Cascaded failures in weighted networks

EH086 PRE October 9, 20 5:24 2 PHYSICAL REVIEW E 00, 00600 (20) Cascaded failures in weighted networks 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 Baharan Mirzasoleiman, Mahmoudreza Babaei, Mahdi Jalili, * and MohammadAli Safari Department of Computer Engineering, Sharif University of Technology, Tehran, Iran (Received 4 August 20; published xxxxx) Many technological networks can experience random and/or systematic failures in their components. More destructive situations can happen if the components have limited capacity, where the failure in one of them might lead to a cascade of failures in other components, and consequently break down the structure of the network. In this paper, the tolerance of cascaded failures was investigated in weighted networks. Three weighting strategies were considered including the betweenness centrality of the edges, the product of the degrees of the end nodes, and the product of their betweenness centralities. Then, the effect of the cascaded attack was investigated by considering the local weighted flow redistribution rule. The capacity of the edges was considered to be proportional to their initial weight distribution. The size of the survived part of the attacked network was determined in model networks as well as in a number of real-world networks including the power grid, the internet in the level of autonomous system, the railway network of Europe, and the United States airports network. We found that the networks in which the weight of each edge is the multiplication of the betweenness centrality of the end nodes had the best robustness against cascaded failures. In other words, the case where the load of the links is considered to be the product of the betweenness centrality of the end nodes is favored for the robustness of the network against cascaded failures. 20 DOI: 0.03/PhysRevE.00.00600 PACS number(s): 89.75.Fb, 89.75.Hc 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 37 38 39 40 4 42 43 44 45 46 47 48 49 50 5 52 53 I. INTRODUCTION Network science has attracted much attention in recent years, primarily due to its application in many areas ranging from biology to medicine, engineering, and social sciences [,2]. Research in network science starts by observing a phenomenon in real data and then tries to construct models to mimic its behavior. Many real-world networks share some common structural properties such as scale-free degree distribution, small-worldness, and modularity. The dynamic behavior of networks largely depends on their structural properties [3,4]. One of the topics that has attracted much attention in this context is the robustness of networks against random and systematic component failures [5 7]. Networks might undergo failures in a number of their components (i.e., nodes and edges) and consequently lose proper functionality [8,9]. The failure in a network can be random or systematic. When a random failure (i.e., error) occurs in a network, a number of its components are randomly removed from the network. While, in a systematic failure (i.e., attack) the components are systematically broken down [5,0]. For example, the hub nodes might be targets for attacks. When the intrinsic dynamics of network flows are taken into account, the systematic removal of the components can have a much more devastating consequence than random removal []. The modern societies are largely dependent on networked structures such as power grids, information communication networks, the internet, and transportation networks. Failure in such networks might collapse normal daily life and result in chaos in the society. Evidence has shown that locally emerging random or systematic failures in networks can influence the entire network, often resulting in large-scale collapse in the network. Examples include large blackouts in the United States due to failure in the power grid [2] and breakdowns of the * Corresponding author: mjalili@sharif.edu internet [3]. Indeed, a cascaded failure has happened in such 54 cases [9,]. 55 Several studies have been devoted to the concept of 56 controlling cascaded failures [4 7]. For example, islanding 57 or separating the survivable parts of a grid has long been 58 used to allow a transmission grid to continue its functionality. 59 Building a system for allocating competing resources during 60 an extended failure is another solution for controlling such 6 failures. While these are applicable solutions, they cannot 62 eliminate all failures [5]. Therefore, it is desired to design a 63 network that is robust against cascaded failures. The influence 64 of the cascaded failure in the size of the largest connected 65 component has been investigated in a number of network 66 models including preferential attachment scale-free [8], 67 Watts-Strogatz small-world [9], and modular networks [20]. 