Localized attack on networks with clustering

Network systems with clustering have been given much attention due to their wide occurrence in the real world. One focus of these studies has been on robustness of single clustered networks and interdependent clustered networks under random attack (RA) or hub-targeted attack. However, infrastructure networks could suffer from a damage that is localized, i.e. a group of neighboring nodes attacked or fail, a topic that was not studied earlier on clustered networks. In this paper, we analytically and via simulations study the robustness under localized attack (LA) of single Erdős–Rényi clustered network and interdependent clustered network. For generating networks with clustering we use two models: (i) double Poisson distribution (DPD) and (ii) fixed degree distribution (FDD). For the LA case, the DPD model shows a second order phase transition behavior for a single clustered network, while for dependent networks, the system undergoes a change of percolation phase transition from a first order (abrupt transition) to a second order (continuous) transition when the coupling strength q decreases below a critical value qc. Our results imply that single networks become significantly more vulnerable with increasing clustering coefficient c with respect to LA. This is in contrast to RA where the robustness is almost independent of c. We obtain similar results when testing different real networks. For LA on dependent networks, we also observe that the system becomes more vulnerable as c increases. This is again in contrast to RA, where for, q < qc, the system robustness is almost unaffected by increasing clustering. We also solved analytically the case of LA on random regular networks which are clustered and interdependent and find that as m (the number of clustered networks that each network depends on) or c increases, the system becomes significantly more vulnerable. We also analyzed via simulations the case of generating clustering in networks for the model of keeping a FDD, and find that the influence of clustering on the robustness of two partially interdependent networks under LA is smaller than for DPD, which is very different from these cases under RA.


Introduction
Over the past two decades, the study of complex networks has gained increasing attention. The main reason is that many real systems in our daily life can be described and better understood when represented as complex networks. Examples include the Internet and World Wide Web, food webs, social networks, transportation systems, electricity distribution networks, genetic networks, brain networks and many others [1][2][3][4][5][6][7][8][9][10][11]. An important concern in the study of complex networks is their robustness, which is important for many fields, such as ecology, biology, economics and engineering [12][13][14][15][16][17][18]. Network robustness deals usually with the question of the response of the network to random failures and targeted attacks. This question can be analyzed and characterized using percolation theory by studying the critical thresholds or the integrated size of the largest cluster during the attack process [19][20][21][22][23].
Many useful results have been obtained by analyzing the robustness of single isolated networks. However, in many real scenarios, critical infrastructures rarely appear in isolated state but usually depend on other infrastructures for functioning. This has led to the emerging sub-field of research in network science, called interdependent networks or more general, networks of networks (NON). Many constructive conclusions have been obtained which improve our understanding of the robustness of interdependent networks. Buldyrev et al [24] developed a framework for understanding the robustness of two fully interdependent networks under random failures, and found that interdependent networks become, due to cascading failures, significantly more vulnerable compared to their single networks counterparts and undergo an abrupt (first order transition) collapse. Subsequently, a system of two partially interdependent networks (where a fraction of q nodes in both networks depend on each other) under random failures has been studied by Parshani et al [25]. It was found, both analytically and numerically that reducing the coupling strength below a critical value q c , yields to a change from a first order to a second order percolation transition. Gao et al developed a general framework to study the percolation behavior of n interdependent networks, suffering from random failures [26][27][28]. The above studies reveal that dependency links between networks make the system highly vulnerable to random failures that may yield cascading failures and understanding their mechanisms might help to design resilient infrastructures and improve existing infrastructures.
