Efficient collective influence maximization in cascading processes with first-order transitions
Sen Pei, Xian Teng, Jeffrey Shaman, Flaviano Morone, Hernán A. Makse
aa r X i v : . [ phy s i c s . s o c - ph ] M a r Efficient collective influence maximization incascading processes with first-order transitions
Sen Pei , Xian Teng , Jeffrey Shaman , Flaviano Morone , and Hern ´an A. Makse Department of Environmental Health Sciences, Mailman School of Public Health, Columbia University, New York,NY 10032, USA Levich Institute and Physics Department, City College of New York, New York, NY 10031, USA * [email protected] ABSTRACT
In many social and biological networks, the collective dynamics of the entire system can be shaped by a small set of influentialunits through a global cascading process, manifested by an abrupt first-order transition in dynamical behaviors. Despiteits importance in applications, efficient identification of multiple influential spreaders in cascading processes still remains achallenging task for large-scale networks. Here we address this issue by exploring the collective influence in general thresholdmodels of cascading process. Our analysis reveals that the importance of spreaders is fixed by the subcritical paths alongwhich cascades propagate: the number of subcritical paths attached to each spreader determines its contribution to globalcascades. The concept of subcritical path allows us to introduce a scalable algorithm for massively large-scale networks.Results in both synthetic random graphs and real networks show that the proposed method can achieve larger collectiveinfluence given the same number of seeds compared with other scalable heuristic approaches.
Introduction
Cascading process lies at the heart of an array of complex phenomena in social and biological systems, including failurepropagation in infrastructure, adoption of new behaviors, diffusion of innovations in social networks and cascading failuresin brain networks, etc. In these cascading processes, a small number of influential units, or influencers, arise as a consequenceof the structural diversity of the underlying interacting networks. In different fields, it has been accepted that the initialactivation of a small set of such “superspreaders”, who usually hold prominent locations in networks, is capable of shapingthe collective dynamics of large populations.
In practice, identification of superspreaders can help to control the entirenetwork’s dynamics with a low cost, e.g., a company can boost product popularity by targeted advertisement on influencersin viral marketing, or we can maintain the robustness of infrastructure systems by protecting structurally pivotal units. Givenits great practical values in a wide range of important applications, the problem of locating superspreaders has attracted muchattention in various disciplines.
In the simple case of finding single influential spreaders, centrality-based heuristic measures such as degree, Between-ness, PageRank and K-core are routinely adopted. Beyond this non-interacting problem of finding single spreaders, itbecomes more complicated when trying to select a group of spreaders, due to the collective effects of multiple agents. In fact,searching for the optimal set of influencers in cascading dynamics is an NP-hard problem and remains to be a challengingconundrum in network science. To address the many-body problem, several approaches designed for the influence maximiza-tion in different models have been proposed. In the case of percolation model, a framework for optimal percolation based onthe stability analysis of zero solution was developed.
More recently, message passing algorithms for optimal decyclingin statistical physics further pushed the critical point toward its optimal value.
For susceptible-infected-recovered (SIR)model, the problem of finding influential spreaders is also explored using the percolation theory in recent works.
In theabove models, cascading processes can be transformed to the percolation model with a continuous phase transition. Whileoptimal percolation theory applies only to systems with second order phase transitions, here we treat the case of cascad-ing models which present first order discontinuous transitions. Such transitions cannot be treated with the stability analysismethods based on the non-backtracking matrix as done in for models with continuous transitions, so a new approach isneeded.In a large variety of contexts, the cascading process is properly described by the Linear Threshold Model (LTM) in whichthe states of nodes are determined by a threshold rule.
That is, a node will become active only after a certain number of itsneighbors have been activated. The choice of threshold m i = k i − k i is the degree of node i . In this case, the cascading dynamics of LTM can be mapped to the classical percolation process, for jk m =2 i1 k k I =0 k iij I =1 k iij b d ef Figure 1.
Subcritical paths and collective influence of spreaders. (a), Three combinations of neighbors P , P and P corresponding to ν i → j in message passing equation. Node i has a threshold m i =
2. The full activation of at least onecombination will lead to ν i → j = i → j with an active neighbor k and inactive ones k and k , I k → i , i → j = i has 0 ( < m i −
1) active neighbor excluding k and j , while I k → i , i → j = i has 1 ( = m i −
1) active neighbor k excluding k and j . (c), Illustrations of subcritical paths ending with link i → j for L = , ,
2. Red dots stand for seeds,while squares represent m − i to k ν → k exerted through subcritical paths of length L = , ,
2. (e), Calculation method ofCI-TM L ( i ) . Subcritical paths starting from i with length ℓ ≤ L are displayed by different colors. (f), An example ofsubcritical cluster. Assuming a uniform threshold m =
3, nodes inside the circle are subcritical since they all have 2 activeneighbors, represented by blue nodes. Activation of the red node will trigger a cascade covering all subcritical nodes.which the influence maximization problem can be solved by various algorithms designed for optimal percolation. Nevertheless,for other choices of threshold, LTM exhibits a first-order (i.e., discontinuous) phase transition. In fact, influence maximizationin LTM corresponds to finding nontypical trajectories of cascading processes that deviate from the average ones. Altarelli etal. analyzed the statistics of large deviations of LTM dynamics with a belief propagation algorithm, and further developeda Max-Sum (MS) algorithm to explicitly find the optimal set of seeds in terms of a predefined energy function. Guggiolaand Semerjian obtained the theoretical limit of the size of minimal contagious sets for random regular graphs, and used asurvey propagation like algorithm to locate the minimal set of seeds. Given these recent progresses in searching for optimalinfluencers in LTM, it is a challenging task to apply these methods to massively large-scale