68 In many of the studies, as a component fails, the loads 69 are recalculated and the components whose loads exceed 70 their capacity are removed from the network. The process is 7 repeated until the loads of all remaining components are below 72 their capacity [8 20]. However, this might not be realistic in 73 some applications. For example, let us consider the internet. It 74 is natural that the load passing through a failed component is 75 redistributed among its neighboring components. To this end, 76 a local weighted flow redistribution rule has been proposed 77 [2]. In this model, the cascaded failure is triggered by 78 removing the edge with the maximal load. As an edge is 79 removed from the network, its load is redistributed among 80 the neighbors. Studying model networks with scale-free and 8 small-world properties and by applying the local weighted 82 flow redistribution rule, Wang and Chen found the strongest 83 robustness against cascaded failure at a specific weighting 84 strength [2]. 85 In this paper we investigated a number of factors influencing 86 the robustness of the networks against cascaded failures. An 87 important question in this context is which component has 88 the largest cascaded effect on the network. We considered 89 the cascaded effect of failures in the edges. Furthermore, the 90 539-3755/20/00(0)/00600(8) 00600-20 American Physical Society ACC. CODE EH086 AUTHOR Mirzasoleiman

EH086 PRE October 9, 20 5:24 MIRZASOLEIMAN, BABAEI, JALILI, AND SAFARI PHYSICAL REVIEW E 00, 00600 (20) 9 92 93 94 95 96 97 98 99 00 0 02 03 04 05 06 07 08 09 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 37 38 39 40 4 networks were weighted according to different rules due to the fact that many real-world networks are inherently weighted. We used three weighting strategies: the betweenness centrality of the edges, the product of the degrees of the end nodes, and the product of their betweenness centrality. The numerical simulations were performed on a number of model networks such as preferential attachment scale-free [22], Newman-Watts small-world [23], Erdős-Rényi [24], and modular networks [20]. Furthermore, we considered a number of real-world networks including the power grid, the internet in the level of autonomous systems, the railway network of Europe, and the United States airports network. We found that the networks whose weights are the product of the betweenness centrality of the end nodes have the most robustness against cascaded failures. This study suggests that in order to enhance the robustness of the networks against cascaded failures, one could take the loads (weights) of the edges as the product of the betweenness centrality of the end nodes. II. LOCAL WEIGHTED FLOW REDISTRIBUTION RULE The local weighted flow redistribution rule (LWFRR) has been recently introduced and studied by Wang and Chen [2]. In this model, when an edge is subject to an attack and removed from the network, the flow passing through this edge is redistributed to its nearest-neighbor edges [2,25]. As a consequence, the load of each neighbor edge increases proportional to its weight. More precisely, F im = F ij w im a Ɣ i w ia + b Ɣj w jb where e ij is the attacked edge, Ɣ i and Ɣ j are the set of neighbors of nodes i and j, respectively. F ij is the flow on e ij before being broken and F im is the additional flow that the e im receives. Every edge e ij has some limited capacity C ij determining the maximum load that the edge can handle. The capacity C ij of the edge e ij is assumed to be proportional to the initial load of the edge w ij (i.e., C ij = Tw ij ). That is, there exists a constant threshold value T such that if () F im + F im Tw im = C im (2) Then, the edge e im cannot tolerate the additional flow and will break apart. As a result, the network faces further redistribution of the flows, and consequently, more edges might break. Cascading failure continues as long as there is no edge e uv whose flow dominates its capacity (i.e., F uv Tw uv ). The lower the number of broken edges, the more robust the network is against attacks. There exists a minimum threshold at which removal of an edge does not lead to cascading failure anymore. A phase transition is occurred at this critical threshold (T c ), where for T < T c the network preserves its robustness against any random or systematic failure. On the other hand, for T < T c failure of a part can trigger the failure of successive parts of the network and cascading failure suddenly emerges. T c is a significant measure in determining a network s robustness; the lower the value