced into the percolation model to show the robustness of networks under attacks[228, 387, 404–408]. When the intentional attacks are dependent on spatial distances, the spatial networks are morefragile than expected [409]. Moreover, the finite-size scaling analysis suggests that for some of the intentional attackstrategies, the critical exponents seem to deviate from the mean field, but the evidence is not conclusive [410].The percolation model is a random process, and the corresponding theory only characterizes the results on aver-age. Nevertheless, two di ff erent realizations with the same magnitude of removal could result in very di ff erent giantclusters. For the study of the robustness of real networks, the fluctuations of di ff erent percolation realizations becomeextremely important. Bianconi introduced a message-passing algorithm able to predict the fluctuations in a singlenetwork, and an analytic prediction of the expected fluctuations in ensembles of sparse networks [411, 412]. The fol-lowing works also contributed to the stability of the giant cluster [413], the large deviation of percolation [414, 415],and the largest biconnected cluster for random graphs [216]. The results mainly showed that the large-deviationapproach to percolation can provide a more accurate characterization of system robustness.As shown in Sec.2.7, the percolation transition can also be well defined on directed networks, so that with similarconsideration the robustness of directed networks can also be studied in the framework of percolation transition.It mainly demonstrates that the GWCC is more vulnerable, and the directed network may have the GWCC and,simultaneously, may not have the GSCC [182–190].In addition, if the communication of nodes in the network is e ff ective only if the shortest path between nodes i and j after removal is shorter than α l i j , where l i j is the shortest path before removal, a much smaller failure of the networkcan lead to an e ff ective network breakdown. This problem was also studied in a percolation form called limited pathpercolation [207, 416]. Note that this mechanism was also used to feature the entanglement percolation in quantumcomplex networks [417]. In recent years, multiplex network has become a research hotspot. The consideration of multiplex network natu-rally introduces the interaction between di ff erent network layers, thus constructs a cascading picture, which changesthe nature of percolation transition, see Sec.3.2. Consequently, the robustness of multiplex networks under the frame-work of percolation transition has its own characteristics, or even in a completely opposite manner as the singlenetwork. Topological structure.
As shown in Sec.3.2, multiplex networks are more fragile than single networks due to thecascading failures induced by the interdependence between nodes from di ff erent network layers. More interestingly,the e ff ects of the topology structure on the robustness of the network are also very di ff erent from those of the singlenetwork.In Refs.[120, 122], researchers found that a network with a broader degree distribution results in a larger p c ,which is opposite to ordinary networks. As we know, the hub nodes have a dominant role in the robustness of anetwork. However, when the node dependence is involved, a hub node could depend on a small degree node, whichis quite vulnerable during the iterated removal process triggered by the initial node removal. Moreover, a broaderdegree distribution with the same average degree implies that there are more small degree nodes. Therefore, theadvantage of a broad degree distribution for the ordinary network becomes a disadvantage for the network withdependence links. For this reason, the assortativity and high clustering structures will also make such networks fragile[418, 419]. These features are in consonance with the robustness of single networks under intentional attack [391,392, 395–397, 404]. For directed networks, the in-degree and out-degree correlations could increase the robustnessof interdependent networks with heterogeneous degree distributions, but decrease the robustness of interdependentnetworks with homogeneous degree distributions and with strong coupling strengths [420].48 nter similarity. In contrast to single networks, the multiplex networks are di ffi cult to defend by strategies suchas protecting the large degree nodes (intentional attack with α <
0) that have been found useful to significantlyimprove the robustness of single networks [398, 421, 422]. This is mainly because the dependence partners couldhave very di ff erent degrees, and defend strategies based on degrees cannot protect the two nodes at the same time.By this consideration, the system will obviously become robust, when the dependence partners have identical degrees[423, 424]. Such correlations between dependent partners are called inter similarity of di ff erent network layers.Parshani et al. introduced two parameters to evaluate the level of inter similarity between networks: the interdegree-degree correlation and the inter clustering coe ffi cient. The simulation results demonstrated that the inter-similar multiplex networks are significantly more robust to random failures than a randomly interdependent system[425]. Theoretically, Hu et al. developed an approach to analyze the robustness of multiplex networks with intersimilarity [426]. In their model, there are some probabilities that the dependence nodes of two adjacent nodes inone network are also connected. The studies showed that this inter similarity can improve the robustness of theinterdependent networks and change the critical behaviors of the percolation transition. In addition, a similar work hasalso been done on this problem [427]. The inter-similarity of dependence partners is also studied in single networks.The results show that the local dependence, i.e. , dependence partners are adjacent, can result in a robust network[428]. Dependence relations.
The dependence links were proposed to reflect the dependence relationships in reality, butnot all the nodes in a networked system have such dependence. In this way, the percolation on networks with weakdependence has been studied [121, 422, 429]. In this model, instead of each node having a dependence partner, only afraction q of nodes have dependence links. When q =
0, the multiplex networks reduce to some single networks, while q =
1, the fully dependence model discussed above is covered. Thus the system becomes robust with the decreasingof q .Besides, without reducing the number of dependence links, the dependence can also be adjusted by the so-calleddependence / coupling strength α ∈ [0 ,
1] [430–432]. When a node loses its dependence partner in the pruning process,each of its links is removed from the network with a probability α . With this mechanism, heterogeneous dependencecan be realized by assigning di ff erent dependence links with di ff erent dependence strength, and the results demonstratethat the heterogeneous dependence strength makes the system more robust [431]. In addition, Liu et al. proposed amodel that integrates interdependent networks and single networks with dependence links [433]. They found thatwhen most of the dependence links are inner- or inter-ones, the coupled network system is fragile and shows adiscontinuous percolation transition. However, when the two types of dependence links have nearly the same number,the system is robust and shows a continuous percolation transition.The one-to-one dependence in the original model is also generalized to multiplex dependence or group depen-dence. Obviously, the multiplex / group dependence makes the spreading of failures easier, so that the vulnerabilityof interdependent network under various dependence mechanisms is demonstrated [122, 434–436]. Wang et al. fur-ther showed that a larger dependence group did not always make the network fragile, which is also dependent onthe grouping mode and the cascading rules in the group [437, 438]. Li et al. also pointed out that the asymmetricdependence can break the cascading picture between dependence partners, thus results in a robust network [439]. Network of networks.