Due to the broad degree distribution of real networks, it was proposed to analyze the vulnerability with respect to a targeted attack on the high degree nodes. Such attacks have dramatic structural effects on single networks and can lead easily to network fragmentation [18,20,[29][30][31][32][33]. By introducing a probability function of node degree to fail, Gallos et al found that for the targeted attacks case, even little knowledge of the highly connected nodes can reduce significantly the robustness compared to the random attack (RA) case [34]. By mapping the targeted-attack problem to the RA problem, Huang et al [35] studied the robustness of two fully interdependent networks under targeted attack. Later, Dong et al [36,37] studied the robustness of two partially interdependent networks against targeted attack, and further proposed a general theoretical framework for understanding the targeted-attack problem in a NON system. However, in many real scenarios, attacks are neither random nor targeted, but localized, which means a group of neighboring nodes in a network are attacked or fail due to natural disasters like earthquakes or floods. For example, when an earthquake occurs, it releases energy in the form of seismic waves that spread from the epicenter in all directions. According to local amplification effect, even for low-intensity earthquakes, local geological features can induce high levels of shaking ground surface in a certain radius around the center, which can destroy locally the infrastructures. Only few studies on such localized attacks (LAs) strategy have been reported. Shao et al developed a theoretical and numerical approach to study the robustness of complex networks against LA [38]. Berezin et al described and predicted the effects of LA on spatially embedded systems with dependencies, and found that a LA can cause substantially more damage compared to an equivalent RA [39]. By mapping the LA problem to a RA problem, Yuan et al showed how the broadness of the degree distribution affects the fragility of interdependent networks due to LA [40]. Zhao et al finds a mapping between overload failures and dependency links [41]. Dong et al proposed a modified partially LA strategy, and studied the network robustness against this attack analytically and numerically [42].
As one of the key issues in complex networks, clustered networks, which is a realistic feature appearing frequently in real network, have attracted much attention in both theoretical research and in various applied fields [43]. However, networks with clustering were studied only with respect to random failures or high degree attacks [44,45], while the effect of LAs on clustered networks has not been studied earlier.
In this paper we study the percolation behavior due to LAs in two types of clustered networks models in single clustered network, as well as in network of interdependent networks with clustering. The two models are: (a) we generate networks for which the degree distribution of the clustered network follows double Poisson distribution (DPD) [46]. (b) We generate networks with fixed degree distribution having a Poisson distribution (FDD) [45,47]. The results for single networks and real networks are described in section 2. The robustness of NON with clustering are analyzed in section 3.

Single networks with clustering
In a network, the clustering feature can be characterized by specifying the fraction of nodes connected to s single links and having t triangles (clustering). As a special case, we consider an Erdős-Rényi (ER) type network with clustering having a probability density which obeys a DPD [46] = á ñ á ñ e e , 1 s s t t where á ñ s and á ñ t are the average numbers of single links and triangles per node, respectively. The average degree of a node is thus, á ñ = á ñ + á ñ k s t 2 . The generating function of the DPD can be expressed as [46], The clustering coefficient is defined as = á ñ á ñ + á ñ c t t k 2 2 2 , which implies the average of the clustering coefficient of all nodes by using the probability that two edges share a node in the network. For c=0, the network does not have clustering, which is the limit of the ER network.
The LA is performed as follows. We randomly choose a node as a 'root' node and denote all nodes distances from this root, shell by shell according to increasing distance. Next, the LA is performed by the following two stages. We first remove around the root node all nodes shell by shell according to increasing distance, and remove all the links connecting all pairs of the removed nodes until a fraction of 1−p nodes from the whole network is removed. In this stage we keep the links between the removed nodes and the remaining nodes. The distribution of nodes with s single links and t triangles in the remaining network is [35,38] where A p (s, t) denotes the number of nodes with s single links and t triangles. When one more node is being removed, we get By differentiating equation (3) with respect to p and substituting it in equation (5), we get, The solution of equation (6) can be written as, The generating function of the residual network is The probability of a link to end at an unremoved node in the remaining network can be expressed as [38], In the second stage we remove all remaining links from the removed nodes, which are connected to the remaining non removed nodes. The generating function of the remaining network is If we find a clustering networkÃ of generating function˜( ) G x y , 0 , after a RA of removing 1−p fraction of nodes than the generation function of the remaining network becomes 45,47]. Next, we map the LA problem on network A to a RA problem on networkÃ. By using Thus, the fraction of the giant component of the remaining clustered network is  (13) and simulations for several values of c, which support well the theory. In addition, when comparing different c, we find that p c increases with c and ¥ P increases continuously from zero at the critical threshold p c to a finite value, which means that the system undergoes a second order phase transition. As , the critical threshold of the second order phase transition p c can be found as [38], Moreover, the dependence of p c on c and á ñ k are shown in figures 1(b) and (c). As can be seen, p c increases as c increases and á ñ k decreases. Thus, the network becomes more vulnerable with increasing clustering coefficient. This is in marked contrast to RA, where the system robustness does not change with increasing clustering coefficient, as seen in figures 1(b) and (c).