of T c is the stronger the robustness of the network is against removal of its components. In real-world networks, cascading failure is often studied 42 in order to protect many infrastructure networks. Computer 43 networks and the internet are such examples that should 44 be protected against cascaded failures [26,27]. Protecting 45 electrical grids against failures and a society against spread 46 of an infectious disease are other examples where the studies 47 in this context can be beneficial. Let us consider a computer 48 network. If a few important cables break down, the traffic 49 should be rerouted either globally or locally towards the 50 destination. This will lead to redistribution of the traffic in the 5 network. When a line receives extra traffic, its total flow may 52 exceed its bandwidth (threshold) and cause congestion. As a 53 result, an avalanche of overloads emerges on the network and 54 cascading failure might occur. As another example, suppose a 55 disease appears in a region. It might spread to other regions 56 through infected individuals traveling across the regions. It 57 is obvious that immunization of individuals who travel from 58 populated regions prevents the widespread distribution of the 59 disease. Consequently, spending more money for vaccinating 60 these individuals seems a reasonable action. In the power grid 6 example, when an element (completely or partially) fails, its 62 load shifts to nearby elements in the system. Some of those 63 nearby elements might be pushed beyond their capacity and 64 become overloaded; thus get broken and shift their load onto 65 other neighboring elements. This surge current can induce 66 the already overloaded nodes into failure, setting off more 67 overloads and thereby taking down the entire system in a very 68 short time. Under certain conditions, a large power grid might 69 collapse after the failure of a single transformer. All these 70 networks are examples of weighted networks in which the 7 weight of each edge can be interpreted as either its capacity 72 or cost of immunization and failure of an edge causes an 73 immediate increase of the load of its nearest-neighbor edges. 74 III. WEIGHTING METHODS 75 In network characterization, the centrality of an element 76 is a significant measure and plays a fundamental role in 77 studying cascading failure [28]. The degree of a node is an 78 obvious topological metric that can be used for determining 79 its connectivity as well as centrality. The degree of the node i 80 is defined as 8 k i = N a ij, (3) where N is the size of the network and A = (a ij ), i, j =,..., 82 N, is the adjacency matrix of an undirected and unweighted 83 network. However, there may exist some nodes that play a 84 crucial role in connecting different parts of the network despite 85 their small degree. Such nodes are called bridges or local 86 bridges that connect parts of the network that would become 87 disconnected otherwise. Because of their topological positions 88 in the network, many shortest paths (often the only plausible 89 route between many pairs of nodes) pass through these nodes. 90 These reasons motivated the introduction of another measure 9 for centrality of a node in the network (i.e., node betweenness 92 centrality). Node betweenness centrality is defined as the 93 number of shortest paths between pairs of nodes that pass 94 00600-2

EH086 PRE October 9, 20 5:24 CASCADED FAILURES IN WEIGHTED NETWORKS PHYSICAL REVIEW E 00, 00600 (20) 95 96 97 98 99 200 20 202 203 204 205 206 207 208 209 20 2 22 23 24 25 26 27 28 29 220 22 222 223 224 225 226 227 228 229 230 23 232 233 through a given node [29]. More precisely, B i = [Ɣ pq (i)/ɣ pq ] (4) p i q where Ɣ pq is the number of shortest paths from the node p to node q and Ɣ pq (i) is the number of these shortest paths making use of node i. The larger the betweenness centrality of a node is, the more its significance in the formation of the shortest paths in the network. Another measure of centrality is edge betweenness centrality, which has been widely used to model the traffic load or weight of an edge; it is defined similar to node betweenness centrality. The edge betweenness centrality of an edge is the number of shortest paths between pairs of nodes that pass through the edge e ij [29], that is, B ij = [Ɣ pq (ij )/Ɣ pq ] (5) p q where Ɣ pq (i) is the number of shortest paths that go through the edge ij. These centrality measures can be used to determine the loads in an unweighted network or estimate the weights in a weighted real network. Wang and Chen [2] used node degrees to model the traffic on a network and studied cascading failure. They used the power-law function of degrees of the two ends of an edge as measure