Considering the real situation that more than two networks are connected by the dependencelinks, Gao et al. extended the model of the interdependent networks into the model of a network of networks [440–446]. In this model, each network is composed of a set of nodes and connectivity links, then the dependence linksconnect them to form a larger network. The common factors that influence the network robustness, such as clustering[447], and targeted attack [448], have also been explored in the network of networks. Liu et al. proposed a “weak”interdependence mechanism capturing the di ff erences of networks, where network layers at di ff erent positions withinthe multiplex system experience distinct percolation transitions [306].In general, the robustness of network of networks decreases with the increasing of network layers. Due to thecascading failures caused by the dependence of networks, the percolation transition of a network of networks couldbe discontinuous and even a single node failure may lead to an abrupt collapse of the system [440–442]. Based ona network model composed of spatially embedded networks, Shekhtman et al. showed the extreme sensitivity ofcoupled spatial networks and emphasized the susceptibility of these networks to sudden collapse [449]. In real life,infrastructural networks are governed and operated separately, and interactions are only allowed within well-defined49 a) k =3, l =2 (b) k =4, l =2 Figure 19: (color online) Schematics of the clique percolation. Two k -cliques are regarded as adjacent, if they share l ( ≤ k −
1) nodes. Di ff erentclusters are distinguished by di ff erent colors. (a) k = l =
2. (b) k = l = boundaries. Therefore, for di ff erent situations, the percolation on network of networks may be very di ff erent. Forexample, Radicchi et al. introduced a model of percolation where the condition that makes a node functional is thatthe node is functioning in at least two network layers of the system, and found that the addition of a new layer ofinterdependent nodes to a preexisting multiplex network will improve its robustness [450].In addition, the network of networks is also studied as interacting networks [319, 451, 452], in which the connec-tions between the network layers are ordinary links. A system of this kind is therefore equivalent to a single modularnetwork, for which the system can be easily separated into isolated modules, and di ff erent modules might percolate atdi ff erent thresholds [389, 453–455]. It should mention that there are some special reviews on the resilience of networkof networks [456, 457]. One can refer to those for more details. Community structures are the subsets of nodes within a network that have a high density of within-group con-nections but a lower density of between-group connections. Traditional community detection methods are usuallydeterministic and used for finding separated communities, whereas most of the actual networks are made of highlyoverlapping cohesive groups of nodes. To identify the overlapping communities in large real networks, Der´enyi andco-workers have proposed a dependent percolation model, called clique percolation [170, 171].A clique is a fully connected subset of nodes of a network; such a subset with k nodes is often called a k -clique. Intheir model, two k -cliques are regarded as adjacent, if they share l ( = k −
1) nodes. With this connection rule, a nodecan belong to more than one clique cluster, see Fig.19. Then, by identifying these clique clusters as communities, theoverlapping communities are thus detected.This method also has a clear limitation. That is the network must have a large number of cliques, so it may failto give meaningful community structures for networks with just a few cliques. For more discussions on this problem,one can refer to Ref.[458].Aside from the analysis of network structure, clique percolation transition itself provides a set of very interestingproblems. As classical percolation in ER networks, k -clique percolation considers the behaviors of clusters formedby connected k -cliques for a di ff erent connection probability p of links. The simulation results in Ref.[170] indicatethat the fraction of nodes in the largest k -clique cluster makes a discontinuous percolation transition with increasingconnection probability p ; however, the fraction of k -cliques in the largest k -clique cluster demonstrates a continuouspercolation transition. Bollob´as and Riordan gave a rigorous mathematical result of the critical point for a moregeneral percolation model, in which two k -cliques are regarded as adjacent if they share l ( = , , ..., k −
1) or morenodes [123]. The Monte Carlo simulation indicates that for di ff erent k and l , this general clique percolation modelcould demonstrate rich critical phenomena [459]. 50n Ref.[124], Li et al. proved theoretically that the fraction of nodes in the giant clique cluster φ for l > l =
1. Moreinterestingly, they showed that the order parameter at the critical point φ c is neither 0 nor 1, but a constant dependingon k and l . In addition, the fraction of cliques in the giant clique cluster ψ always makes a continuous phase transitionas the classical percolation. Through analysis of the clique cluster number distribution, they found that there is noessential distinction between the two processes measured by ψ and φ , and the di ff erent behaviors are mainly causedby the quantitative relation between the total numbers of cliques and nodes. Besides, the clique percolation has alsobeen studied in two-dimensional systems [208]. The finite-size scaling analysis shows that, in contrast to the cliquepercolation on an ER network where the critical exponents are parameter dependent, the two-dimensional cliquepercolation simply shares the same critical exponents with traditional site or bond percolation, independent of theclique percolation parameters [208]. The heterogeneity of complex networks shows up in the di ff erent roles that nodes play in the structure resilienceand the dynamical behaviors. The vital nodes identification classifies nodes based on their roles played in networkstructure or dynamics [245]. As a model of cluster forming, percolation model has also been employed to identifyvital nodes.In Ref.[460], Piraveenan et al. proposed a centrality measure based on the percolation state of networks, calledpercolation centrality. In this method, percolation state of node i is denoted as x i . Specifically, x i = i is in the fully percolated state, and x i = < x i <
1. Accordingly, the percolation centrality of a node i in a network with size N is defined as PC ( i ) = N − X s , i , r σ s , r ( i ) σ s , r x s P j x j − x i , (117)where σ s , r is the number of shortest paths between nodes s and r , and σ s , r ( i ) is that pass through node i . With this,nodes in large clusters are considered to be more important. If all nodes are in the same cluster, x s / ( P j x j − x i ) = / ( N − i.e. , randomly removing some links. In the resulting network, the largest degree nodes in the top- n largestclusters are selected and one score is assigned. With di ff erent realizations, the average score of each node can beobtained, and the ones with large scores are suggested to be the influential spreaders. Furthermore, Hu et al. o ff eredan analogy of the correlation length in percolation transition to explain the local length scale in spreading dynamics,which could also be used to measure a node’s or a group of nodes’ global influence [464].In addition, percolation process is also employed as a criterion of the node ranking algorithm [245]. In thiscriterion, the size of the giant cluster after a node’s removal is assumed to be the structural importance of the node,such that the smaller the obtained giant cluster is the more important the node is. For a node ranking, one can removenodes from the network one by one in the order of the ranking, then the average size of the giant clusters for eachstage can be expressed as R = P Ni = S i N . (118)Here, N is the number of nodes in the network, and S i is the size of the giant cluster when the i -th node has beenremoved. Obviously, a good node ranking will give a small R . From Eq.(118), if the network is large enough, R isapproximately the area surrounded by function S i and the horizontal axis.51 .5. Network observability In Ref.[197], a fundamental problem in network science, namely, the determination of the state of the systemfrom limited measurements, was abstracted as a percolation model, called network observability transition. In thismodel the initial node occupation is understood as placing a sensor device on the corresponding node, making boththe node and all of its neighbors observable. Then, network observability transition focuses on the emergence of thegiant cluster formed by observable nodes.Obviously, the threshold of such percolation must be smaller than that of the classical percolation. By generatingfunction technique, this model can also be solved theoretically. The percolation threshold depends dominantly on theaverage degree and the variance of degree distribution. The transition occurs earlier in denser and more heterogeneousnetworks, meaning that only a small amount of sensor devices can observe the key structure of such networks. How-ever, Allard et al. further found a nontrivial coexistence of an observable and of a nonobservable extensive cluster inthis percolation model [465]. This suggests that monitoring a macroscopic portion of a network does not prevent amacroscopic event to occur or be unknown to the observer. Further studies also showed that both the negative degreecorrelation and the betweenness-based sensor placement can make networks more observable [466, 467], and wereapplied to many real networks [468, 469]. In Ref.[469], a discontinuous observability transition was found in syntheticmultiplex networks.There is another percolation model called l -hop percolation [198], following a very similar percolation rule. Inthis model when a node is removed all the nodes in the l -hop distance, i.e. , the nodes no farther than l away fromthis node, will also be removed, therefore it can cause more damage than the normal robustness model based onthe classical percolation. For the study of networks robustness, such a removal rule is often named localized attack[228, 387, 404–408].