networks with tens of millionsnodes encountered in modern big-data analysis. Therefore, the problem of developing an efficient scalable algorithm ofinfluence maximization in cascading models with discontinuous transitions that is feasible in real-world applications stillneeds to be further explored.Here, we examine the collective influence in general LTM, and develop a scalable algorithm for influence maximization.By analyzing the message passing equations of LTM, we formulate the form of interactions between spreaders and provide ananalytical expression of their contributions to cascading process. Each seed’s contribution, defined as the collective influencein threshold model (CI-TM), is determined by the number of subcritical paths emanating from it. Since the subcritical pathsare such routes along which cascades can propagate, CI-TM can be considered as a reliable estimation of seeds’ structuralimportance in LTM. CI-TM is the generalization of the CI algorithm of optimal percolation for second order transitions treatedin to the present case of first order transitions. To apply our method to large-scale networks, we present an efficientadaptive selection procedure to achieve collective influence maximization. Compared with other competing heuristics, ourresults on both synthetic and realistic large-scale networks reveal that the proposed mechanism-based algorithm can producea larger cascading process given the same number of seeds. Results
Collective influence in threshold models: CI-TM
We present a theoretical framework to analyze the collective influence of individuals in general LTM. For a network with N nodes and M links, the topology is represented by the adjacency matrix { A i j } N × N , where A i j = i and j are connected,and A i j = n = ( n , n , · · · , n N ) records whether a node i is chosen as a seed ( n i =
1) or not ( n i = he total fraction of seeds is therefore q = ∑ i n i / N . During the spreading, the state of each node falls into the category ofeither active or inactive. The spreading starts from a q fraction of active seeds and evolves following a threshold rule: a node i becomes active when m i neighbors get activated. This process terminates when there are no more newly activated nodes. Weintroduce ν i as node i ’s indicator in active ( ν i =
1) or inactive ( ν i =
0) state at the final stage, and denote Q ( q ) as the size ofthe giant connected component of active population.For a directed link i → j , we introduce ν i → j as the indicator of i being in an active state assuming node j is disconnectedfrom the network. If n i =
1, then ν i → j =
1. Otherwise, ν i → j = m i active neighbors excluding j . Since there exist many possible choices of these m i neighbors, we define P m i ∂ i \ j as the set of all combinations of m i nodesselected from ∂ i \ j , where ∂ i \ j is the set of nearest neighbors of i excluding j . Clearly, if i has k i connections emanatingfrom it, there are (cid:0) k i − m i (cid:1) combinations, so the set P m i ∂ i \ j contains (cid:0) k i − m i (cid:1) elements, denoted by P h , h = , · · · , (cid:0) k i − m i (cid:1) . Each element P h has the form P h = { p h , · · · , p h mi } where { p h , · · · , p h mi } are the m i nodes in the h th combination. Figure 1a illustrates allthree combinations P , P and P corresponding to ν i → j for node i with a threshold m i =
2. Should at least one combination isfully activated, we have ν i → j = ν i → j = n i + ( − n i )[ − ∏ P h ∈ P mi ∂ i \ j ( − ∏ p ∈ P h ν p → i )] . (1)The final state of i is given by ν i = n i + ( − n i )[ − ∏ P h ∈ P mi ∂ i ( − ∏ p ∈ P h ν p → i )] . (2)The above equations Eq. (1-2) describe the general cases of LTM. For the special choice of threshold m i = k i −
1, there isonly one combination in P m i ∂ i \ j , and the transition becomes continuous, and then it can be treated with the stability methods ofoptimal percolation as done in. We note that, Eq. (1-2) are only valid under the locally tree-like assumption. For syntheticrandom networks, this assumption holds since short loops appear with a probability of order O ( / N ) . Nevertheless, aconsiderable number of short loops may exist in real-world networks. For those networks with clustering, many prior workshave confirmed that results obtained for tree-like networks apply quite well also for loopy graphs. For instance, Melnik etal. found that, for a series of problems, the tree-like approximation worked well for clustered networks as long as the meanintervertex distance was sufficiently small. As most of real-world networks are small-world, the approximation of Eq. (1-2)should be reasonable provided the density of loops is not excessively large.For all the 2 M directed links i → j , Eq. (1) is a nonlinear function of ν → = ( · · · , ν i → j , · · · ) T : ν → = n → + G ( ν → ) . (3)In Eq. (3), n → = ( · · · , n i → j , · · · ) T in which n i → j = n i for link i → j , and G = ( · · · , G i → j , · · · ) T where G i → j is the nonlinearfunction of vector ν → for link i → j . Given the initial configuration of seeds n , the final state of ν → is fully determined by theself-consistent Eq. (3). Unfortunately, it cannot be solved directly due to the exponentially growing number of combinationsin P m i ∂ i \ j . Therefore, for a small number of seeds, we adopt the iterative method to estimate the solution. In this point of view,Eq. (3) can be treated as a discrete dynamical system ν t + → = n → + G ( ν t → ) (4)with the initial condition ν → = n → .To simplify the calculation, we approximate the nonlinear function G i → j by linearization. Define G ′ i → j ( ν → ) = ( · · · , ∂ G i → j ∂ν k → ℓ , · · · ) .By Eq. (1), we know that ∂ G i → j ∂ν k → ℓ = ℓ = i . While in the case of ℓ = i and k = j , we have ∂ G i → j ∂ν k → i = ( − n i ) ∏ ¯ P h ∈ P mi ∂ i \ j , k ¯ P h ( − ∏ p ∈ ¯ P h ν p → i ) × ∑ P h ∈ P mi ∂ i \ j , k ∈ P h [( ∏ p ∈ P h \ k ν p → i ) ∏ P ′ h = P h , k ∈ P ′ h ( − ∏ p ∈ P ′ h ν p → i )] . (5)Although Eq. (5) has a complex form, in fact it is only determined by a simple quantity a k → i , i → j = ∑ p ∈ ∂ i \ ( k , j ) ν p → i , whichis interpreted as the number of i ’s active neighbors excluding k and j when i is absent from the network. On one hand, if a k → i , i → j ≥ m i , at least one term of ∏ p ∈ ¯ P h ν p → i equals one, since we are selecting m i elements from a set containing at least m i elements of value 1. Under such condition, ∂ G i → j ∂ν k → i =
0. On the other hand, if a k → i , i → j ≤ m i −
2, all the terms ∏ p ∈ P h \ k ν p → i re zeros because we are selecting m i − m i − ∂ G i → j ∂ν k → i =
0. When a k → i , i → j = m i −
1, all the terms ∏ p ∈ ¯ P h ν p → i and ∏ p ∈ P ′ h ν p → i are zeros, and only the exact combinationof these m i − ∏ p ∈ P h \ k ν p → i =
1. Therefore, we have ∂ G i → j ∂ν k → i = − n i . Based on the abovereasoning, we define a quantity I k → ℓ, i → j for links k → ℓ and i → j as follows: I k → ℓ, i → j = ( ℓ = i , k = j , a k → i , i → j = m i − , . (6)The definition of I k → ℓ, i → j is reminiscent of the Hashimoto non-backtracking (NB) matrix B . In the case of m i = k i −