Clustering has an important effect in epidemic processes, information spread, network resilience, and dynamical systems on the networks. For example, in a social network, two friends of an individual have a high probability to become friends. From above, we analyze and compare the robustness of several real networks under LA and RA, and analyze them while changing the actual clustering in the real network using rewiring algorithm [48][49][50][51][52]. Figure 2 demonstrates that S as a function of p for several real networks with different clustering c under LA and RA. For real networks, we use a small S cutoff to find p c . Simulation results suggest that p c is almost unchanged for RA but p c dramatically increases for LA, as seen in figure 3. Thus again, clustering coefficient almost have no effect on system robustness for the case of RA, but it has a significant effect on system robustness for LA. As clustering coefficient increases, the system becomes more and more vulnerable and significantly more difficult to protect for LA, similar with above theoretical results. Note that the effect of clustering in the real networks (figure 3) on their vulnerability is significantly more than found in our theory ( figure 1(b)). The reason is that the model we solved analytically is for Poisson degree distribution (equation (1)) but the real networks shown in figure 3 are of scale free type, where we expect a stronger effect in LA. This since neighbors of nodes in scale free networks are usually high degree and a LA will remove them in the first stages [53] while RA removes mostly low degree nodes.
Nextly, we consider the FDD model for single clustered networks, which preserves the total degree distribution P(k) for different c using the method proposed by Hackett et al as given by equation (15) below, i.e. changing c but keeping a FDD [45] å å 0, 1 and⌊ ⌋ . is the floor function [45,47,54]. According to above expression, equation (15), clustered network are assumed to have joint distribution P st from a given degree distribution P(k) by randomly choosing a fraction f of nodes to be connected to maximum possible number of triangles while the  (14). Results are also compared to RA. (c) p c as a function of k with different c=0, 0.1 and 0.15 from equation (14). Results are also compared to RA. remaining fraction 1−f of nodes are attached to single edges only. From this definition, we can get the above equation (15) for the clustering coefficient c as a function of f [45].
From figure 4, one can observe the peaks of the second largest cluster, ¥ P ,2 for different c corresponding to the phase transition point from simulation results. It is also seen that p c under LA is almost unchanged with increasing c, but increases for RA as seen in both figures 4(b) and (c). This means that changing clustering coefficient for a single FDD has a little effect for LA but increasing clustering coefficient can make single network under FDD more vulnerable, which is in marked contrast to the case of DPD.

