Fractal scale-free networks resistant to disease spread
aa r X i v : . [ q - b i o . P E ] A p r Fractal scale-free networks resistant to diseasespread
Zhongzhi Zhang a , b , ∗ Shuigeng Zhou a , b Tao Zou a , b Guisheng Chen c a Department of Computer Science and Engineering, Fudan University, Shanghai200433, China b Shanghai Key Lab of Intelligent Information Processing, Fudan University,Shanghai 200433, China c China Institute of Electronic System Engineering, Beijing 100840, China
Abstract
In contrast to the conventional wisdom that scale-free networks are prone to epi-demic propagation, in the paper we present that disease spreading is inhibited infractal scale-free networks. We first propose a novel network model and show thatit simultaneously has the following rich topological properties: scale-free degree dis-tribution, tunable clustering coefficient, “large-world” behavior, and fractal scaling.Existing network models do not display these characteristics. Then, we investigatethe susceptible-infected-removed (SIR) model of the propagation of diseases in ourfractal scale-free networks by mapping it to bond percolation process. We find anexistence of nonzero tunable epidemic thresholds by making use of the renormal-ization group technique, which implies that power-law degree distribution does notsuffice to characterize the epidemic dynamics on top of scale-free networks. We arguethat the epidemic dynamics are determined by the topological properties, especiallythe fractality and its accompanying “large-world” behavior.
Key words:
Complex networks, Disease spread, Fractal networks, Scale-freenetworks
PACS: ∗ Corresponding author.
Email addresses: [email protected] (Zhongzhi Zhang), [email protected] (Shuigeng Zhou), [email protected] (Tao Zou).
Preprint submitted to Elsevier 15 November 2018
Introduction
In recent years, there has been much interest in the study of the structure anddynamics of complex networks [1,2,3,4]. One aspect that has received consid-erable attention is the epidemic spreading taking place on top of networks [5],which is relevant to computer virus diffusion, information and rumor spread-ing, and so on. In the study of epidemic spreading, the notion of thresholds isa crucial problem since it finds an intermediate practical application in diseaseeradication and vaccination programs [6,7]. In homogeneous networks, there isan existence of nonzero infection threshold, if the spreading rate is above thethreshold, the infection spreads and becomes endemic, otherwise the infectiondies outs quickly. However, recent studies demonstrate that the threshold isabsent in heterogeneous scale-free networks [8,9,10,11]. Thus, it is importantto identify what characteristics of network structure determine the presenceor not of epidemic thresholds.To date the influences of most structural properties on disease dynamics havebeen studied, which include degree distribution [9,10,11], clustering coeffi-cient [12], and degree correlations [13]. However, these features do not sufficeto characterize the architecture of a network [14]. Very recently, by introduc-ing and applying box-covering (renormalization) technique, Song, Havlin andMakse found the presence of fractal scaling in a variety of real networks [15,16].Examples of fractal networks include the WWW, actor collaboration network,metabolic network, and yeast protein interaction network [20]. The fractaltopology is often characterized through two quantities: fractal dimension d B and degree exponent of the boxes d k , both of which can be calculated by thebox-counting algorithm [17,18]. The scaling of the minimum possible numberof boxes N B of linear size ℓ B required to cover the network defines the fractaldimension d B , namely N B ∼ ℓ − d B B . Similarly, the degree exponent of the boxes d k can be found via k B ( ℓ B ) /k hub ∼ ℓ − d k B , where k B ( ℓ B ) is the degree of a boxin the renormalized network, and k hub the degree of the most-connected nodeinside the corresponding box. Interestingly, for fractal scale-free networks withdegree distribution P ( k ) ∼ k − γ , the two exponents, d B and d k , are related toeach other through the following universal relation: γ = 1 + d B /d k [15].Fractality is now acknowledged as a fundamental property of a complex net-work [14]. It relates to a lot of aspects of network structure and dynamicsrunning on the network. For example, in fractal networks the correlation be-tween degree and betweenness centrality of nodes is much weaker than thatin non-fractal networks [19]. In addition, several studies uncovered that