5. Applications to network dynamics
The origin and development of percolation are inextricably bound up with the spreading process, therefore it iscompletely natural that in network science the percolation mechanism has been widely used to embody the spreadingdynamics. In this section we will first review some typical models of spreading dynamics and their relations withthe percolation transition. Besides, the percolation theory has also been borrowed to analyzing the cluster forming ofindividuals in the dynamics that the node or individual group plays a key role in the evolution of systems, such as thecascading process, tra ffi c jam, and cooperation evolution. These works will also be reviewed in this section. The modeling of epidemics is a very active field of research across di ff erent disciplines [470], and complex net-works provide a powerful framework to describe the contact structure among individuals [471]. As mentioned inSec.2.1, the epidemic spreading on networks is closely related to percolation process. When the spanning cluster doesnot exist or the susceptible individuals are isolated into small clusters, infectious diseases will be constrained to somesmall areas and a global outbreak is impossible. Similarly, when some individuals are immunized to the disease, i.e. ,recovered from the disease, being quarantined or get vaccinated, the susceptible individuals can be also separated assmall isolated clusters and thus the disease cannot spread across the network. From this perspective, regardless ofthe specifics, the outbreak of the epidemics corresponds to a percolation transition among connected individuals, andthere is a direct correlation between the percolation threshold and the epidemic threshold.In this section, we introduce the framework of epidemic modeling in complex networks from a percolation per-spective, although most of them are not inspired by percolation theory initially. Due to the focus of this article, we willnot deeply explore various models and the corresponding analytic methods. For details, one can refer to the specialreviews [23, 472, 473].It should be noted that the spreading of epidemics is a dynamic process, while the percolation process results ina static configuration. Because of this, one often employs the master equation to feature the system state during thespreading process, instead of considering the static configuration of the system. Nevertheless, we will find that theresults obtained by the master equation directly reflect the nature of percolation transition. With some mathematicaltechniques, some spreading models can also be exactly mapped to a percolation process [23, 87].52enerally, the epidemic model assumes that the individuals can be categorized as di ff erent classes dependingon the stage of the disease in the population, such as susceptible ( S , those who have not got infected), infectious( I , those who have been infected and are contagious), and recovered ( R , those who have recovered from the dis-ease or death) [23]. Based on this classification, three epidemiological models, susceptible–infected (SI), suscepti-ble–infected–susceptible (SIS) and susceptible–infected–recovered (SIR) are usually used to study the dynamics ofdisease evolution.The simplest one is the SI model in which individuals can only exist in two discrete states, namely, S and I . Theprobability that an S node gets infected by any given neighbor in an infinitesimal time interval dt is β dt , where β isthe infection rate of the epidemic. The SIS model considers that individuals change stochastically through a reversibleprocess: S → I → S , where the infection rate is described as in the S I model, but infected individuals may recoverand become susceptible again with a recovery rate µ . In the SIR model, infected individuals are removed permanentlyfrom the network with a rate µ and labeled as the state R corresponding to immune or death. There is a second versionof SIS / SIR model, where infected individuals remain in state I for τ time steps, and then become either R in SIRmodel or S again in SIS model [474]. As the setting of SI model, even for a very small infection rate β , all the individuals will be finally infected. There-fore, the epidemic threshold is meaningless for SI model, and the interest of research is the e ff ect of the heterogeneityof degree distributions on the spreading velocity [475].Let i k ( t ) be the fraction of infections in nodes with degree k at time step t , then the growth of i k ( t ) per time stepcan be expressed as β ks k ( t ) Θ k ( t ) = β k [1 − i k ( t )] Θ k ( t ), where s k ( t ) is the fraction of susceptible in nodes with degree k ,and Θ k ( t ) is the probability that a neighbor of a node of degree k is infected. Note that i k ( t ) cannot decrease with timefor this model, so the rate equation reads di k ( t ) dt = β k [1 − i k ( t )] Θ k ( t ) = β k Θ k ( t ) − β ki k ( t ) Θ k ( t ) . (119)For uncorrelated networks, Θ k ( t ) ≡ Θ ( t ) is independent of k . Considering that an infected node at least has an infectedneighbor, from which the disease has been transmitted, we have Θ ( t ) = P k ′ ( k ′ − p k ′ i k ′ ( t ) h k i . (120)In the early epidemic stages, i k ( t ) is small, then one can neglect the second term of the right hand side of Eq.(119),which is in an order of O ([ i ( t )] ). Together with Eq.(120), we have di k ( t ) dt = β k Θ ( t ) , (121) d Θ ( t ) dt = β ( κ − Θ ( t ) , (122)where κ = h k i / h k i . For a uniform initial condition i k ( t = ≡ i (0), the two di ff erential equations have solution i k ( t ) = i (0) " + k h k i h k i − κ − (cid:16) e t /τ − (cid:17) , (123)with the outbreak’s time scale τ = β ( κ − . (124)The average prevalence can also be obtained, i ( t ) = X k p k i k ( t ) = i (0) " + h k i − κ − (cid:16) e t /τ − (cid:17) . (125)53hese results indicate that the prevalence shows an exponential growth in the early stages, and larger degree nodesdisplay larger prevalence levels. By the way, the spreading dynamics on the networks with small-world e ff ects alsoreported a power-law growth with a tunable exponent [476, 477], which has already been observed in the epidemicdata of COVID-19 [478–481]. Such growth patterns are often referred to the sub-exponential growth dynamics in thephenomenological models [482].Equation (124) implies that the outbreak’s time scale of an epidemic is related to the heterogeneity of the degreedistribution as measured by κ . A strong heterogeneity corresponds to a large κ , and thus the time scale of outbreak τ is very small, which signals a very fast spreading of the infection. The physical reason is that once the diseasehas occupied the hubs, it can spread very fast among the network following a “cascade” of decreasing degree classes[475, 483].Note that the percolation threshold corresponds to p c = / ( κ −
1) [72]. When the infection rate β > p c , theoutbreak’s time scale becomes τ <
1. This means that an epidemic with an infection rate larger than the percolationthreshold can quickly spread far and wide. Moreover, a special case is SF networks with an exponent 2 < λ <
3, forwhich any infected rates β >
The above discussion can be easily extended to the SIR model. To describe the transition from I state to R state, anextra term − µ i k ( t ) defining the rate at which infected individuals of degree k recover and permanently removed mustbe added into Eq.(119) [484], di k ( t ) dt = β ks k ( t ) Θ k ( t ) − µ i k ( t ) , (126) ds k ( t ) dt = − β ks k ( t ) Θ k ( t ) , (127) dr k ( t ) dt = µ i k ( t ) , (128)where r k ( t ) is defined as the density of removed individuals of degree k and s k ( t ) + i k ( t ) + r k ( t ) =
1. With the initialconditions s k (0) ≃ i k (0) ≃
0, and r k (0) =
0, the solution of Eq.(127) can be written as s k ( t ) = e − β k φ ( t ) . (129)Here, considering the uncorrelated networks, i.e. , Θ k ( t ) ≡ Θ ( t ), φ ( t ) reads φ ( t ) = Z t Θ ( t ′ ) dt ′ = h k i X k p k ( k − Z t i k ( t ′ ) dt ′ = µ h k i X k p k ( k − r k ( t ) . (130)In the last equality we have utilized the integral of Eq.(128).Di ff erent from SI model, for t → ∞ , the infections vanish and the individuals are either in state S or in state R ,depending on the infection rate β and the recovery rate µ . We can use d φ ( t ) / dt = = d φ ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t →∞ = h k i X k p k ( k − i k ( t ) = h k i X k p k ( k −
1) [1 − r k ( t ) − s k ( t )] = h k i − h k i − µφ ( t ) − h k i X k p k ( k − s k ( t ) . (131)54ogether with Eq.(129), we find a self-consistent equation for φ ( ∞ ), φ ( ∞ ) = h k i − µ h k i − µ h k i X k p k ( k − e − β k φ ( ∞ ) . (132)Solving this equation, we can find φ ( ∞ ), and thus all other parameters for the steady state can be found.It is easy to find that if dd φ ( ∞ ) h k i − µ h k i − µ h k i X k p k ( k − e − β k φ ( ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ ( ∞ ) = < , (133)Eq.(132) only has a solution φ ( ∞ ) =
0, indicating the epidemic cannot break out. This yields the epidemic threshold δ c = β c µ c = κ − , (134)above which the epidemic incidence attains a finite value. Note that 1 / ( κ −