Two interdependent clustered networks
In this subsection, we study the robustness under LAs of two partially interdependent clustered networks A and B, which obey the DPD P(s, t), with parameters á ñ = á ñ = á ñ s s s and N A =N B =N, respectively. We assume a fraction q A (q B ) of nodes in network A (B) depends on nodes in network B (A). This means that a node in network B which depends on a failed node in network A, will also fail, and vice versa. We start by removing a fraction 1−p of nodes from network A and B separately through LA, cascading failures occur, until the system reaches a steady state. At this time, the remaining fraction of nodes in network A and network B are equal to X and Y [25,36],  where g A (X) and g B (Y) satisfy = -= - and p=G 0 (h, h 2 ). Figure 5(a) shows that simulation results for several values of c, are in good agreement with the theoretical results obtained from equations (16)- (18) We can see in figure 5(a) that ¥ P A , continuously increases from zero to finite value at the second order critical threshold p c II for q=0.2, but abruptly jumps from zero to a finite value at the first order critical threshold p c I for q=0.8. These results suggest that the phase transition nature changes from second order to first order at a critical coupling strength q c . By combining equations (16)-(18), we can obtain p c as a function of q, as shown in figure 5(b). The upper panel in figure 5(b) shows that p c increases with increasing c, which means that increasing either clustering coefficient or coupling strength makes the network more vulnerable to LA. In contrast, as seen in the lower panel, for RA, the system robustness almost remains the same for q<q c and increasing c and becomes more vulnerable for q>q c . In figures 5(c) and (d), we see also that increasing c or/and decreasing á ñ k will enlarge p c , which means increasing clustering coefficient make networks more vulnerable. From comparing the two kinds of attacking strategies in figure 5(b), we can also see that q c is almost constant when changing clustering for RA, but increases as c increases for LA.
Next, we perform a LA by simulated removing a fraction 1−p of nodes from network A. Let ¥ P ,1 denote the size of the giant component of network A at the steady state. Figures 6(a) and (b) compare simulation results of FDD with theoretical results of DPD from equations (16)-(17) of [45]. Figures 6(a) and (b) indicate that for LA, second and first order phase transition behaviors can be observed, respectively, for weak (q=0.2) and strong (q=0.8) coupling strength with different c. Note that p c increases with increasing c for both, FDD and DPD. This suggests that robustness of both cases of clustered networks decreases with increasing clustering coefficient. Note, however, that, p c of DPD is larger than that of FDD except for c=0. As seen in figures 6(c) and (d), that the  figure 6(c) for FDD under LA illustrates that increasing clustering coefficient has almost no effect on robustness for weak coupling strength. But for DPD, we notice that the system become more vulnerable as c increases for all coupling strengths as seen in the right panel of figure 6(c). In marked contrast, if the FDD system suffers from RA, it becomes more vulnerable as c increases for both weak and strong coupling strength as shown in the left panel of figure 6(d). However for DPD, the system gradually becomes vulnerable as c increases only for strong coupling strength as seen in right panel of figure 6(d).

Star-like NON of ER networks with clustering
We generalize our results for two interdependent networks with clustering analyzed in above subsection, to a system whose dependence structure is a network and each node is a clustered network, i.e. NON [26]. For simplicity, we assume that all clustered networks satisfy a joint degree distribution with the same á ñ s and á ñ t . Here, we adopt the non-feedback condition [27] for dependency structure like in the above section. In this subsection, we study the cases of a star-like NON formed of ER networks with clustering (as demonstrated in figure 7(a)) and random regular (RR) of ER networks with clustering (as shown in figure 7(b)).