frac-tal networks are not assortative [16,20,21]. The peculiar structural nature offractal networks make them exhibit distinct dynamics. It is known that frac-tal scale-free networks are more robust than non-fractal ones against mali-cious attacks on hub nodes [16,21]. On the other hand, fractal networks and2heir non-fractal counterparts also display disparate phenomena of other dy-namics, such as cooperation [23,22], synchronization [21], transport [24], andfirst-passage time [26,25]. Despite of the ubiquity of fractal feature and itsimportant impacts on dynamical processes, the dynamics of disease outbreaksin fractal networks has been far less investigated.In this current paper, we focus on the effects of fractality on the dynamicsof disease in fractal scale-free networks. Firstly, we propose an algorithm tocreate a class of fractal scale-free graphs by introducing a control parame-ter q . Secondly, we give in detail a scrutiny of the network architecture. Theanalysis results show that this class of networks have unique topologies. Theyare simultaneously scale-free, fractal, ‘large-world’, and have tunable cluster-ing coefficient. Thirdly, we study a paradigmatic epidemiological model [6,7],namely the susceptible-infected-removed (SIR) model on the proposed fractalgraphs. By mapping the SIR model to a bond percolation problem and usingthe renormalization-group theory, we find the existence of non-zero epidemicthresholds as a function of q . We also provide an explanation for our findings. This section is devoted to the construction and the relevant structural proper-ties of the networks under consideration, such as degree distribution, clusteringcoefficient, average path length (APL), and fractality.
Fig. 1. (Color online) Iterative construction method of the fractal networks. Eachiterative link is replaced by a connected cluster on the right-hand side of the arrow.The red link is a noniterated link.
The proposed fractal networks have two categories of bonds (links or edges):iterative bonds and noniterated bonds which are depicted as solid and dashedlines, respectively. The networks are constructed in an iterative way as shown3n Fig. 1. Let F t ( t ≥
0) denote the networks after t iterations. Then thenetworks are built in the following way: For t = 0, F is two nodes (vertices)connected by an iterative edge. For t ≥ F t is obtained from F t − . We replaceeach existing iterative bond in F t − either by a connected cluster of links onthe top middle of Fig. 1 with probability q , or by the connected cluster on thebottom right with complementary probability 1 − q . The growing process isrepeated t times, with the fractal graphs obtained in the limit t → ∞ . Figure 2shows a network after three-generation growth for a specific case of q = 0 . Fig. 2. (Color online) Sketch of a network after three iterations for the particularcase of q = 0 . Next we compute the numbers of total nodes and edges in F t . Let L v ( t ), L i ( t ) and L n ( t ) be the numbers of nodes, iterative edges, and noniteratededges created at step t , respectively. Note that each of the existing iterativeedges yields two nodes and four new iterative edges; at the same time thisoriginal iterative edge itself is deleted, which means that L i ( t ) is also the totalnumber of iterative edges at time t . Then we have L v ( t ) = 2 L i ( t −
1) and L i ( t ) = 4 L i ( t −
1) for all t >
0. Considering the initial condition L i (0) = 1,one can obtain L i ( t ) = 4 t and L v ( t ) = 2 × t − . Thus the number of totalnodes N t present at step t is N t = t X t i =0 L v ( t i ) = 2 × t + 43 . (1)On the other hand, at each construction step, each of the existing iterativeedges may yield one noniterated link with probability 1 − q , so the expected4alue of L n ( t ) is (1 − q ) L i ( t −
1) for t ≥
1, i.e., L n ( t ) = (1 − q ) 4 t − . Therefore,the total number of edges E t present at step t is E t = L i ( t ) + t X t i =1 L n ( t i ) = (4 − q ) 4 t − (1 − q )3 . (2)The average node degree after t iterations is h k i t = E t N t , which approaches4 − q in the infinite t limit. When a new node u enters the system at step t u ( t u ≥ − q one noniterated edge is createdand linked to node u . Let L i ( u, t ) be the number of iterative links emanatedfrom node u at step t , then L i ( u, t u ) = 2. Notice that at any subsequent stepeach iterative edge of u is broken and generates two new iterative edges linkedto u . Thus L i ( u, t ) = 2 L i ( u, t −