1) is just the percolation threshold of thenetwork. This suggests that there is a tight correspondence between the emergence of the giant cluster in the bondpercolation and the outbreak of epidemic in the system.Similar to SI model, Eq.(134) also suggests that a high level of connectivity heterogeneity (large κ ) facilitates thespreading of epidemics. When κ diverges (SF networks with 2 < λ < β can break out in the system [475, 483, 485]. This is analogous to those concerning the network resilience underrandom damages, which can be studied by the standard percolation on networks.The above discussion focuses the steady state of spreading process. The time evolution of SIR model can besolved by other approaches, such as the e ff ective degree-based method [486] and the pair-based mean-field method[487, 488]. The extended degree-based method is used to derive an expression for the basic reproduction ratio R [486], and also provides an excellent agreement with numerical simulations for both the temporal evolution and thefinal outbreak size [489]. The threshold condition derived from the e ff ective degree-based method turns out to be equalto the one obtained by percolation theory. The pair-based method was proven to be an exact deterministic descriptionof the infection probability time course for each individual in the tree-like network [490]. For networks with loops, aprecise closures can be also derived according to the detailed loop structure [491]. At the same time, the pair-basedapproach can be easily extended to models with heterogeneous infection and recovery rates [487].The correspondence between the steady properties of SIR model and the bond percolation is also studied inRefs.[1, 87, 492, 493]. Considering the second version of SIR model with a uniform infection time τ , i.e. , infectednodes will recover after being infected for τ time steps. For continuous-time dynamics, the transmissibility T , definingthe probability that the disease will be transmitted from an infected node to a susceptible neighbor before recoverytakes place, can be computed as T = − lim ∆ t → (1 − β ∆ t ) τ/ ∆ t = − e − τβ . (135)With this relation, we can exactly map the steady state of an SIR model to a bond percolation with occupied probability T . Then, from the critical transmissibility T c = G ′ (1) = h k ih k i − h k i = κ − , (136)one can find the critical infection rate for a given τ , that is β c = τ ln κ − κ − . (137)For discrete time, particularly computer simulations, the transmissibility T = − (1 − β ) τ , thus β c = − κ − κ − ! /τ . (138)These results also show that di ff erent combinations of β and τ could result in the same spreading result, related to thepercolation cluster under occupied probability T [87]. 55 .1.3. SIS model Based on the degree-based mean-field theory [494], the complete evolution equation for the SIS model on anetwork with arbitrary degree distribution can be written as di k ( t ) dt = β [1 − i k ( t )] k Θ k ( t ) − µ i k ( t ) . (139)The creation term considers the density 1 − i k ( t ) of susceptible nodes with degree k that can get the infection froma neighboring individual [495]. For uncorrected networks, Θ ( t ) = P k ′ k ′ p k ′ i k ′ ( t ) / h k i is used in Ref.[494], which isdi ff erent from Eq.(120). This is because the infected nodes can recover to be a susceptible, and all the excess-degreeof an infected node at the end of a link could lead to a node in state S . However, we think that around or above theepidemic threshold, on average an infected individual is very unlikely to recover before the individuals infected by ittransmit the disease to others. Thus, we still use Eq.(120) to represent Θ k ( t ) here. It must be pointed out that both thetwo choices are an approximation for Θ k ( t ), and do not a ff ect the basic conclusion.With the t → ∞ limit, the system can reach the stationarity state, i.e. , di k ( t ) / dt =
0. This yields i k ( ∞ ) = k β Θ ( ∞ ) µ + k β Θ ( ∞ ) . (140)This equation shows that the larger the degree of a node, the higher its probability to be infected. Substituting Eq.(120)into Eq.(140), a self-consistent equation can be found Θ ( ∞ ) = h k i X k p k ( k − k β Θ ( ∞ ) µ + k β Θ ( ∞ ) . (141)This equation always has a trivial solution Θ ( ∞ ) =
0, and the nontrivial solution exists only for dd Θ ( ∞ ) h k i X k p k ( k − k β Θ ( ∞ ) µ + k β Θ ( ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Θ ( ∞ ) = ≥ . (142)Thus, the epidemic threshold reads δ c = β c µ c = κ − . (143)This result is the same as that found in SIR model, the qualitative analysis is not repeated here. Note that for SFnetworks with 2 < λ <
3, neither assortativity nor disassortativity can reconstruct a non-vanished epidemic thresholdfor SIS model [496].For the version that infected nodes remain in state I for τ time steps, and then become S again, the epidemicthreshold of SIS model can be also derived by the branching process in percolation theory [474]. In SIS model, it hasbeen found that there exist dynamic correlations coming from that if one node is infected, its neighbors are likely tobe infected, which has been neglected by the degree-based mean-field theory. To account for this, e ff ective degreeapproach [489, 497] and the combination of degree-based mean-field theory and e ff ective degree approach [498] havebeen proposed to obtain accurate expressions for the average prevalence and the epidemic threshold. Not only the epidemic model itself but also the immunization strategy for suppressing the spreading is relatedto percolation process. The simplest immunization strategy is the random one, which chooses immune individualsrandomly from a population. After immunization, the infected individuals cannot transmit disease to the immuneindividuals. Thus it is equivalent to a spreading process on a diluted network. If there is no giant cluster in the dilutednetwork, the epidemic will not break out, regardless of infection rate. The corresponding immunization ratio is calledimmunization threshold. A detailed discussion for this problem is the same as that for the network robustness, andthus immunization can be treated as a percolation process.56he introduction of a fraction g of immune individuals chosen randomly is equivalent to dilute nodes with proba-bility 1 − g . According to Eqs.(15) and (16), the immunization process results in κ − → ( κ − − g ) [499]. Thus,the epidemic threshold is rescaled as δ c = β c µ c = − g κ − . (144)This generally covers the common sense that the more immune individuals, the larger the epidemic threshold. How-ever, for SF networks with 2 < λ < κ is divergent, implying that random immunization cannot bring the epidemicinto healthy region except for the case g =
1. From Eq.(144), we can also find that if 1 − g < / ( κ − ≡ p c , even if β = − g c = p c , the epidemic cannot break out [499]. This is mainly because below this thresholdthe susceptible individuals are segregated in smaller regions by immune individuals.Obviously, whoever the immune individuals are, they can suppress the infection locally. However, an apparentinadequacy of random immunization strategies is that it gives the same importance to both high-connected nodes (withthe largest infection potential) and low-connected nodes, and thus is unable to find the critical individuals to makethem immune for the eradication of infection. Targeted immunization is referred as the immunization of most highlyconnected individuals (potentially the largest spreaders), i.e. , a fraction g of the individuals with the highest degreesis immunized [470]. The equivalent diluted result depends on the details of the targeted immunization, which is verysimilar to the intentional attack in the study of network robustness (see Sec.4.2).In Ref.[499], the authors showed that the immunization threshold takes the form g c ∼ e − µ/ m β for SF networkswith λ =
3, where m is the minimum degree. This clearly highlights that the targeted immunization program isextremely convenient, with an immunization threshold that is exponentially small in a wide range of infection rates.Based on the global knowledge of network, the nodes with the highest recalculated degree or betweenness centralityare selected for immunization during the selection procedure [500]. Schneider et al. sought to minimize a properlydefined size for the connected cluster of susceptible individuals, which is found to be more e ff ective than the methodsbased on immunizing the highest-betweenness links or nodes [501]. Similarly, seeking to fragment the network intoconnected clusters of approximately the same size, “equal graph partitioning” strategy can achieve the same degree ofimmunization as targeted immunization by fewer immunization doses [502].An e ffi cient targeted immunization requires global information about the network, which makes it to be impracticalin many real situations. Acquaintance immunization can make immune a small fraction of random acquaintances ofrandomly selected nodes [503]. In this way, the highly connected nodes are more likely to be chosen and immunized,and the process prevents epidemics with a small finite immunization threshold and without requiring global knowledgeof the network. In acquaintance immunization, a random fraction q of nodes is chosen and then selects their randomneighbors, i.e. , acquaintances. The acquaintances, rather than the originally chosen nodes, are the ones immunized.The fraction q may be larger than 1, for a node might be queried more than once, on average, while the fraction ofnodes immunized g is always less than or equal to 1. The critical threshold q c for a complete immunization can besolved analytically based on a branching process on a network as that in percolation model.If more local information is available, the acquaintance immunization strategy can be further improved in ef-ficiency. For example, allowing initial selected node to have knowledge of the degrees of its nearest neighbors,immunizing the neighboring nodes with the largest degree is smart for e ffi ciency [504]. Similarly, if the informationwithin a distance d of initial selected node is available, one can consider the immunization of the nodes with thehighest degrees within that scope. It has been widely recognized that there are interdependencies or hidden interactions among the nodes in a net-worked system, which allows the viruses, information, opinion or failures to propagate from node to node via a certaintopology. Although the specific scenarios might be di ff erent, most models present an iterative process: each individualin a population must decide a state between two or more alternatives and their decisions are made based on the optionsof their neighbors rather than relying on their own information about the problem, thus after an initial state setting thestate flipping of individuals will occur recursively until no node meets the flipping criterion. This type of processes isoften called cascading process.With a given amount of