for edge centrality and obtained several experimental results on different real-world networks. According to their definition, the weight of an edge is modeled by w ij = (k i k j ) θ, (6) where k i is the degree of node i and θ is a tuning parameter. They showed that θ = leads to the strongest robustness on various networks [2]. We introduce a weighting method based on node betweenness centrality. Our studies showed that this weighting method is in accordance with the weights of many real networks. The intuition for this weighting method is based on the observation that an edge is important in a network when its two end nodes are important. As an example, assume one is flying from London to Melbourne. He probably chooses some central cities such as Dubai or Kuala Lumpur and flies through them on his way to Melbourne. Therefore, an edge is chosen when its two ends have high centrality. A similar observation can be made for packet routing on the internet. The links between central points are more probable to be chosen when sending a packet. Based on the above observation, one can take into account the centrality of both end nodes of an edge and define the weight 234 of an edge e ij as 235 w ij = (B i B j ) θ. (7) In this method, the weight of an edge has a power-law 236 dependence on the product of betweenness centrality of its two 237 end nodes. This is indeed somehow the case in many real-world 238 networks, where the weights of the links do not follow the 239 betweenness centrality of the edges. However, it shows high 240 correlation with the weights introduced through Eqs. (6) and 24 (7). We showed the correlation between the above weighting 242 strategies in a number of real-world networks including: 243 US airlines. An airlines connection network in the USA 244 consists of 332 nodes and 063 edges. The weights correspond 245 to the number of seats available on the scheduled flights [30]. 246 US airports network. This is the network of the 500 busiest 247 commercial airports in the USA. The weights correspond to 248 the number of seats available on the scheduled flights [3]. 249 This network has 2980 edges. 250 Lesmis. Coappearance network of characters in the novel 25 Les Miserables. This network has 77 nodes and 27 edges [32]. 252 Netscience. Coauthorship network of scientists working in 253 the field of network theory and experiment. This network 254 contains 589 nodes and 37 edges [33]. 255 Bkham. The network of human interactions in bounded 256 groups and on the actors ability to recall those interactions. 257 This network consists of 44 nodes and 53 edges [34]. 258 Table I shows the Pearson correlation coefficients between 259 the real weights and different metrics including the between- 260 ness centrality of the edges B ij, the product of the betweenness 26 centrality of the end nodes, B i B j, and the product of the degree 262 2 of the end nodes k i k j. As it is seen, except for Netscience, the 263 edge betweenness centrality has almost no correlation with the 264 real weights, whereas, the product of the degrees and the node 265 betweenness centralities showed significant correlation with 266 the real weights. The results indicate that these two measures 267 could be a good candidate for the weights of the edges. 268 This issue is important especially in designing technological 269 networks where the link weights (or loads) are not necessarily 270 the edge betweenness centrality and can be appropriately de- 27 signed. Next we investigate which of the weighting strategies 272 has the best robustness against cascaded failures. 273 IV. NETWORK DATA 274 In this section, the cascaded failure is investigated in 275 artificially constructed model networks as well as in a number 276 of real networks, weighted through different strategies. 277 TABLE I. Person correlation coefficients between real weights and different metrics including the betweenness centrality of the edges B ij, the product of the betweenness centrality of the end nodes, B i B j, and the product of the degree of the end nodes k i k j in a number of real-world networks. Network Correlation with B i B j Correlation with k i k j Correlation with E ij USAir97 0.24 0.28 0.08 USAirport500 0.29 0.6 0.04 Lesmis 0.25 0.36 0.04 Netscience 0.2 0.0 0.9 Bkham 0.54 0.63 0.05 00600-3