1, our quantity I k → ℓ, i → j can be transformed to the corresponding element of NB matrix B k → ℓ, i → j . In fact, I k → ℓ, i → j isclosely related to the concept of subcritical nodes. Recall that a node i is subcritical if it has m i − Thisimplies that one more activation of its neighbors will cause i activated. From Eq. (6) we know that I k → ℓ, i → j = k → ℓ and i → j are connected, non-backtracking, and additionally, node i is subcritical in the absence of node k and j .In Fig. 1b, node i has an active neighbor k and two inactive ones k and k . By definition, for a threshold m i =
2, we conclude I k → i , i → j = i has no active neighbor excluding k and j , while I k → i , i → j = i has 1 (= m i − ) active neighborexcluding k and j .For a small ν → , a standard linearization around origin gives G i → j ( ν → ) ≈ G i → j ( )+ G ′ i → j ( ) ν → . However this will causedegeneracy since Eq. (5) constantly gives G ′ i → j ( ) = . Therefore, we approximate G i → j ( ν → ) by G i → j ( ) + G ′ i → j ( ν → ) ν → given ν → is close to . In Methods, we prove that this linearization has an approximation accuracy of O ( | ν → | ) ( | · | is thevector norm), same as the linear Taylor expansion. To account for the increasing network size as N → ∞ , we define the vectornorm as | ν → | ≡ ∑ i j ν i → j / M (2 M is the number of directed links), so that | ν → | is always bounded below 1 for all networksizes. The linear approximation is valid when the number of links attached to initial seeds is small compared with all directedlinks. As we will see, the fraction of seeds at the discontinuous transition is small for both synthetic and realistic networks.Therefore, the linear approximation before the critical point should be valid. In Methods, we compare | ν → | calculated by linearapproximation with its real value on a scale-free network assuming LTM is initialed by one single seed. The approximationresults agree well with their true values. See the detailed discussion in Methods.Combining all direct links, Eq. (4) can be approximated by a linear equation ν t + → = n → + F t ν t → , (7)where F t = ( · · · , G ′ i → j ( ν t → ) , · · · ) T is a 2 M × M matrix defined on the directed links k → ℓ , i → j with elements F tk → ℓ, i → j = ∂ G i → j ∂ν k → ℓ (cid:12)(cid:12)(cid:12)(cid:12) ν t → . (8)With the notion of I k → ℓ, i → j , we can write F t as: F tk → ℓ, i → j = ( − n i ) I tk → ℓ, i → j . (9)Now we update the state of ν t → following Eq. (7). In the following calculation, we simplify F tk → ℓ, i → j and I tk → ℓ, i → j to F tk ℓ i j and I tk ℓ i j respectively for notation convenience. We put the matrix F t in a higher-dimensional space: F tk ℓ i j = ( − n i ) A k ℓ A i j δ i ℓ ( − δ jk ) I tk ℓ i j , (10)where function δ i ℓ is 1 if i = ℓ , and 0 otherwise. The indices k , ℓ, i , j run from 1 to N . Starting from ν → = n → , ν → = n → + F n → gives ν i → j = n i + ( − n i ) A i j ∑ k A ki ( − δ jk ) I kii j n k . (11)The physical meaning of Eq. (11) can be interpreted as follows. If node i is a seed, ν i → j =
1. Otherwise, ν i → j is nonzero if i is subcritical ( I kii j =
1) and at least one of its corresponding neighbors k is a seed ( n k = i is not a seed, thecontribution of a neighboring seed k is conveyed by the directed path k → i → j that satisfies n k = , n i = I kii j =
1, whichis shown in the second panel of Fig. 1c.For t =
2, we have ν → = n → + F n → + F F n → . Therefore, ν i → j = n i + ( − n i ) A i j ∑ k A ki ( − δ jk ) I kii j n k + ( − n i ) A i j ∑ k ( − n k ) A ki ( − δ jk ) I kii j ∑ s A sk ( − δ is ) I skki n s . (12) he last term in Eq. (12) is actually the contribution of node i ’s 2-step neighbors s to ν i → j . The contribution of a seed s isconducted through a directed path s → k → i → j that satisfies n s = , n k = , n i = I skki = , I kii j = subcritical paths . For a directed link i → j , the path i → i → · · · → i L → i → j is a subcritical path of length L if n i = , n i = , · · · , n i = I i i i i = , · · · , I L − i L ii j =
1, and any two consecutive linksare non-backtracking. If i = i , we set the path’s length L =
0. The subcritical paths of length L =
0, 1 and 2 are displayed inFig. 1c. Notice that, the calculation of L -length subcritical paths is in fact implemented by the multiplication of F L − · · · F .In fact, the concept of subcritical path has a clear physical meaning. A subcritical path is composed of connected subcriticalnodes. So once the node i at the beginning of the subcritical path is activated, the cascade of activation will propagate alongthe path and lead to ν i → j = i → j at the tail. Therefore, the long-range interaction between node i and node i isrealized through the subcritical path connecting them. Following this idea, we can generalize Eq. (12) to ν Ti → j at a given time T . The exact formula for ν Ti → j is ν Ti → j = n i + ( − n i ) A i j T ∑ L = L ∏ ℓ = h ∑ k ℓ ( − n k ℓ ( − δ ℓ L )) A k ℓ k ℓ − ( − δ k ℓ − k ℓ ) I T − ℓ k ℓ k ℓ − k ℓ − k ℓ − (( n k ℓ − ) δ ℓ L + ) i , (13)where k − = j , k = i and k ℓ runs from 1 to N for ℓ >
0. Notice that the form of ν Ti → j is nothing but n i plus the contribution ofseeds connected to i through subcritical paths with length L ≤ T when n i = CI-TM Algorithm
To quantify the active population in LTM, we define k ν → k ≡ M | ν → | = ∑ i j ν i → j , where 2 M is the total number of directedlinks. Starting from k ν → k = k ν → k increases as more seeds are activated. Therefore, we expectthat the collective influence of a given number of seeds can be optimized by maximizing k ν → k .Based on the form of each element in ν → , we learn that the contribution of a seed i to k ν → k is composed of all its collectivecontributions to every potential element, exerted through the subcritical paths attached to i . Therefore, we employ a seed’scontribution to k ν → k to define its Collective Influence in Threshold Model (CI-TM) to find the best influencers. For the trivialcase of subcritical paths with length L =
0, we define CI-TM ( i ) = k i , where k i is the degree of node i . Thus, at the zero-orderapproximation we recover the high-degree heuristic. The first panel of Fig. 1d illustrates CI-TM ( i ) = i . For L ≥ L = ( i ) = k i + ∑ j ∈ ∂ i ( − n j ) ∑ k ∈ ∂ j \ i I i j jk . (14)As shown in Fig. 1(d), the contribution of node i to k ν → k through subcritical paths of length L = ( i ) =
5. For L = ( i ) = k i + ∑ j ∈ ∂ i ( − n j ) ∑ k ∈ ∂ j \ i I i j jk + ∑ j ∈ ∂ i ( − n j ) ∑ k ∈ ∂ j \ i ( − n k ) I i j jk ∑ ℓ ∈ ∂ k \ j I jkk ℓ (15)In Fig. 1d, the additional 2-length subcritical paths also contribute to CI-TM ( i ) , leading to CI-TM ( i ) =