excitation sources, the emergence of a giant cluster formed by individuals with identicalstates is just a dependent percolation process. Most of the models reviewed in Sec.4 belong to this category, i.e. , the57ystem evolves in generations. Here we mainly concentrate on another model widely used for the cascading process,called threshold model.Threshold model was first proposed by Watts for the study of cascading failures in networks [194]. In this model,there are two states for nodes, namely, functional and failed, to be more generic, we use state 0 and state 1 instead.At each time step, the binary decision rule with externalities is outlined as follows: each node in state 0 observes thecurrent states (either 0 or 1) of all its k neighbors, and adopts state 1 if at least a threshold fraction β of its k neighborsare in state 1, else remains unchanged. No transition from state 1 back to state 0 is possible. With a small fraction ofseed nodes (state 1) in a population which is initially set as state 0, the system evolves at successive time steps withall nodes updating their states according to the threshold rule above.In a su ffi ciently large random network with sparse connections and only a few seeds, it can be assumed that thenetwork is locally tree-like, hence, no node has more than one neighbor in state 1. Under this condition, only thenodes with degrees k ≤ /β have the potential to flip their states to be 1, called vulnerable nodes. If the vulnerablenodes percolate in the system, the global cascades can be triggered by the initial seeds; otherwise, the propagation ofcascades will be limited by the stable nodes and global cascade is impossible.That is to say, this dynamic problem can be mapped to a static percolation problem, which can be solved by thegenerating function approach, see Sec.2.2. According to percolation threshold of random networks G ′ (1) = k is vulnerable as θ k , the largest vulnerable cluster percolates when1 h k i X k θ k p k k ( k − = . (145)With the above analysis, we know that θ k = k ≤ /β , otherwise θ k =
0. For large thresholds β , Eq.(145)has no solution, meaning that the vulnerable nodes cannot percolate and there is no global cascade in the system. Forsmall thresholds β , Eq.(145) has two solutions resulting in two phase transitions. Below the smaller critical point, thenetwork is composed of some small clusters and the propagation of cascades is thus constrained by these segregatedclusters. Above the larger critical point, the network becomes dense enough, so that the propagation of cascades isblocked by the local stability of nodes. In summary, the global cascades can only occur in the network with a structurelying between the two phase transition points. Moreover, the transition of the cascade upon the average degree iscontinuous at the smaller critical point and discontinuous at the larger critical point.The above discussion is based on that the seeds are scarce. Gleeson and Cahalane further studied the e ff ects ofseed size on the cascading process [253]. Their method is based on a level-to-level update process, the final cascadesize ρ (rescaled by the system size) of active nodes (state 1) is given by ρ = ρ + (1 − ρ ) ∞ X k = p k k X m = km ! q m (1 − q ) k − m F (cid:18) mk (cid:19) , (146)where ρ is seed size, and F ( m / k ) gives the probability that the node with degree k and m active neighbors meets theexcitation condition. In addition, q is the probability that a link leads to an active node, which satisfies the recursionrelation q = ρ + (1 − ρ ) ∞ X k = p k k h k i k − X m = k − l ! ln ! q m (1 − q ) k − m − F (cid:18) mk (cid:19) . (147)With these equations, the cascade size can be solved. The results show that the cascade transition at low h k i may infact be discontinuous in certain parameter regimes, and the seed size ρ as low as 0 .
1% has dramatic e ff ects on thelocation of cascade transition points, that is1 h k i X k = p k k ( k − " F k ! − F (0) > − ρ . (148)Obviously, this result reduces to Eq.(145) found in Ref.[194] by letting ρ →
0, for which F (0) = θ k = F (1 / k ).This analysis method is also valid for large seed ρ , which has been also extended to analyze networks withcommunity structures and degree-degree correlations [256, 505]. For networks with clusterings [258], it turns out thatfor large and small values of h k i clustering reduces the size of cascades, while the converse occurs for intermediate58alues of the average degree. In Ref.[195], Liu et al. showed that using the seed size ρ as the control parameter, thecrossover of the continuous and discontinuous phase transitions can be found.Watts’s threshold model can be seen as a particular instance of a more generalized model of contagion [506], andhas been used to study the e ff ects of network properties that may influence the spreading process, such as booleannetworks [507], SW networks [254], multiple layers [261], temporal networks [508] where the associated burstyactivity of individuals may either facilitate [509] or hinder [510] the spreading process. ffi c and transportation As travel demand increases in modern cities, tra ffi c congestion becomes more frequent. As a result, the studyon the dynamical transition from free flow to congestion in tra ffi c flow has attracted much attention. To understandthe tra ffi c transition in a network scale (representing a city or an urban region), the percolation model has also beenapplied to the urban tra ffi c network.With percolation theory, the typical question of how the tra ffi c in the network collapses from a global e ffi cient stateto isolated local flows in small clusters is clearly demonstrated, which might be useful for improving transportatione ffi ciency, reducing tra ffi c delay, and facilitating emergency evacuation. Percolation theory also provides useful toolsfor urban planning. Tra ffi c network connectivity under disasters, such as earthquake or rain, has also been investigatedby percolation theory. ffi c networks Li et al. considered the percolation process in Beijing road tra ffi c network, and identified bottleneck roads thatconnect di ff erent functional clusters [511]. The velocity data of 5-min segment records measured in a central regionof Beijing is used to construct the road network, where nodes represent the intersections and links represent the roadsegments between two intersections. For each road, the velocity varies with time during a day. A dynamic functionaltra ffi c network can be constructed by the links with velocity higher than a given threshold q . With a small threshold q ,a giant cluster of functional network can extend to almost the full scale of the road network. As q increases, the size ofthe giant component decreases, and the second-largest cluster becomes maximal at a threshold q c , which signifies thepercolation transition from the connected phase to the fragmented phase of the functional tra ffi c network. The criticalvalue q c can be seen as an indicator of the e ffi ciency of global tra ffi c. Obviously, an e ffi cient tra ffi c system will havea higher q c .In a static network, bottleneck links are identified usually based on structural connectivity information. Tra ffi c, asa dynamical non-equilibrium system, evolves with time and shows varying q c during the day. The bottleneck linksof global tra ffi c will also evolve accordingly and appear in di ff erent locations in di ff erent hours. By investigating thefunctional cluster at percolation criticality, Li et al. identified some bottleneck links that play a critical role in bridgingdi ff erent functional clusters of tra ffi c [511]. With the congestion of these links, the giant cluster will disintegrate andresult in some small clusters. Conversely, improvement of these critical bottleneck roads can significantly increasethe threshold q c of the tra ffi c network, thus benefit the global tra ffi c. This kind of improvement of q c is higher thanthat by increasing the velocity of a random link or the link with the highest path-based betweenness. This proves theuniqueness of the bottleneck links found by percolation theory.Zeng et al. further identified two modes of percolation behaviors in Beijing and Shenzhen tra ffi c network [512].To investigate the di ff erences of the dynamic tra ffi c network in rush hours and nonrush hours, they performed apercolation analysis by calculating the size of the functional clusters. At criticality, a well-defined Fisher exponent τ as that in percolation theory can be observed. With this, Zeng et al. found that the disintegration transition of urbantra ffi c can be characterized by two sets of exponents. During rush hours on workdays, the critical exponent τ is ingeneral smaller than that during nonrush hours or days o ff . During workday rush hours, τ is close to the theoreticalresult of the two dimensional lattice ( τ = / ≈ . τ = / ffi c network can be generally seen as a systemembedded in two dimensions with some long-range connections corresponding to urban highways. During nonrushhours, all the roads are in the free flow state, the system is thus a two dimensional lattice with long-range connec-tions. In contrast, during rush hours, the highways become congested and the urban tra ffi c network reduces to a twodimensional system. With the aid of tra ffi c management methods, the tra ffi c system can be tuned to the desired class59y adjusting the amount of e ff ective long-range connections. Using the corresponding signal control or road pricingon urban highways, this mechanism can help to create a significantly better tra ffi c system.The above investigations are based on the analysis of the empirical tra ffi c data. Obviously, the percolation prop-erties vary with tra ffi c parameters, such as the level of tra ffi c load and the detailed behaviors of drivers. Such analysiscan also be performed by modeling simulations. Wang et al. studied the tra ffi c percolation properties on a tra ffi csystem based on an agent-based model [513]. Using the tra ffi c model in Ref.