1) = 2 t − t u +1 .We define k u ( t ) as the degree of node u at time t , then we have k u ( t ) = 2 t − t u +1 , with probability q,k u ( t ) = 2 t − t u +1 + 1 , with probability 1 − q, (3)where the last term 1 in the second formula represents the noniterated linkconnected to node u . For the initial two nodes created at step 0, neither ofthem has a noniterated link, both nodes have a degree of 2 t .Equation (3) indicates that the degree spectrum of the networks is not con-tinuous. It follows that the cumulative degree distribution [27,28] is given by P cum ( k ) = N t,k N t , where N t,k is the number of nodes whose degree is not less than k . When t is large enough, we find P cum ( k ) ≈ k − . So the degree distributionfollows a power-law form with the exponent γ = 3, which is independent of q .The same degree exponent has been obtained in the famous Barab´asi-Albert(BA) model [8]. The clustering coefficient [29] of a node u with degree k u is the probabilitythat a pair of neighbors of u are themselves connected, which is given by C ( k u ) = 2 b u / [ k u ( k u − b u is the number of existing connections5etween the k u neighbors of u . For our networks, the clustering coefficient C ( k ) for a single node with degree k can be evaluated exactly. Note thatexcept for those nodes born at step t , all existing links among the neighbors ofa given node are noniterated ones, whose number is ease to calculate. For theinitial two nodes, the expected existing noniterated links among the neighborsis (1 − q )2 t − . For each of those nodes created at step φ (0 < φ < t ), thereare average (1 − q )2 t − φ noniterated links among its neighbors. Finally, forthe nodes generated at step t , some of them have a degree of 3, the numberof links between the neighbors of each is 2; the others have a degree of 2,their clustering coefficient is zero. Thus, there is a one-to-one correspondencebetween the clustering coefficient C ( k ) of the node and its degree k : C ( k ) = (1 − q ) / ( k −
1) for k = 2 m (2 ≤ m ≤ t ) , (1 − q ) /k for k = 2 m + 1(2 ≤ m ≤ t ) , k = 2 , /k for k = 3 . (4)Using Eq. (4), we can obtain the clustering ¯ C t of whole the network at step Fig. 3. (Color online) The average clustering coefficient C of the whole network asa function of q . It can be tunable systematically by changing q . The squares arethe simulation results, and the line represents the analytical expression given byEq. (5). t , which is defined as the average clustering coefficient of all individual nodes.Then we have¯ C t = 1 − qN t L v (0)2 t − t − X r =1 qL v ( r ) K r − t − X r =1 (1 − q ) L v ( r ) K r + 1 + 2 L v ( t ) K t + 1 , (5)6here K r = 2 t − r +1 and K t = 2. In the infinite network order limit ( N t → ∞ ),¯ C t converges to a nonzero value C as a function of parameter q . In the twoextreme cases of q = 0 and q = 1, C are 0 and 0.5435 [30], respectively.When q increases from 0 to 1, C grows from 0 to 0.5435, see Fig. 3. Thus,the parameter q in our model introduces the clustering effect by allowing theformation of triangles. Furthermore, the relation between C and q is almostlinear, as depicted in Fig. 3. Shortest paths play an important role both in the transport and communica-tion within a network and in the characterization of the internal structure ofthe network [31,32,33,34]. Let d ij represent the shortest path length from node i to j , then the average path length (APL) d t of F t is defined as the mean of d ij over all couples of nodes in the network, and the maximum value D t of d ij is called the diameter of the network.For general q , it is difficult to derive a closed formula for the APL d t ofnetwork F t . But for two limiting cases of q = 0 and q = 1, both the networksare deterministic ones, we can obtain the analytic solutions for APL, denotedas d t and d t , respectively. In the large t limit, d t ≈ t [30] and d t ≈ t [23].On the other hand, for large t limit, N t ∼ t , so both d t and d t grow as asquare power of the number of network nodes.By construction, it is obvious that d t and d t are the lower and upper bound ofAPL for network F t , respectively. As a matter of fact, if we denote the specific q = 1 case of the network as F t (it has no noniterated edges, and thus themaximum APL), then one can obtain F t from F t by adding noniterated edgesin a certain way. The less the parameter q , the more the added noniteratededges. In the case of q = 0, the network is the densest one with the minimumAPL. Therefore, the APL is a decreasing function of q . As q drops from 1 to0, d t increases from t at q = 1 to t at q = 0. From above arguments,we can conclude that for the full range of q between 0 and 1, the APL d t hasan exponential