Here we study star-like NON formed of n clustered networks, in which a central network is linked via dependency links with other n−1 networks. I.e. the n−1 networks are mutually dependent on the central network but do not depend on each other, see figure 7(a). We assume that a fraction q i,1 (i=2, 3, K, n) of nodes in network A i and vice versa. depends on nodes in the central network A 1 . If one of a pair of interdependent nodes fail, the other node that depends on it also fail to function. The initial attack is exerted on each network by removing locally a fraction 1−p of nodes and this damage spreads in this system back and forth until no node depends on a disabled node, and the remaining network is stable or fully collapsed. For simplicity, but without loss of generality, we set q 2,1 =q 3,1 =...=q n,1 =q. The fraction of left nodes in A is equal to X and in the other n−1 networks A i (i=2, 3, K, n) is equal to Y, following the expressions [27, 37], The size of the giant components of network A and the other (n−1) networks A i , can be expressed as ¥ P ,1 and ¥ P , where g 1 and g 2 satisfy   Figure 8(a) shows that simulation results agree well with theoretical predictions obtained from equation (22). Additionally, we observe in figures 8(a) and (b) that as q decreases, the system changes from an abrupt first order phase transition to a continuous second order transition at a critical coupling strength q c . And, the results show that the system becomes more vulnerable for larger c. Moreover, it is seen that for RA as q   approaching to 0, p c lines for different c gradually become closer, while for LA, as c increases, p c becomes larger, suggesting that increasing clustering within networks makes the system more vulnerable to LA. This is seen even more clearly in figures 8(c) and (d) comparing p c as functions of c and á ñ k for different q under LA and RA. As seen in figure 8(c) the system under LA and RA shows very different robustness as c increases for weak coupling strength q=0.2, 0.5 and á ñ = k 4. The system is significantly more robust under RA compared to LA for weak coupling strength. But for strong coupling strength q=0.7, the system shows similar robustness for both attacking strategies. From figure 8(d), one can see that increasing á ñ k enhances the robustness for both attack strategies, but LA has a more destructive power, that makes the system more vulnerable compared to RA. Note that as á ñ k increases, the robustness of system gradually shows differences for different attacking strategies even for strong coupling strength, as shown in figure 8(d). Figure 9 shows the dependence of the critical coupling strength q c on n from equation (22) for several c. The results show that q c monotonically decreases as n increases, implying that the NON system more easily shows first order phase transition, when n increases. Additionally, q c increases as c increases for different n, which imply that the region of occurring second phase transition becomes larger as clustering coefficient increases.

RR NON of clustered networks
Next we study RR NON formed of ER networks with clustering. The ER networks are clustered networks while the NON is a RR network with same degree m for all the nodes, which means that every ER clustered network is partially dependent (with q dependency nodes) on other m clustered networks, see figure 7(b) [55]. Assuming that the initial attack is exerted on each ER network through locally removing a fraction 1−p of nodes of the network by removing all nodes shell by shell around a randomly chosen node. For simplicity, we let each clustered network to have the same average number of single links á ñ s , average number of triangles á ñ t and coupling strength q. When the cascading failures process reaches a stable state, the remaining fraction of nodes in any clustered network is X [55]

=
. 24 The size of giant component in each network is Simplifying equations (23)- (25), we obtain, The critical threshold of the second order phase transition p c II can be obtained from  ¥ ( ) P p 0 c II in equation (26). And, by equating the first derivative of both sides of equation (26) with respect to ¥ P , we obtain the critical threshold at q c . Thus, the critical coupling q c , where the first order phase transition changes to a second order phase transition can be obtained analytically as follows

Conclusions
Here we study both theoretically and numerically the robustness of clustered networks under LA of three cases, single clustered network, two interdependent clustered networks and NON (star-like NON and RR of clustered networks). We also studied, two types of clustering models. One is the DPD and the second is fixed degree distribution obeying Poisson distribution (FDD). For the DPD case, the LA on a single network shows a second order percolation behavior. When considering several real networks, we also find that system becomes more vulnerable and significantly more difficult to protect against LA, but the system robustness does not almost change in RA as clustering coefficient increases. For two interdependent clustered networks and RR NON of clustered networks, increasing q leads to a change from a second order to a first order phase transition. For the case of a single clustered network, p c increases as c increases and k decreases. This is in marked contrast to RA, where p c almost keep constant. For the case of dependent networks, the results demonstrate that p c becomes larger as c and q increase. Besides, for star-like NON, as n increases, p c becomes larger and q c decreases gradually, which indicates that the system becomes more vulnerable and is easier to show a first order phase transition. Furthermore, we studied two generalized cases for NONs and found that higher clustering coefficient causes the system to be more vulnerable. For FDD case, by taking two interdependent networks as an example, we found that p c is almost unaffected for q<q c when c increases, but for q>q c , p c increases as c increases, which is in marked contrast to RA.