EH086 PRE October 9, 20 5:24 MIRZASOLEIMAN, BABAEI, JALILI, AND SAFARI PHYSICAL REVIEW E 00, 00600 (20) 278 279 280 28 282 283 284 285 286 3 287 288 289 290 29 292 293 294 295 296 297 298 299 300 30 302 303 304 305 306 307 308 309 30 3 32 33 34 35 36 37 38 39 320 32 322 323 324 325 326 327 328 329 330 33 332 333 334 335 A. Model networks We considered a number of models to produce artificially constructed networks. Random scale-free networks were generated using the original preferential attachment algorithm proposed by Barabasi and Albert in their seminal paper [22]. Starting with a number of all-to-all connected nodes, the network grows by adding new nodes. These nodes are connected to the old nodes with probability proportional to their degree. The resulting network has degree distribution that is a power law with exponent 3. Scale-free networks are widely observed in natural and man-made systems, including the internet, the world wide web, citation networks, and some social networks. The degree distribution of scale-free networks is heterogeneous; however, many real networks have a homogeneous degree sequence. In order to construct such networks, we considered two other models, namely, Newman-Watts and Erdős-Rényi models. The Newman-Watts networks were constructed as follows [22]. Starting with a regular ring graph with nodes connected to their m nearest neighbors, the nonconnected nodes get connected with a probability P.In order to construct Erdős-Rényi random networks, in a network with N nodes, the nodes are connected with a probability P, where for the values of P = an all-to-all connected network is obtained [22]. It has been shown that many real networks have modular structure [22,35]. We also considered modular networks constructed through an algorithm as follows [20]. First n isolated modules each with preferential attachment scale-free structure are built. Then, with probability P the intramodular links are disconnected and intermodular connections are created. B. Real networks Although model networks are useful in understanding how real systems behave, they cannot capture many of the structural properties of the real networks. Therefore, we also considered a number of real networks and applied the cascaded failure to them. We considered four technological networks where the weights of the links (and the traffic as well) can be designed. US airports network. We analyze the USA airports network containing the 500 busiest commercial airports in the United States [3]. A link between two airports indicates that a flight was scheduled between them in 2002. The network has 500 nodes and 2980 links. Internet. Sometimes, the internet is considered as a network of routers connected by links. Each router belongs to some autonomous system (AS), this dataset simultaneously studies the router and AS level topology; and thus, both routers and ASs are considered as nodes. It contains 2062 nodes and 4233 links [36]. Transportation network. The railway network used in this work is formed by major trains and stations in the central European region [37]. Two stations are considered to be connected by a link when there is at least one vehicle that stops at both stations. The network consists of 2488 nodes and 669 edges. Electrical power grid. This network is an undirected and unweighted network representing the topology of the western states high-voltage power grid of the United States [38]. In construction of the network, the transformers, substations, and 336 generators are considered as nodes, and the links are high- 337 voltage transmission lines. The network is composed of 494 338 nodes and 6549 links. 339 V. SIMULATION RESULTS 340 In order to investigate the profile of robustness against 34 cascaded failures, the behavior of the network is studied as a 342 function of the threshold parameter T. We considered different 343 weighting strategies in the networks as follows: 344 (i) w ij = (B i B j ) θ, where w ij is the weight of edge e ij and 345 B i is the betweenness centrality of nodes i, 346 (ii) w ij = B ij, where B ij is edge betweenness of edge e ij, 347 (iii) w ij = (k i k j ) θ, where k i is the degree of nodes i. 348 In order to study the effect of a small initial attack on 349 the cascading model, we cut an edge e ij and computed the 350 number of broken edges once the process of cascading failure 35 is over. Then, we computed the expected value according to 352 the following formula: 353 S N = i j s ij, (8) E where E is the number of edges in the network, s ij is the 354 normalized avalanche size (i.e., the number of broken edges) 355 by cutting edge e ij and S N is averaged over all normalized 356 avalanche sizes. Note that as one edge gets broken and its 357 weight is locally redistributed, other edges get broken if their 358 new load becomes more than their capacity that is proportional 359 to their initial weight. In other words, the initial load of the 360 edge is considered to be its weight. 