[514], the level of tra ffi c demand androuting preference can be tuned to verify their impacts on the percolation property. It has been widely proved that thetra ffi c will undergo a phase transition from the free state to the congested state with the increase of tra ffi c load in thenetwork. In Ref.[513], the functional network is constructed by the free state nodes with load lower than a threshold.The emergence of the giant cluster indicates the percolation transition and the formation of global tra ffi c flow. It isalso found that the threshold of the tra ffi c percolation q c shows a sharp minimum value at the tra ffi c transition thresh-old, indicating that the global flow can be maintained with minimal number of local flows. The value of q c is alsoinfluenced by the routing choice. In the giant percolation cluster, optimal paths will accumulate to a high level. Thiscan provide a possible routing choice to mitigate tra ffi c congestion.In the modern urban tra ffi c system, di ff erent transportation modes have been widely constructed, including bus,metro, and road networks, which can be represented as a multiplex network. In such a system, a fraction of nodefailures in one layer can trigger a cascade of failures that propagate in the multiplex network, see Sec.3.2. Baggag et al. studied the resilience and robustness of multi-modal transportation networks using percolation theory [515].In this study, percolation is used to investigate the impact of link failures in the multiplex transportation network.By calculating the size of the mutually connected giant cluster under random or targeted failures, the robustness ofmultiplex transportation networks can be evaluated. The four cities studied in Ref.[515] behave similarly in terms ofcoverage degradation under random failure, with Paris network being the most robust. Connected Vehicles (CV) technology is the leading edge of intelligent tra ffi c systems [516], and has great potentialto mitigate tra ffi c congestion through the creation of a CV network [517]. Once the vehicle cloud network is formed,travel information such as speed, density, and signal timing can be retrieved from the network. This informationwould help the driver’s decision-making and eventually enhance the mobility of CV in terms of travel time. In anurban tra ffi c system, the benefits of the corresponding applications will depend on the CVs’ communication range andthe market penetration, which can be featured by the percolation model.The study of CV network was pioneered by the analysis of network connectivity for wireless sensor networksand ad-hoc networks. Ammari and Das studied the sensing-converge and network-connectivity in wireless sensornetworks using percolation theory [518]. A probabilistic approach is created to compute the covered area fraction atpercolation threshold. Under various scenarios, the critical percolation density and radius of the covered componentsare identified. Similarly, Khanjary et al. conducted percolation studies in the two-dimensional fixed-orientationdirectional sensor network [519]. The critical density of the nodes can be analytically computed from the percolationtheory framework.Jin et al. first assessed the connectivity of Vehicular Ad-hoc NETwork (VANET) through a percolation framework[520]. The relationship among network connectivity, vehicle density and communication range was analyzed. Whenvehicle density or communication range is big enough, VANET will show a jump of network connectivity, corre-sponding to the percolation transition. Given vehicle density, the minimum communication range can be calculatedto achieve good network connectivity. As a large transmission range can cause serious collisions in wireless links, thepercolation analysis can help to determine the tradeo ff and guide the deployment of VANETs in real world.Talebpour et al. investigated the e ff ect of CV information availability on the stability of tra ffi c flow throughcontinuum percolation theory [521]. The impact of CV density and communication range on the connectivity of thenetwork is determined by percolation properties. The results show that as the communication range increases, thetra ffi c flow system becomes more stable.Mostafizi et al. investigated the impacts of CV network on the mobility of tra ffi c systems at varying levels ofmarket penetration and communication range [522]. When more CVs are present, the network tends to be moreconnected. When the percentage of CV decreases, there exists a percolation threshold where the giant cluster breaksapart. It has been found that the percolation transition is a function of both market penetration and communicationrange. The emergence of mobility benefits is highly related to the percolation transition in CV network. Only when60he system reaches the percolation threshold, the benefits of reducing mean travel time begin to appear. Below thethreshold, although small clusters can form in the network, they will not create a significant impact on the mean traveltime due to the scarce information available from the CV network. ffi c planning The analysis of the inter-relations between tra ffi c flows and urban street networks can provide a meaningful toolfor urban planning processes. From this perspective, Serok et al. identified functional spatio-temporal street clustersof London and Tel Aviv using network percolation analysis [523]. The dynamics of these clusters and their spa-tial stability over time are analyzed, which can help to develop new, adaptive, decision-making tools for urban andtransportation planners. Traditionally, urban planning is a long-term and static process. The application of tra ffi cpercolation will enable planners to keep their role in the flexible and dynamic urban system.Behnisch et al. studied the settlement connectivity in Germany using the percolation concept [524]. The perco-lation is introduced to quantify the connectivity of buildings. They found that at a critical distance of 830 ±
10 mthe buildings settlement network in Germany will undergo a transition from isolated clusters to a country-spanningbuilding cluster. This rather short critical distance indicates that the landscape is already overbuilt in Germany. Forurban planning, limiting urban sprawl and further land consumption is crucial. The application of percolation canprovide a useful monitoring tool for settlement and landscape degradation.Transport accessibility is an important issue for industrial relocation and the related sustainable development ofeconomics. Jiang et al. studied the impact of transportation accessibility on industry relocation pattern of China’sYangtze River Economic Belt using percolation theory [525]. In the network, the nodes are cities, and the links aredefined according to the Yangtze River’s waterways and the highways in the Yangtze River Economic Belt. Theirresults proved the existence of percolation transitions during the process of industry relocation, and identified thebottlenecks for industry relocation as cities located in border regions.Piovani et al. studied the relation of urban retail location distribution and road network properties in Londonfrom percolation theory. Road network and retail location are two interwoven factors in an urban ecosystem. Theemergence of clustering in retail centers depends greatly on road network [526]. Piovani et al. compared the roadnetwork’s hierarchical percolation structure with the retail location distribution. To evaluate the road network, thesubgraph is constructed by the road segments with length below a threshold, and the transition is defined at thethreshold that generates the maximal entropy configuration. The results show that clusters of the road network andretail location are very similar in spatial characteristics. ffi c networks Post-disaster tra ffi c networks are very important since they o ff er access to a ff ected areas, sustain evacuation andmedical operations after a disaster. If a disaster is very severe and many roads are malfunctioned, the tra ffi c networkcould collapse into isolated subnetworks, and there will be no route connecting some locations. Therefore, the connec-tivity of post-disaster tra ffi c networks also provide an important index for the assessment of vulnerability, robustness,and resilience of the road system.Zhou et al. studied the connectivity of post-earthquake road networks using percolation theory [527]. To assessthe road network, they proposed the concepts of “global connectivity” and “local connectivity”. Global connectivitymeasures the extent to which the whole network is connected, and local connectivity measures the distances betweeneach node to its neighbors. Specifically, the global connectivity is quantified by the percolation threshold, the size ofthe giant subnetwork, and average sizes of small subnetworks, while local connectivity is quantified by the numberof neighbors. In particular, Zhou et al. proposed an integrated percolation model combining localized attacks andrandom failures to capture the feature of earthquake-impacted [527]. Namely, the impact of the earthquake on theinner network is modeled as a localized attack (where node disruption is dominant), while it is modeled as randomlink disruption in the outer network. The model can be used to assess the connectivity of road networks impacted bydi ff erent magnitudes of earthquakes.Extreme weather, such as torrential rain and hurricane, could greatly influence urban tra ffi c system. Althoughthe tra ffi c flows under extreme weather have been widely studied, little work has been done to understand whetherand how the destruction of local roads can degrade the global tra ffi c. Guo et al. studied the impact of torrential rainon tra ffi c percolation properties in Beijing and Wuhan [528]. In Beijing, whose street layout is more symmetrical,the dysfunctional roads are scattered uniformly. However, for Wuhan with a river valley, the dysfunctional roads61ppear in a cluster way and partially along the riverbank. Interestingly, the percolation threshold of tra ffi c network isstable against weather perturbation, even if some roads are significantly influenced. This is mainly because extremeweather (torrential rain) will not only damage road conditions in the supply end, but also reduce the tra ffi c demandcorrespondingly. Evolutionary game theory o ff ers a powerful mathematical framework to study the emergence of cooperationamong selfish individuals. When the contact network of individuals is taken into account, i.e. , the spacial evolu-tionary game, the focus of the research just translates to the emergence of the clusters formed by adjacent cooperators.Therefore, the percolation theory is also employed to explore the spacial evolutionary game.In Ref.[529], Yang et al. studied the cooperation percolation in spatial prisoner’s dilemma game. Their modelis defined on a two-dimensional square lattice. Initially, with probability f a node is occupied by a cooperator andwith probability 1 − f a node is occupied by a defector. At each time step, each individual plays the prisoner’sdilemma game with its four nearest neighbors. Specifically, if both players cooperate, then both get the payo ff