7. Moreover, inFig. 1d, we can observe that for the tree structure, the activation of node j in the first-step update will not affect I jkk ℓ in thesecond-step update, which means I jkk ℓ = I jkk ℓ . More generally, I jkk ℓ is not affected by the activation of k ’s any precedent nodeson the subcritical path. Therefore, we will leave out the superscript t in the definition of CI-TM for locally tree-like networks.We can generalize the above CI-TM calculation to any given L . In summary, the definition of node i ’s influence CI-TM in anarea of length L is:CI-TM L ( i ) = number of subcritical paths starting from i with length 0 ≤ ℓ ≤ L . (16)Figure 1e illustrates the calculation of node i ’s CI-TM for L =
2, in which subcritical paths with length ℓ ≤ L are distinguishedby colors.For a given fraction q of seeds, our goal is to maximize k ν → k . As we have explained, the CI-TM value of a seed dependson the choice of other seeds. Therefore, it is hard to obtain the optimal configuration { n | ∑ i n i / N = q } without turningto extremely time-consuming algorithms. To compromise and obtain a scalable algorithm, we propose an adaptive CI-TMalgorithm following a greedy approach. Define C ( i , L ) as the set of node i plus subcritical vertices belonging to all subcriticalpaths originating from i with length ℓ ≤ L . Beginning with an empty seed set S , we remove the top CI-TM nodes as follow.The calculation proceeds following the CI-TM algorithm. lgorithm 1 CI-TM algorithm Initialize S = /0 Calculate CI-TM L for all nodes for l = qN do Select i with the largest CI-TM L S = S S { i } Remove C ( i , L ) , and decrease the degree and threshold of C ( i , L ) ’s existing neighbors by 1 Update CI-TM L for nodes within L + C ( i , L ) end for Output S In the above algorithm, we remove C ( i , L ) once i is added to the seed set. The reason lies in that it is unnecessary toselect nodes in C ( i , L ) as seeds in later calculation, because the activation of i will definitely active C ( i , L ) (See Fig. 1f).Besides, C ( i , L ) can be identified during the computation of CI-TM L without additional cost. In traditional centrality-basedmethods, seeds may have significant overlap in their influenced population. It has been reported that the performance of thesemethods, such as K-core, suffers a lot from this phenomenon. On the contrary, in our algorithm, this problem is alleviatedby the removal of subcritical nodes in C ( i , L ) , which successfully reduces the overlap and improves the efficacy of eachseed. Although such greedy strategy is not guaranteed to give the exact optimal spreaders, we expect a good performancein comparison with other heuristic methods in large-scale networks. For extreme sparse networks with large numbers offragmented subcritical clusters, a simple modified algorithm can find a smaller set of influencers (See Methods).More importantly, the CI-TM algorithm is scalable for large networks with a computational complexity O ( N log N ) as N → ∞ . On one hand, computing CI-TM L is equivalent to iteratively visiting subcritical neighbors of each node layer by layerwithin L radius. Because of the finite search radius, computing CI-TM L for each node takes O ( ) time. Initially, we have tocalculate CI-TM L for all nodes. However, during later adaptive calculation, there is no need to update CI-TM L for all nodes.We only have to recalculate for nodes within L + O ( ) compared to thenetwork size as N → ∞ as shown in Ref. On the other hand, selecting the node with maximal CI-TM can be realized bymaking use of the data structure of heap that takes O ( log N ) time. Therefore, the overall complexity of ranking N nodesis O ( N log N ) even when we remove the top CI-TM nodes one by one. In addition, considering the relative small number ofsubcritical neighbors, the cost of searching for subcritical paths is far less than that when scanning all neighbors. This permitsthe efficient computation of CI-TM for considerably large L . In our later experiments on finite-size networks, we do not puta limit on L so as to calculate CI-TM thoroughly. But remember that we can always truncate L to speed up CI-TM algorithmfor extremely large-scale networks. Test of CI-TM Algorithm
We first simulate LTM dynamics on synthetic random networks, including Erd¨os-R´enyi (ER) and scale-free (SF) networks. Inthe models, we adopt a fractional threshold rule m i = ⌈ tk i ⌉ , which means that a node will be activated once t fraction of itsneighbors are active ( ⌈·⌉ is the ceiling function). Here we choose this special form of threshold setting. But we note that thealgorithm can apply to other more general choices of threshold in LTM. In order to verify the efficacy of CI-TM algorithm, wecompare its performance against several widely-used ranking methods, including high degree (HD), high degree adaptive(HDA), PageRank (PR) and K-core adaptive (KsA). As a reference, we also display the result of random selection of seeds,as well as the size of optimal seed set identified by Max-Sum algorithm (MS). Details about these strategies are explainedin Methods.Figure 2a presents Q ( q ) versus q on ER networks ( t = . N = × , h k i = Q ( q ) first undergoes a continuous transition from Q ( q ) = q c . Remarkably, compared with competing heuristics, CI-TM algorithm achieves a largeractive population for a given number of seeds. It not only brings about an earlier continuous transition, but also activatesthe total population with a smaller seed set. Among all the strategies, random selection represents the average behavior ofcascade initiated by randomly chosen seeds, with a critical value q Random c = . the adaptive version KsA gives a better result similar to HDA.For different threshold t , the critical values q c for CI-TM and other heuristic methods are shown in Table 1. We also providethe first-order critical value q c for CI-TM and HDA on ER networks with different average degree h k i in the inset of Fig. 2a.With the growth of h k i , q c increases and so does the difference between CI-TM and HDA. In some cases, q c can be furtherimproved by a simple modification on CI-TM (See Methods).We then examine CI-TM’s performance on SF networks with power-law degree distributions P ( k ) ∼ k − γ in Fig. 2b. Wegenerate SF networks of size N = × and power-law exponent γ = It can be seen b Figure 2.
Performance of CI-TM algorithm on random networks. (a), Size of active giant component Q ( q ) versus thefraction of seeds q for ER networks with size N = × and mean degree h k i =
6. Different methods are distinguished bydistinct markers and colors. Threshold is set as fractional t = .
5. The CI-TM algorithm is run without limitation on L . MS isimplemented by using T =
40 and a reinforcement parameter r = × − . For CI-TM, the identified critical value is q CI-TM c = . ( ) while for Random selection q Random c = . ( ) . Inset presents the critical values q c identified by HDAand CI-TM for different mean degree h k i . (b), Comparison for scale-free networks with size N = × , power-lawexponent γ =
3, minimal degree k min = k max = t = .