36 Let us first study the influence of the parameter θ on the 362 critical threshold T c. It has been shown that the values of θ 363 = are optimal for the weighting method [Eq. (3)], where the 364 weights of the edges are the product of the degrees of the end 365 nodes [2]. Figure shows T c as a function of θ in various 366 network structures with the weights assigned as the method 367 () (i.e., the product of the betweenness centralities of the 368 end nodes). This figure shows that weighting networks with 369 any structure considering θ = results in optimal T c. Although 370 with θ, which assigns stronger weights to the links between 37 central nodes, it might make the networks more robust against 372 cascading failure for smaller values of T ; it increases the 373 critical threshold T c that is not desired. Since optimal T c is 374 important in the robustness of the networks against cascaded 375 failures, we adopted θ = for the numerical simulations. Next 376 we also derive some theoretical foundation for this choice for 377 networks of any structure. 378 In order to avoid cascading failure in a network, the flow 379 passing from each edge after flow redistribution should remain 380 less than its capacity [see Eq. (2)]. From Eqs. () (2) and using 38 the weighting method as w ij = (B i B j ) θ, one can derive 382 4 (B i B j ) θ (B i B m ) θ a Ɣ i (B i B a ) θ + b Ɣ j (B j B b ) + (B ib θ m ) θ <T(B i B m ) θ (9) where Ɣ i and Ɣ j are the set of neighbors of nodes i and j, 383 respectively. 384 00600-4

EH086 PRE October 9, 20 5:24 CASCADED FAILURES IN WEIGHTED NETWORKS PHYSICAL REVIEW E 00, 00600 (20) From Eqs. (0) and () wehave 408 B max (B i B a ) θ = Bi θ k i B θp(b ) = Bi θ k i B θ. (2) a Ɣ i B =B min Therefore, one may rewrite Eq. (9)as 409 B θ i Bθ j B θ Bi θk i + Bj θk j + <T (3) Using geometric inequality 40 Bi θ k i + Bj θ k j (B i B j ) (θ)/2, (4) Eq. (3) can be approximated as 4 (B i B j ) θ/2 2 + <T (5) k i k j B θ 385 386 387 388 389 390 39 392 393 394 395 396 397 398 399 400 40 402 403 404 405 406 407 FIG.. (Color online) The critical threshold T c as a function of θ for different average degrees k on (a) scale-free networks with 000 nodes, (b) Newman-Watts networks with 000 nodes and p = 0.3, (c) Erdős-Rényi network with 000 nodes, and (d) modular networks that has three modules with 200, 300, and 500 nodes. Data shows averages over 0 realizations. Now let us define (PB B i ) as the conditional probability that a node with betweenness centrality B is connected to a node with betweenness centrality B i. Using Bayes rule [39] one can get B max (B i B a ) θ = Bi θ k i P (B B i )B θ (0) a Ɣ i B =B min where B max and B min are the maximum and minimum node betweenness centralities and k i is the number of neighbors of node i. It has been shown that networks constructed through preferential attachment and Newman-Watts algorithms do not show assortative or disassortative behavior (i.e., no degree-degree correlations) [40,4]. Similarly we numerically computed the betweenness-betweenness correlations for the networks. The scale-free networks were constructed with N = 000 and m = 3, and the Newman-Watts networks with N = 000, m = 3, and P = 0.. Then, the correlation coefficient among the betweenness centrality of the nodes was computed. Averaging over 00 realizations, the betweenness-betweenness correlations were obtained as -0.03 for scale-free networks and 0.0 for Newman-Watts networks. Similar results were obtained for networks of other sizes and topological parameters (data not shown here). Therefore, these networks do not show any significant betweenness-betweenness correlations, and thus, one can write P (B B i ) = P (B ). () Noting that Eq. (5) gives us a greater estimation for T c and 42 we can use it instead of Eq. (3). 43 Assuming θ ineq.(5), we can derive the solution for 44 T c by replacing B i and B j with B max as 45 (B max B max ) θ/2 B θ max( (B ib j ) θ/2 B θ ) <T B θ max B θ = B θ max N N i= Bθ i N B max B max N N i= B i N i= B i(b θ i /Bmax θ ) B max B = T c(θ = ) (6) where B max is the maximum node betweenness centralities. As 46 we can see, T c (θ) T c (θ = ). 47 For θ 0, we can derive the solution for T c by replacing 48 B i and B j with B min in Eq. (5). 49 (B min B min ) θ/2 B θ max( (B ib j ) θ/2 B θ ) <T Bmin θ B θ = Bmin θ 2 k i k N j N i= Bθ i N N i= (Bθ i /Bθ max ) k i + k j = T c (θ = 0) (7) As we can see, T c (θ) T c (θ = ) and T c (θ<0) 420 T c (θ = 0). Apparently, the system reaches its strongest 42 robustness level at 0 <θ<. 422 Although we cannot derive an equation for approximating 423 the T c for θ [0,], we can estimate the T c for θ [0,] 424 numerically. As Fig. shows, the strongest robustness level 425 results from considering θ =. 426 00600-5