growth with network size, which indicates that the considerednetworks F t are not small worlds. To determine the fractal dimension, we follow the mathematical frameworkintroduced in Ref. [16]. We are concerned about three quantities, network size N t , network diameter D t , and degree k u ( t ) of a given node u . By construction,we can easily see that in the infinite t limit, these quantities grow obeying the7ollowing relation: N t ≃ N t − , D t = 2 D t − , k u ( t ) ≃ k u ( t − N t , k u ( t ) and D t increase by a factor of f N = 4, f k = 2 and f D = 2, respectively.From above obtained microscopic parameters demonstrating the mechanismfor network growth, we can derive the scaling exponents: the fractal dimension d B = ln f N ln f D = 2 and the degree exponent of boxes d k = ln f k ln f D = 1. According tothe scaling relation of fractal scale-free networks, the exponent of the degreedistribution satisfies γ = 1 + d B d k = 3, giving the same γ as that obtained inthe direct calculation of the degree distribution.The fractality behavior can be also easy to see by tiling the system using arenormalization procedure as follows. we can ‘zoom out’ (i.e. renormalize) thenetwork by replacing each connected cluster of bonds on the right of Fig. 1by a single ‘super’-link, in a way that reverses the process of network growth,see Fig. 1. This has the effect of rescaling diameter, it reduces the diameterby a factor of 2 by carrying a cluster of bonds with diameter 2 into a single‘super’-link with unit length. At the same time, the number nodes of in therescaled network decreases by a factor = 4. Thus, the fractal dimension is d B = ln 4ln 2 = 2. As demonstrated in the preceding section, the networks exhibit simultaneouslymany interesting properties: scale-free phenomenon, tunable clustering, “large-world” behavior, and fractality. To the best of our knowledge, these featureshave not been reported in previous network models. Hence, it is of scientificinterest and great significance to investigate dynamic processes taking placeupon the model to inspect the effect of network topologies on the dynamics.Next we will focus on studying the SIR model of epidemics, which is one ofthe most important disease models and has attracted much attention withinphysical society, see [5] and references therein.The standard SIR model [6,7] assumes that each individual can be in anyof three exclusive states: susceptible (S), infected (I), or removed (R). Thedynamics of disease transmission on a network can be described as follows.Each node of the network stands for an individual and each link representsthe connection along which the individuals interact and the epidemic can betransmitted. At each time step, an infected node transmits infection to each ofits neighbors independently with probability λ ; at the same time the infectednode itself becomes removed with a constant probability 1, whereupon it cannot catch the infection again, and thus will not infect its neighbors any longer.8he above-described SIR model for disease dynamics is equivalent to a bondpercolation process with bond occupation probability equal to the infectionrate λ [35,36]. The connected clusters of nodes in the bond percolation processthen correspond to the groups of individuals that would be infected by a dis-ease outbreak starting with any individual within the cluster it belongs to. Itwas shown that in scale-free networks such as Barab´asi-Albert (BA) network,for any vanishingly small λ , there exists an extensive spanning cluster or “gi-ant component”, implying that scale-free networks demonstrate an absence ofepidemic thresholds. However, we will present that for our networks, epidemicthresholds are present, the values of which depend on the parameter q .According to the mapping between the SIR dynamics and the bond percola-tion problem, the epidemic threshold corresponds precisely to the percolationthreshold. In our case, the recursive construction of the networks make it pos-sible to study the percolation problem by using the real-space renormaliza-tion group technique [37,38,39] to obtain an exact solution for the percolation(epidemic) thresholds. Here we are interested in only the percolation phasetransition point, excluding the size of giant cluster. Let us describe the pro-cedure in application to the considered network. Supposing that the networkgrowth stops at a time step t → ∞ , then we spoil the network in the followingway: for an arbitrary present link in the undamaged network, it is retained inthe damaged network with probability λ . Then we invert the