5. Inset shows the critical values q c for different exponents γ . All the results are averaged over 50 realizations. t CI-TM HDA HD KsA PR Random0.3 0.0197(2) 0.0258(3) 0.0266(3) 0.0260(3) 0.0264(3) 0.0566(8)0.4 0.0562(2) 0.0630(4) 0.0679(5) 0.0630(3) 0.0682(4) 0.1322(8)0.5 0.1042(2) 0.1083(3) 0.1222(5) 0.1086(4) 0.1210(6) 0.220(1)0.6 0.2049(3) 0.2099(4) 0.279(1) 0.2099(5) 0.282(1) 0.435(2)
Table 1.
Critical points for different threshold values.
The critical values q c found by CI-TM and other heuristicsincluding HDA, HD, KsA, PR, and Random strategies for ER networks ( N = , h k i =
6) with different threshold values t .Results are averaged over 50 realizations, and the numbers in parentheses are standard deviations of the last digit.that the critical value of first-order transition becomes rather small for SF networks, due to the existence of highly connectedhubs. Still, CI-TM algorithm outperforms other heuristic approaches by producing a larger active component Q ( q ) for a givenfraction of seed q . Since most nodes have a quite small number of connections in SF networks, the cascade triggered byrandomly selected seeds is limited to a local scale, even with a relatively large number of seeds, as shown by the grey line atthe bottom of Fig. 2b. This implies, compared with homogeneous networks, the deviations of the optimized trajectories fromtypical ones are much more extreme in heterogeneous networks. Moreover, as SF networks become more heterogeneous witha smaller power-law exponent γ of the degree distribution, the minimal number of seeds required for global cascade decreasesaccordingly, as shown in the inset of Fig. 2b.In applications, we are frequently faced with large-scale networks which exhibit more complicated topological character-istics than random graphs. Thus, it is more necessary and challenging to find a feasible strategy to efficiently approximate theoptimal spreaders for those networks. Next, we explore CI-TM algorithm’s performance for real networks. We examine tworepresentative datasets: Youtube friendship network ( N = , , , M = , , , c = . , l = . and Internetautonomous system network ( N = , , , M = , , , c = . , l = . Here N is the network size, M is thenumber of undirected links, c is the average clustering coefficient, and l is the average shortest path length. Youtube networkrepresents the undirected friend relations between users in the famous video sharing website Youtube. The Internet networkrecords the communications between routers in different autonomous systems. The links between routers are constructed fromthe Border Gateway Protocol logs in an interval of 785 days. This provides an example of infrastructure network on whichmalicious attack and failure propagation may occur. Both networks are treated as undirected in the analysis.Figure 3a displays Q ( q ) for different numbers of seeds | S | for Youtube network. CI-TM is able to trigger the globalcascade with a smaller group of seeds, whose size is quite small compared to the entire network due to the heavy-tailed degreedistribution. We also discover that, in the setting of first-order transitions, some nodes with moderate numbers of connectionsplay a crucial role in the collective influence of LTM. As shown in the inset figure, we present the percentage of influencers b Figure 3.
Performance of CI-TM algorithm on large-scale real-world networks. (a), The relationship between the sizeof active giant component Q ( q ) and the number of seeds | S | for Youtube friendship network, calculated by different methods.Inset displays the percentage of influencers predicted by CI-TM that HDA and HD have identified.The vertical dash lineindicates the critical point of CI-TM. (b), Same analysis for Internet network.predicted by CI-TM that HDA and HD have identified, with the vertical dash line indicating CI-TM’s q c . At q c , HDA andHD locate nearly 80% overlapping seeds with CI-TM algorithm, most of which are tagged as hubs. However, due to thecollective nature of LTM, seeding the set of privileged nodes in the non-interacting view does not guarantee the maximizationof collective influence. The other proportion of spreaders with lower degree, although may be inefficient as single spreaders,are responsible for bridging the collective influence of hubs. With the help of both hubs and bridging low degree nodes, CI-TMcan expand the collective influence with a smaller number of seeds. The Internet network also exhibits a similar phenomenonin Fig. 3b. In this case, HDA and HD can only find 80% influencers at the first-order transition of CI-TM algorithm, missinga substantial amount of nodes with lower degree but indispensable in integrating the collective influence of high-degree seeds.Although the performance of K-core can be improved by adaptive calculation in Fig. 2, for large-scale real networks, wedo not display the result of KsA due to its O ( N ) computational complexity and only show the curve of K-core. One causefor the unsatisfactory result of K-core is that it is not designed as a multiple spreaders finder since high K-core nodes tend toform densely connected clusters in the same shell, which prevents the expanding of information cascade.In Methods, we further compare CI-TM algorithm with other methods, including Betweenness Centrality (BC), Close-ness Centrality (CC) and Greedy Algorithm (GA). Results from ER and SF networks suggest that CI-TM algorithm alsooutperforms computationally expensive BC, CC and GA.
Analysis of subcritical paths
With the CI-TM algorithm, we present an analysis of subcritical paths in cascading process. In Fig. 4a, we first display theevolution of the number of subcritical paths during the sequential activation process based on CI-TM ranking. We run LTMmodel for t = . N = and average degree h k i =
6. Nodes are activated sequentially accordingto their ranks in CI-TM algorithm. At the time of each activation, the number of subcritical paths attached to the node iscalculated. After the activation, nodes on the subcritical paths are activated automatically, as we did in the CI-TM algorithm.In Fig. 4a, the evolution of subcritical path number for CI-TM ( L = , ,
20) is displayed. For all L values, the number ofsubcritical paths peaks at the critical point, where the first-order transition occurs. In addition, as L increases, the peak timeof CI-TM is slightly shifted forward, while the peak value increases dramatically. As large numbers of subcritical paths implya heavy computational burden, the majority of computation is concentrated around the critical point. Therefore, if we wantto optimize the influence before the discontinuous transition, which is common in real-world applications, CI-TM algorithmbecomes much more efficient since it avoids counting extremely long subcritical paths near the critical point.We also examine the distribution of nodes’ activation time in a global cascading. In Fig. 4b, we report the distributionof activation time at the critical point for CI-TM ( L = , ,
20) and random selection of seeds. All the curves first decreaseand then develop a second peak. Compared with the distribution of random selection, CI-TM has a much larger number ofnodes getting activated at the second peak. More importantly, the increase of L in CI-TM algorithm will postpone the arrivalof the second peak, which is similar to the previous finding on regular networks. In CI-TM algorithm with a larger L , longersubcritical paths are allowed during the calculation, as shown in the inset of Fig. 4b. Considering the size of optimal seeds inFig. 4a and the distribution of activation time in Fig. 4b, a smaller L in CI-TM algorithm can expedite the global cascading, b Figure 4.