EH086 PRE October 9, 20 5:24 MIRZASOLEIMAN, BABAEI, JALILI, AND SAFARI PHYSICAL REVIEW E 00, 00600 (20) TABLE II. Critical threshold of scale-free network with N = 000 nodes and m = 3, Newman-Watts network with N = 000 nodes, k = 3, and p = 0.3, Erdős-Rényi network with N = 000 nodes and p = 0.006, and modular network that has three modules with 200, 300, 500 nodes and 3000 edges. The results are shown at θ = for weighting strategies 3. Each data point is averaged over 0 different network realizations. Network w ij = B i B j w ij = B ij w ij = k i k j Scale-free.09.42.09 Newman-Watts.08.74.2 Erdős-Rényi.7.223.29 Modular scale-free.093.85.3 427 428 429 430 43 432 433 434 435 436 437 438 439 440 44 442 443 444 445 446 447 448 449 450 45 452 453 454 455 456 457 458 459 460 46 462 463 464 465 466 467 468 469 470 47 To confirm that our weighting strategy (strategy ) can result in the strongest robustness of the network with any structure against random or systematic failures, we compared the critical threshold of our weighting method at θ = with the other strategies (note that the θ = has been previously obtained as the optimal case for the weighting strategy 3 [2]). The critical thresholds for each weighting strategy are listed in Table II. As it is seen the weighting based on the product of the betweenness centrality of the end nodes resulted in the least critical threshold for these networks (the less the critical threshold of a network is, the more desirable its behavior is against cascaded failures). In other words, the strongest level of robustness for the networks results from the weighting strategy, and hence, weighting strategy results in the strongest level of robustness for any choice of parameter T. We investigated the profile of cascaded failure in model networks including scale-free, Newman-Watts, Erdős-Rényi, and modular scale-free networks. Figure 2 shows the value of S N as a function of the threshold parameter for networks with size N = 000. These networks were weighted with the three weighting strategies discussed above. To have more reliable statistics, each simulation was repeated 0 times and the average was shown. As it is seen, the networks weighted through the product of the node betweenness centralities (i.e., the weighting strategy ) have the best robustness against cascaded failure regardless of their structure. In other words, for a fixed value of the threshold T,theS N for the networks weighted based on strategy is smaller than the other two cases; the smaller the value of S N is, the greater the robustness of the network against the cascaded failure is. Interestingly, the difference between the profiles of various weighting methods was more pronounced for scale-free and modular scale-free networks as compared to Newman-Watts and random graphs (Fig. 2). This can be explained by the fact that unlike random and small-world networks that have homogeneous betweenness centrality distribution, scale-free networks have heterogeneous distribution of betweenness centrality. Thus, there are some nodes that have much higher betweenness centrality than other nodes in such networks. This means a high number of shortest paths in the network passing through these central nodes, and thus, extremely overloading them. Consequently, assigning more weight to the links between these central nodes is a more realistic solution and significantly improves the robustness of the network against random or systematic failures. This explanation may FIG. 2. (Color online) Normalized average size of the removed edges (S N ) as a function of the threshold parameter (T ) for (a) scalefree network with 000 nodes and m = 3, (b) Newman-Watts network with 000 nodes, k = 3, and p = 0.3, (c) Erdős-Rényi network with 000 nodes and p = 0.006, and (d) modular networks that has three modules with 200, 300, and 500 nodes and 3000 edges. The red triangle, blue square and green circle lines show the changes in thes N for weighing methods based on edge betweenness centrality, degree multiplication, and multiplication of node betweenness centrality, respectively. Data shows averages over 0 realizations. raise this question that if assigning higher weights to those 472 edges whose end nodes have higher betweenness centrality 473 improves the robustness of the network against cascading 474 failure, why doesn t increasing θ in Eqs. (6) and (7) have 475 the same effect. Indeed, increasing θ makes the network more 476 robust for T<T c (results not shown here); however, as shown 477 in Fig., T c increases by increasing θ that is not desirable. 478 FIG. 3. (Color online) S N as a T for the US airports network. The cyan diamond line shows the changes in thes N for real weights (RW). Other designations are as in Fig. 2. 00600-6