transformationshown in Fig. 1 and define n = t − τ for this inverted transformation, which isin fact a decimation procedure [39]. Further, we introduce the probability λ n that if two nodes are connected in the undamaged network at τ = t − n , thenat the n th step of the decimation for the damaged network, there exists a pathbetween these vertices. Here, λ = λ . According to the network constructionand the analysis in [39], it is easy to obtain the following recursive relation for λ n : λ n +1 = q (cid:16) λ n + λ n − λ n × λ n (cid:17) + (1 − q ) (cid:20) λ n + 5 λ n (1 − λ n )+ 8 λ n (1 − λ n ) + 2 λ n (1 − λ n ) (cid:21) . (6)Equation (6) has five roots (i.e., fixed points), among which two are invalid:one is greater than 1, the other is less than 0. The other three fixed points areas follows: two stable fixed points at λ = 0 and λ = 1, and an unstable fixedpoint at λ c that is the percolation threshold. We omit the expression of λ c asa function of q , because it is very lengthy. We show the dependence of λ c on q in Fig. 4, which indicates that the threshold λ c increases almost linearly as q increases. When q grows from 0 to 1, λ c increases from 0.5 to √ − ≈ . λ c such that for λ > λ c a giant9 ig. 4. The dependence relation of percolation threshold λ c on the parameter q . component appears, for λ < λ c there are only small clusters. This means thatfor the SIR model the epidemic prevalence undergoes a phase transition at anonzero epidemic threshold λ c . If the infection rate λ > λ c , the disease spreadsand becomes persistent in time; otherwise, the infection dies out gradually. Theexistence of epidemic thresholds in our networks is in sharp contrast with thenull threshold found in a wide range of stochastic scale-free networks of theBarab´asi-Albert (BA) type [9,10,11].Why are the present scale-free networks not prone to disease propagation aspreviously studied uncorrelated BA type scale-free networks? We argue thatthe presence of finite epidemic thresholds in our networks lies with their two-dimensional fractal structure with diameter increasing as a square power ofnetwork order, a property analogous to that of two-dimensional regular lat-tice [40]. The ‘large-world’ feature stops the diffusion of diseases, and makesthe behaviors of disease spreading in our networks similar to those of reg-ular lattices. Thus, the fractal topology provides protection against diseasespreading. In the present work, we have introduced a new model for fractal networks andprovided a detailed analysis of the structural properties, which are relatedto the model parameter q . The model exhibits a rich topological behavior.The degree distribution is power-law with the degree exponent asymptoticallyapproaching 3 for large network order. The clustering coefficient is changeable,which can be systematically tunable in a large range by altering q . Particularly,the networks are topologically fractal with a fractal dimension of 2 for all q . Along with the fractality, the networks display ‘large-world’ phenomenon,their average path length increases approximatively as a square power of the10umber of nodes.We have also investigated the effects of the particular topological character-istics on the SIR model for disease spreading dynamics. Strikingly, We foundthat epidemic thresholds are recovered for all networks regardless the value of q , which is in contrast to the conventional wisdom that being prone to diseasepropagation is an intrinsic nature of scale-free networks. We concluded thatthe dominant factor suppressing epidemic spreading is the fractal structure ac-companied by a ‘large-world’ behavior. The peculiar structural properties andepidemic dynamics make our networks unique within the category of scale-free networks. Our study is helpful for designing real-life networked systemsrobust to epidemic outbreaks, and for better understanding of the effluencesof structure on the propagation dynamics. Acknowledgment
We thank Yichao Zhang for preparing this manuscript. This research wassupported by the National Basic Research Program of China under grantNo. 2007CB310806, the National Natural Science Foundation of China un-der Grant Nos. 60496327, 60573183, 90612007, 60773123, and 60704044, theShanghai Natural Science Foundation under Grant No. 06ZR14013, the ChinaPostdoctoral Science Foundation funded project under Grant No. 20060400162,the Program for New Century Excellent Talents in University of China (NCET-06-0376), and the Huawei Foundation of Science and Technology (YJCB2007031IN).
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