Analysis of subcritical paths. (a), Comparison of the number of subcritical paths attached to each node when itis activated sequentially according to CI-TM ranking. Results for CI-TM with L = L =
10 and L =
20 are displayed. LTMfor t = . N = and average degree h k i =
6. (b), Distribution of activation time in theglobal cascading for CI-TM ( L = , ,
20) and Random strategy at q c . Inset shows the length of subcritical paths for eachnode when activated in CI-TM ranking. The curve for Random seed selection is averaged over 1,000 independent LTMrealizations.but at the expense of a few more seeds. Discussion
Identification of superspreaders in LTM has great practical implications in a wide range of dynamical processes. However,the complicated interactions among multiple spreaders prevent us from accurately locating the pivotal influencers in LTM. Inthis work, we propose a framework to analyze the collective influence of individuals in general LTM. By iteratively solvingthe linearized message passing equations, we decompose k ν → k into separate components, each of which corresponds to thecontribution made by a single seed. Particularly, we find that the contribution of a seed is largely determined by its interplaywith other nodes through subcritical paths. In order to maximize the active population, we develop a scalable CI-TM algorithmthat is feasible for large-scale networks. Results show that the proposed CI-TM algorithm outperforms other ranking strategiesin synthetic random graphs and real-world networks. Our CI-TM algorithm can be employed in relevant applications such asviral marketing and information spreading in big-data analysis. Methods
Linearization of G i → j The conventional method to linearize the nonlinear function G i → j ( ν → ) is Taylor expansion around the fixed point : G i → j ( ν → ) ≈ G i → j ( ) + G ′ i → j ( ) ν → . However, for our specific function G i → j , the gradient G ′ i → j ( ) is constantly according to Eq. (5). Toavoid the degeneracy, other linear approximation method should be applied.For a differentiable function f : R n → R and x , y ∈ R n , the mean value theorem guarantees that there exists a real number c ∈ ( , ) such that f ( y ) − f ( x ) = ∇ f (( − c ) x + c y ) · ( y − x ) . Here ∇ is the gradient and · denotes the dot product. Set f = G i → j , x = , and y = ν → , we have G i → j ( ν → ) = G i → j ( ) + G ′ i → j ( c ν → ) ν → . Notice that, if we set c =
0, the above equation becomesthe classical linear Taylor expansion: G i → j ( ν → ) ≈ G i → j ( ) + G ′ i → j ( ) ν → , where the approximation accuracy is O ( | ν → | ) ( | · | is the norm of vectors). In a network with N nodes and M undirected links, we define the norm as | ν → | ≡ ∑ i j ν i → j / M (2 M is the number of directed links) so that | ν → | is bounded below 1 for network size N → ∞ .To deal with the degeneracy of G ′ i → j ( ) , we approximate G i → j ( ν → ) by setting c = ν → : G i → j ( ν → ) ≈ G i → j ( ) + G ′ i → j ( ν → ) ν → . The approximation error can be calculated by e = | G i → j ( ν → ) − G i → j ( ) − G ′ i → j ( ν → ) ν → | ≤ | G ′ i → j ( c ν → ) − G ′ i → j ( ν → ) || ν → | . Recall that G ′ i → j = ( · · · , ∂ G i → j ∂ν k → ℓ , · · · ) . In a finite-size network, for a small ν → with elements | ν i → j | ≤
1, thegradient of each element ∂ G i → j ∂ν k → ℓ is bounded according to Eq. (5). For all 2 M elements, there exists a uniform upper bound forall the gradients ∇ ∂ G i → j ∂ν k → ℓ . Applying the mean value theorem to the differentiable function ∂ G i → j ∂ν k → ℓ , there should be a constant L such that | ∂ G i → j ∂ν k → ℓ ( c ν → ) − ∂ G i → j ∂ν k → ℓ ( ν → ) | ≤ L | ν → | for all the elements of G ′ i → j . Summing up all the elements in G ′ i → j , we conclude -5 -4 -3 -2 -1 | ν → | in linear approximation -5 -4 -3 -2 -1 | ν → | i n r ea l L T M LTM starting from one single seedy=x
Figure 5.
Linear approximation of | ν → | for LTM initiated by one seed. In a scale-free network ( N = , M = γ = t = . | ν → | value and its linear approximation is presented.that | G ′ i → j ( c ν → ) − G ′ i → j ( ν → ) | ≤ | ( · · · , L , · · · ) || ν → | . Therefore, the approximation error e ≤ | ( · · · , L , · · · ) || ν → | . This provesthat the accuracy of the linear approximation is O ( | ν → | ) , which is same as the linear Taylor expansion.In the CI-TM algorithm, we only select one seed at each time step. Here we directly examine the accuracy of the linearapproximation when LTM is initiated by one single seed. Specifically, we run LTM dynamics with threshold t = . N = , M = γ = | ν → | value and its approximation in Fig. 5. As expected, the approximation is generally lower than the real value since loops areneglected. The correlation between real values and approximations is 0.9118, and a higher correlation 0.9437 is obtained inthe logarithmic scale. Therefore, the linear approximation in each step of CI-TM algorithm is accurate. As more seeds areconsidered, the approximation accuracy will decrease gradually. The decreasing rate should be related to the number of shortloops existing in the network. How the density of short loops affects the approximation accuracy will be further explored infuture works. More comparisons with competing methods
A growing number of methods aimed to rank nodes’ influence in networks have been proposed in previous studies. Here weintroduce some of the most widely used competing methods and perform a thorough comparison with CI-TM algorithm.
High degree (HD)
Degree, defined as the number of connections attached to a node, is the most widely-used measure ofinfluence. In HD method, we rank nodes according to their degrees in a descending order, and sequentially select them asinformation sources. For HD method, the selected hubs intend to link with each other due to the assortative mixing property,making their influence areas overlap significantly. In this case, the selected seeds could rarely be optimal. High degreeadaptive (HDA) is the adaptive version of HD method. After each removal, the degree of each node is recalculated. Suchadaptive procedure can usually mitigate the overlapping and improve the performance of HD.
K-core (Ks)
Through a k-shell decomposition process, K-core method assigns each node a k S value to distinguish whetherit locates in the core region or peripheral area. In k-shell decomposition, nodes are iteratively removed from the networkaccording to their current degrees. During the removal, all the nodes are classified into different k-shells. The K-core methodselects nodes within high k-shells as the spreaders. In practice, single influential spreaders can be identified effectively byK-core ranking, which has been confirmed by both simulations and real-world data.