1. Ifone cooperates while the other defects, the defector gets b while the cooperator gets nothing. If both defect, no gainsfor both of them. After the games, individuals update their strategies asynchronously in a random sequential order.A randomly selected player compares its payo ff with its nearest neighbors and changes strategy by following the one(including itself) with the highest payo ff .They found that the percolation transition can be well-defined by the control parameter f , the phase diagram canbe divided into several regions based on whether the cooperator cluster can percolate. With the analysis of finite sizescaling, they claimed that the region 1 < b < / / < b < f , the giant cooperator clusterincreases continuously from zero above a critical point, then at a second critical point the giant cooperator clusterjumps from a low value to a very high value. These rich phenomena highlight the role of the social contact in theemergence of the cooperation.In Ref.[532] Lin et al. also found a scaling feature in the evolutionary game on the interaction graph extractedfrom percolation process. The power-law distribution of cooperator clusters like that in percolation model was alsoobserved in fractal hierarchical networks [533]. Besides, an interaction graph at the percolation threshold has alsobeen proved to be an optimal population density for public cooperation [534, 535]. Peng and Li suggested that thisis due to the fractal structure formed at the percolation threshold [536], which constructs an asymmetric barrier thatthe defection strategy is almost impossible to cross, but the cooperation strategy has a not too small chance. In thesystem with mobile individuals, the geometric percolation of the contact network ( i.e. , irrespective of the strategy)also enhances cooperation [537].
6. Discussion and outlook
Percolation describes the patterns of connections under a random or semi-random connection mechanism, whilenetwork science focuses on the non-trivial topological features inspired largely by the empirical study of real-worldnetworks. In this sense, although the proposal of percolation model predates the rise of the research of complexnetworks by about fifty years, it inherently provides a framework for the studies of complex networks. So it is nosurprise that the application of percolation theory in network science has achieved fruitful results.First of all, the study of percolation process on a non-trivial topology, including strong heterogeneity, high cluster-ing, assortativity or disassortativity, modular, hierarchical structures and so on, has brought new power to percolationtheory, especially for the fractal and non-physical dimension systems. For examples, the heavy-tailed degree distribu-tion of the SF network induces a special mean-field nature of percolation transition (see Sec.2.5.2), and the percolation62n the growing network demonstrates an infinite order transition like that found in the Berezinskii-Kosterlitz-Thoulessphase transition in condensed matter (see Sec.3.4). In recent years, we have also witnessed a wide range of discussionon the explosive percolation and the hybrid percolation transition on network systems (see Sec.3). Besides, quite anumber of pruning rules used in the identification of special network structures are also defined in terms of percolationprocess, such as k -core percolation (see Sec.3.1), clique percolation (see Sec.4.3), and core percolation (see Sec.3.1).These percolation models also attracted a lot of studies on the system with physical dimensions, and enriched theresearch of the percolation theory [20, 21].In addition to providing theoretical supports for the findings from empirical researches, the ideas and conceptionsof percolation theory furnish a quantitative analysis approach to the methodology of network science. A typicalrepresentative is the giant cluster, which has already been widely used to evaluate the state of a network. For thestudy of network resilience, if a giant cluster exists, the network is considered functional, otherwise the network isparalyzed. In this way, the percolation threshold can be used to quantify the robustness of networks (see Sec.4.2).Similarly, if the infections form a giant cluster, it can be concluded that the epidemic breaks out, so that an epidemicwith a smaller percolation threshold is more infectious (see Sec.5). Even for the problems that seemed to be irrelativeto percolation process, the percolation analysis can o ff er a good interpretation for their findings, such as the evolutionof tra ffi c state of an urban network, and the emergence of the cooperation clusters (see Sec.5). Although almost all ofthese studies are not originally motivated by the percolation process and the evolution rules might be far away fromthe standard percolation, the emergence of the giant cluster is employed as a picture of the inherent problem. Becauseof this, some of the modeling of network dynamics just take the form of percolation process, such as the cascadingfailures, and the spreading of information / opinion / epidemics (see Sec.5).The classical percolation in physical dimensions often refers to regular systems, such that the percolation resultsare totally random, and the focus is more on the statistic characteristic of the clusters. However, the important char-acteristic of empirical networks, i.e. , strong heterogeneity, can bring determinacy into the percolation configuration.For example, the hubs of a SF network almost always belong to the giant cluster of di ff erent realizations. Once thehubs are destroyed, the network will be disintegrated. From this perspective, the percolation results in part reflect theconnective features of networks, and thus can be used as an algorithm for identifying the special subsets of networks.Some typical examples have been reviewed in Sec.4, including k -core decomposition, community detection, vitalnode identification, and so on.In summary, the percolation theory has already percolated into many research aspects of network science, rangingfrom structure analysis to dynamics modeling. Loosely speaking, one would hardly evade conceptions, analyticmethods, and algorithms of percolation in the studies of complex networks. For this reason, almost all the reviewarticles on the network science spill some ink on the corresponding percolation problems, such as network structuresand dynamics [3, 5, 6, 538], multiplex networks [8, 317, 445, 446, 457], spatial networks [539], epidemic process[23, 473], explosive transition [206], community detection [458, 540], k -core [214], and vital nodes identification[245]. Besides, the significant findings of network percolation are also included in the articles on recent advances ofpercolation transition [20, 21].By tracing the progress of percolation on networks and its applications, in this paper we briefly review the per-colation on complex networks from models, theoretical methods, to applications for network problems. Overall, thestudies of percolation on networks yield a rich harvest from theory to applications. Meanwhile, many challengesstill remain. First, real networks have a rich mixture of properties, such as degree correlation, clustering, modularity,heterogeneity, small world, and spatial constraint. For these complex structures, only some approximate theory canbe applied to simplify the structural models, while whether these structures can separately or collectively change thenature of the percolation transition or not remains an open question. Moreover, the mixture pattern often refers to thehigh-order structure of networks. This is a new research hotspot of network science in recent years, for which thepercolation theory also plays an important role [10, 541–546].Furthermore, as a random process, the percolation result of a given network can di ff er from realization to realiza-tion, while the network identification often requires a deterministic result. How to solve this conflict when employingpercolation process to analyze network structure? Are there some ways to take advantage of the fluctuation of percola-tion results to give more information on network structures? Moreover, it is known that the percolation clusters in thesubcritical phase are dominated by local connections, and the global connections for the supercritical phase. In thisway, is there a unified algorithm based on percolation process for identifying network structures in di ff erent scales?To answer these questions, more researches should also be done to explore the percolation process on networks.63 cknowledgments We would like to thank Pan Zhang, Manuel S. Mariani, and Xu Na for useful discussions and comments. Theresearch was supported by the National Natural Science Foundation of China under Grant Nos.11622538, 61673150,61773148, and 12072340. L.L. also acknowledges Zhejiang Provincial Natural Science Foundation of China underGrant No.LR16A050001, and the Science Strength Promotion Programme of UESTC.