However, K-core ranking has thedisadvantage of severe overlap of seeds’ influence areas, and therefore performs poorly for multiple node selection. Thislimitation can be alleviated with an adaptive scheme where we recompute the K-core after each removal. Since there existsmany nodes in the same k-shell, we select the node with the largest degree to further distinguish nodes within the highestk-shell. Such K-core adaptive (KsA) method can effectively enhance the performance of K-core.
PageRank (PR)
PageRank is a popular ranking algorithm of webpages which was developed and used by search engineGoogle. Over the years, PageRank has been adopted in many practical ranking problems. Generally speaking, PageRankmeasures a webpage’s stationary visiting probability by a random walker following the hyperlinks in the network. As aspecial case of eigenvector centralities, PageRank evaluates the score of a node by taking into account its neighbors’ scores. N R un t i m e , s e c a b Figure 6.
Comparison of competing methods.
Performance of different methods for (a) ER network ( N = , h k i = t = .
5) and (b) scale-free network ( N = , γ = t = . T =
40 in MS and implement reinforcement with parameter r = − . The insetdisplays the run time of CI-TM ( L =
3) on ER networks ( h k i = t = . Greedy algorithm (GA)
In GA, starting from an empty set of seeds, nodes with the maximal marginal gain are sequentiallyadded to the seed set. Kempe et al. have proven that for a class of LTM with the attribute of submodularity, GA has aperformance guarantee of 1 − / e ≈ This resultrelies on the submodular property defined by a diminishing returns effect: the marginal gain from adding a node to the seedset S decreases with the size of S . It has been proven that several classes of LTM have submodular property, such as a randomchoice of thresholds. However, for LTM with a fixed threshold, it is not generally submodular. As a consequence, GA is notguaranteed to provide a such approximation of the optimal spreaders for general LTM. Furthermore, the greedy search of GArequires massive simulations, which makes GA unscalable and thus limits its application in large-scale social networks. Betweenness centrality (BC)
BC quantifies the importance of node i in terms of the number of shortest paths crossthrough it. Therefore, nodes with large BC usually occupy the pivotal positions in the shortest pathways connecting largenumbers of nodes. In BC method, we select nodes with high BC scores as seeds. Although BC has been widely applied insocial network analysis, its relatively high computational complexity makes BC prohibitive for large-scale networks. A typicalBC algorithm takes O ( MN ) to calculate for a network with N nodes and M links, which is still not applicable to modernsocial networks with millions of users. Closeness centrality (CC)
Closeness centrality quantifies how close a node to other nodes in the network. Formally, CCis defined as the reciprocal of the average shortest distance of a node to others in a network. Thus, nodes with high CC valuestend to locate at the center of network clusters or communities. In CC method, we pick the seeds according to nodes’ CCscores. Same as BC, CC also requires the heavy task of calculating all possible shortest paths. Thus the high computationalcost of CC makes it infeasible for large-scale networks.
Max-Sum (MS)
F. Altarelli et al. developed a Max-Sum algorithm aimed to find the initial configurations that maximizedthe final number of active nodes in threshold models. Precisely, the trajectory of nodes’ states is parametrized by a configura-tion t = { t i , ≤ i ≤ N } where t i ∈ T = { , , · · · , T , ∞ } is the activation time of node i ( t i = ∞ if inactive). By mapping theoptimization onto a constraint satisfaction problem, an energy-minimizing algorithm based on the cavity method of statisticalphysics is proposed. In the algorithm, a convolution process is employed to compute the Max-Sum updates. The technicaldetails of the derivation and implementation of the MS algorithm can be found in Ref. In some cases MS algorithm doesnot converge, then a reinforcement strategy is implemented. By imposing an external field slowly increasing over time witha growth rate r , the system is forced to converge to a higher energy, which increases with r . In addition, it requires O ( r − ) iterations to reach convergence. For a node of degree k and threshold m i , each update takes O ( T k ( k − ) m i ) operations. Pre-computing the convolution can further save a factor of k −
1. Considering the updates of all N nodes for O ( r − ) iterations, theoverall complexity of MS is O ( T ∑ i k i m i / r ) . Therefore, the time complexity of MS depends on both the degree distributionof networks and the choice of threshold. igure 7. Performance of modified CI-TM on ER networks.
For ER networks ( N = × , h k i = t = . Q ( q ) for CI-TMm is lower at first, it exceeds other methods near the critical point and achieves the earliestfirst-order transition.In Fig. 6, we provide the thorough comparisons of different methods on ER and SF networks ( N = ), includingcomputationally expensive methods GA, BC and CC. We set T =
40 and a reinforcement parameter r = × − in MSalgorithm. For both homogeneous and heterogeneous networks, CI-TM shows a larger active population for a given fractionof seed q compared with other heuristic ranking strategies. We display the scaling of run time of CI-TM algorithm for ERnetworks with h k i = t = . N in the inset of Fig. 6a. CI-TM algorithm with L = N = O ( ) within run time of O ( ) seconds, which can be applied to the modern large-scaleonline social networks. Modified CI-TM algorithm
The CI-TM algorithm is essentially a greedy approach based on CI-TM values. The success of CI-TM algorithm dependson whether the currently selected seed has potentials to create more subcritical nodes that are helpful for the early formationof giant subcritical cluster. In LTM, there exists a special case of subcritical nodes with threshold m =
1, which is definedas vulnerable vertices in previous literature. Different from general subcritical nodes, vulnerable vertices are naturallysubcritical since they have threshold m = k becomes vulnerable when its m − k − m + h k i =
4. The critical value q c for CI-TMmis advanced compared to CI-TM algorithm. However, before the first-order transition, CI-TMm cannot optimize the spreadingand has substantially lower Q ( q ) than CI-TM. We should note that, CI-TMm presents a lower q c only in the situation offragmented vulnerable clusters. For ER networks with higher average degrees (e.g., h k i =
6) where relatively large vulnerableclusters emerge, CI-TM still predicts earlier first-order transition. eferences Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependentnetworks.
Nature Watts, D. J. & Dodds, P. S. Influentials, networks, and public opinion formation.
J. Consum. Res. Rogers, E. M.
Diffusion of Innovation (Free Press, New York, 1995). Reis, S. D. et al.
Avoiding catastrophic failure in correlated networks of networks.
Nature Phys. Kleinberg, J.
Algorithmic Game Theory (Cascading Behavior in Networks: Algorithmic and Economic Issues) (CambridgeUniversity Press, Cambridge, 2007) chapter 24, 613-632. Domingos, P & Richardson, M. Mining the network value of customers. In
Proc. 7th ACM SIGKDD Int. Conf. on Knowl-edge Discovery and Data Mining , 57-66 (ACM, 2001). Valente, T. W. & Davis, R. L. Accelerating the diffusion of innovations using opinion leaders.
Ann. Am. Acad. Polit. Soc.Sci. Galeotti, A. & Goyal, S. Influencing the influencers: a theory of strategic diffusion.
The RAND J. Econ. Kempe, D., Kleinberg, J. & Tardos, ´E. Maximizing the spread of influence through a social network. In
Proc. 9th ACMSIGKDD Int. Conf. on Knowledge Discovery and Data Mining , 137-146 (ACM, 2003).
Leskovec, J., Krause, A., Guestrin, C., Faloutsos, C., VanBriesen, J. & Glance, N. Cost-effective outbreak detection innetworks. In
Proc. 13th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining , 420-429 (ACM, 2007).
Chen, W., Wang, Y. & Yang, S. Efficient influence maximization in social networks. In
Proc. 15th ACM SIGKDD Int.Conf. on Knowledge Discovery and Data Mining , 199-208 (ACM, 2009).
Kitsak, M. et al.
Identification of influential spreaders in complex networks.
Nature Phys. Pei, S. & Makse, H. A. Spreading dynamics in complex networks.
J. Stat. Mech.
P12002 (2013).
Pei, S., Muchnik, L., Andrade Jr, J. S., Zheng, Z. & Makse, H. A. Searching for superspreaders of information in real-world social media.
Sci. Rep. Morone, F. & Makse, H. A. Influence maximization in complex networks through optimal percolation.
Nature
Morone, F., Min, B., Bo, L., Mari, R. & Makse, H. A. Collective Influence Algorithm to find influencers via optimalpercolation in massively large social media.
Sci. Rep. Altarelli, F., Braunstein, A., Dall’Asta, L. & Zecchina, R. Large deviations of cascade processes on graphs.
Phys. Rev. E
Altarelli, F., Braunstein, A., Dall’Asta, L. & Zecchina, R. Optimizing spread dynamics on graphs by message passing.
J.Stat. Mech. P09011 (2013).
Guggiola, A. & Semerjian, G. Minimal contagious sets in random regular graphs.
J. Stat. Phys.
Mugisha, S. & Zhou, H. J. Identifying optimal targets of network attack by belief propagation.
Phys. Rev. E
Braunstein, A., Dall’Asta, L., Semerjian, G. & Zdeborov´a, L. Network dismantling.
Proc. Natl. Acad. Sci. USA
Teng, X., Pei, S., Morone, F. & Makse, H. A. Collective influence of multiple spreaders evaluated by tracing real infor-mation flow in large-scale social networks.
Sci. Rep. Pei, S., Muchnik, L., Tang, S., Zheng, Z. & Makse, H. A. Exploring the complex pattern of information spreading inonline blog communities.
PLoS ONE e0126894 (2015).
Radicchi, F. & Castellano, C. Leveraging percolation theory to single out influential spreaders in networks.
Phys. Rev. E
Hu, Y., Ji, S., Feng, L. & Jin, Y. Quantify and maximise global viral influence through local network information. arXivpreprint arXiv:1509.03484 (2015).
Lawyer, G. Understanding the influence of all nodes in a network.
Sci. Rep. Quax, R., Apolloni, A. & Sloot, P. M. The diminishing role of hubs in dynamical processes on complex networks.
J. R.Soc. Interface Tang, S., Teng, X., Pei, S., Yan, S. & Zheng, Z. Identification of highly susceptible individuals in complex networks.
Physica A
Albert, R., Jeong, H. & Barab´asi, A. L. Error and attack tolerance of complex networks.
Nature
Freeman, L. C. Centrality in social networks conceptual clarification.
Soc. Netw. Brin, S. & Page, L. Reprint of: The anatomy of a large-scale hypertextual web search engine.
Computer networks
Seidman, S. B. Network structure and minimum degree.
Soc. Netw. Granovetter, M. Threshold models of collective behavior.
Am. J. Sociol.
Schelling, T. C.
Micromotives and macrobehavior (Norton, New York, 1978).
Valente, T. W.
Network Models of the Diffusion of Innovations (Hampton Press, Cresskill, NJ, 1995).
Watts, D. J. A simple model of global cascades on random networks.
Proc. Natl. Acad. Sci. USA
Baxter, G. J., Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Bootstrap percolation on complex networks.
Phys.Rev. E
Goltsev, A. V., Dorogovtsev, S. N. & Mendes, J. F. F. k-core (bootstrap) percolation on complex networks: Criticalphenomena and nonlocal effects.
Phys. Rev. E
Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. k-core architecture and k-core percolation on complex networks.
Physica D
Schwarz, J. M., Liu, A. J. & Chayes, L. Q. The onset of jamming as the sudden emergence of an infinite k-core cluster.
Europhys. Lett.
560 (2006).
Melnik, S., Hackett, A., Porter, M. A., Mucha, P. J. & Gleeson, J. P. The unreasonable effectiveness of tree-based theoryfor networks with clustering. Phys. Rev. E,
Hashimoto, K. Zeta functions of finite graphs and representations of p-adic groups.
Adv. Stud. Pure Math.
211 (1989).
Martin, T., Zhang, X. & Newman, M.E.J. Localization and centrality in networks.
Phys. Rev. E
Molloy, M. & Reed, B. A critical point for random graphs with a given degree sequence.
Random Structures & Algorithms Mislove, A. Online Social Networks: Measurement, Analysis, and Applications to Distributed Information Systems.
PhDthesis, Rice University (2009).
Leskovec, J., Kleinberg, J. & Faloutsos, C. Graphs over time: densification laws, shrinking diameters and possible expla-nations. In
Proc. 11th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining . 177-187 (ACM, 2005).
Bavelas, A. Communication patterns in tasks oriented groups.
J. Acoust. Soc. Am.
Brandes, U. A faster algorithm for betweenness centrality.
J. Math. Sociol. 25,
Acknowledgements
This work was supported by NIH-NIBIB 1R01EB022720, NIH-NCI U54CA137788/ U54CA132378, NSF-PoLS PHY-1305476,NSF-IIS 1515022, and ARL Cooperative Agreement Number W911NF-09-2-0053, the ARL Network Science CTA (toH.A.M.), as well as US NIH grant GM110748 and the Defense Threat Reduction Agency contract HDTRA1-15-C-0018(to J.S.).
Author contributions statement
S.P., X.T, J.S., F.M. and H.A.M. designed research, performed study, analyzed data, and wrote the paper. All authors reviewedthe manuscript.
Additional information
Competing financial interests