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Bianconi, Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes, arXiv preprintarXiv:2005.02984 (2020).URL https://arxiv.org/abs/2005.02984 able 2: The main notations used in this paper.Notation Meaning d system dimension d f fractal dimension G ( x ) generating function of degree distribution G ( x ) generating function of excess-degree distribution H Hamiltonian of Potts model H ( x ) generating function of π s H ( x ) generating function of ρ s i k ( t ) fraction of infected individuals in all degree- k nodes at time step tk node degree K degree cuto ff of a network h k i average degree h k n i n -th moment of degree distribution M Hashimoto or non-backtracking matrix of graph N node number of a network p occupied probability p c percolation threshold p k degree distribution p s distribution of the cluster sizes q k excess-degree distribution r i → j probability that link i → j belongs to the giant cluster r k ( t ) fraction of removed individuals in all degree- k nodes at time step tR probability that a link belongs to the giant cluster S probability that a node belongs to the giant cluster s cluster size s i probability that node i belongs to the giant cluster s k ( t ) fraction of susceptible individuals in all degree- k nodes at time step ts ξ characteristic size Z partition function of Potts model Z rescaled partition function of Potts model z i partition function of the branching from node i of Potts model β critical exponent for the giant cluster; infection rate γ critical exponent for the mean cluster size δ x , y Kronecker delta function ζ ( x ) Riemann zeta function ζ ( x , a ) Hurwitz zeta function Θ k ( t ) probability that a neighbor of a degree- k node is infected κ ratio of h k i and h k i λ exponent of SF degree distribution, p k ∝ k − λ µ recovery rate ν critical exponent for the characteristic length ξ characteristic length π s size distribution of the cluster that a randomly chosen node belongs to ρ s size distribution of the cluster at the end of a link σ critical exponent for the characteristic size τ Fisher exponent Φ ( x , s , a ) Lerch’s transcendent χ mean cluster size able 3: The percolation thresholds of ER networks for some solvable models. The order parameter of the first part of the table is the occupiedprobability p , i.e. , removing each node with probability 1 − p . In the second part of the table, the order parameter is the link inserting probability p .Model ThresholdClassical percolation 1 / h k i [48]Biconnected cluster 1 / h k i [117]Core percolation 2 . / h k i [118]Greedy articulation point removal 3 . / h k i [119]Two interdependent networks 2 . / h k i [120, 121]Single networks with dependence links √ . / h k i [122]Classical percolation 1 / N [47]Clique percolation { ( k − l )! / [ (cid:16) kl (cid:17) − } / ( k − l )( k + l − N − / ( k + l − [123, 124]Growing network 1 / λ β γ ν σ τ (2 ,
3) 1 / (3 − λ ) − λ − / (3 − λ ) (3 − λ ) / ( λ −
2) (2 λ − / ( λ − ,
4) 1 / ( λ −
3) 1 ( λ − / ( λ −
3) ( λ − / ( λ −
2) (2 λ − / ( λ − , ∞ ) 1 1 3 1 / / ff erent models. In general, there are three kinds of models. The first one is anode removal process triggered by removing a fraction 1 − p of nodes from a network (occupying a fraction p of nodes). This removal process candi ff er from model to model, which could contain multiple steps from one single operation to infinite iteration. It is also worth pointing out that dueto a similar mechanism, although the bootstrap percolation is a recovery process for removed nodes, we also classify it into this class. The secondone is a link insertion process in a set of nodes, i.e. , at each time step a pair of nodes will be connected with probability p . The rule for choosing anode pair is the key of this type of models. For the last one, there could be a special definition of the connection of nodes, and the connection unitis not limited to node. Note that the probability p is used as the order parameter for the first two types, however, the meaning is di ff erent. In thetable these three types of models are separated by a horizontal line. Model Rule Type k -core percolation [193] Removing all the nodes with degrees less than k , iteratively. continuous for k =
2, hybrid for k ≥ k i / k i , < α iteratively, where k i and k i , are the current and the initial degrees of node i , respectively continuous for small α , discontin-uous for large α [195]Bootstrap percolation[196] Removed nodes will be iteratively recovered when they have atleast k neighbors. discontinuous, hybrid [196]Core percolation [118] Removing nodes of degree 1 along with its neighbor, iteratively. continuous for undirected net-works, discontinuous for directednetworks [118]Greedy articulationpoints removal [119] Removing all the articulation nodes iteratively, where the articula-tion node is the one whose removal disconnects the network. hybrid [118]Percolation on interde-pendent networks [120,121] Removing all the failed nodes iteratively, where the failed node isthe one does not belong to the giant cluster, or has a failed depen-dent partner in other layers. hybrid [120, 121]Network observability[197] The initial observable (occupied) node makes both the node and allof its neighbors observable (occupied). continuous [197] l -hop percolation [198] The nodes no farther than l away from the initial removed nodeswill also be removed. continuous [198]History-dependentpercolation [199] Removing all the links between di ff erent clusters of the last gener-ation. continuous for finite generations,hybrid for infinite generation[199, 200]Explosive percolation[21, 201–205] At each time step, more than one potential links are arbitrarily cho-sen. The one that suppresses the emergence of a giant cluster isinserted eventually, and other potential links are discarded. continuous with unusual finitesize behavior [206]Growing network [125] At each time step, a new node is inserted into the system, then twonodes are chosen randomly from all the existing nodes and joinedby a link with probability p . infinite order [125] k -component The nodes in the same cluster must connect to each other by at least k independent paths. continuous for k = / node removal, two nodes are considered as connected, ifthe new shortest path between them is shorter than al ( a ≥ l is the shortest path before removal. continuous [207]Clique percolation [124,170] This model considers the percolation of connected cliques in a net-work, and two cliques are regarded as adjacent, if they share somenodes. continuous [124, 208]Color-avoiding percola-tion [209, 210] Two nodes are considered as color-avoiding connected, only if theyare always connected for the removal of any colored links / nodes. continuous [209, 210]nodes. continuous [209, 210]