EH086 PRE October 9, 20 5:24 CASCADED FAILURES IN WEIGHTED NETWORKS PHYSICAL REVIEW E 00, 00600 (20) 479 480 48 482 483 484 485 486 487 488 489 490 49 492 493 494 495 496 497 FIG. 4. (Color online) S N as a T for the central European rail network. Other designations are as in Fig. 3. The cascaded failure process was also investigated in a number of real-world technological networks, where such a failure can break down the whole network. Such cascaded failures can be due to failure in a central station in the rail transportation network or in a central airport in the network of the airports. In the internet breaking down, a few optical cables can lead to congestion in many other links, and consequently, cascaded failures in the network and significantly delayed information transmission. In a power grid, failure of a single transformer that has a central role in the network may overload the nearby elements and causing the entire system to collapse in a very short time. Figure 3 shows the S N for the US airports network. Since the original version of this network is weighted, we also considered the weighted US airports network and ran the cascaded failure process. As it is seen, the network weighted based on the product of the betweenness centrality of the end nodes has the best robustness among different weighting methods (Fig. 3). For example, for the values of the threshold FIG. 6. (Color online) S N as a T for the electrical power grid. Other designations are as in Fig. 2. as T =.003 05, if the US airports network is weighted 498 according to the product of the betweenness centrality of 499 the end nodes, the average avalanche size is S N = 0.2, 500 while S N 0.6 for other weighting strategies (Fig. 3). The 50 network of the US airports has power-law degree distribution 502 [3] and its profile against cascaded failures, as shown in 503 Fig. 3, is similar to the one obtained for scale-free networks 504 [Fig. 2(a)]. 505 Similar patterns were observed for other real networks 506 including the central European rail network (Fig. 4), the 507 internet in the level of an autonomous system (Fig. 5), and 508 the power grid (Fig. 6). In all these networks, the weighting 509 strategy based on the product of the betweenness centrality 50 of the end nodes resulted in the best robustness against 5 cascaded failures. The European rail network and the internet 52 in the level of an autonomous system are scale-free in their 53 degree distribution, and thus, their profile is similar to that 54 of scale-free networks [Fig. 2(a)]. While, the power grid has 55 Poissonian degree distribution, similar to small-world and 56 random networks, and its behavior (Fig. 6) is also similar 57 to these networks [Figs. 3(b) and 3(c)]. In any of these 58 examples, increasing the weights of the links connecting the 59 most central nodes improves the robustness of the network 520 against cascading failures. 52 This indicates that the robustness of a network with any 522 structure against cascaded attacks can be substantially im- 523 proved by assigning proper weights for the links. Considering 524 this issue while designing networks will make them more 525 robust against cascading failures. 526 FIG. 5. (Color online) S N as a T for the network of the internet in the level of autonomous system. Other designations are as in Fig. 2 VI. CONCLUSION 527 Networks can undergo random or systematic failures in 528 their components. If the components have limited capacity, 529 these failures might lead to a cascade of failures throughout 530 the network and make it lose its proper functionality. In 53 this article, we investigated the profile of the robustness 532 against cascaded failures in weighted networks. A number 533 of model networks as well as real-world networks were 534 considered and weighted through different strategies including 535 00600-7

EH086 PRE October 9, 20 5:24 MIRZASOLEIMAN, BABAEI, JALILI, AND SAFARI PHYSICAL REVIEW E 00, 00600 (20) 536 537 538 539 540 54 542 543 544 545 546 547 548 549 550 55 552 553 the betweenness centrality of the edges, the product of the degrees of the end nodes, and the product of the betweenness centrality of the end nodes. Furthermore, the load of the edges was considered to be as their weights and their capacity as a functional of the initial loads (i.e., the initial weight multiplied by some threshold value). By employing the local weighted flow redistribution rule (i.e., redistribution of the load of the broken edge among the neighboring edges proportional to their load) we investigated the average avalanche size and the critical threshold in the networks weighted through different strategies. We found that the networks weighted through the product of the betweenness centrality of the nodes had the best robustness both in case of avalanche size and the critical threshold. Observing this phenomenon in all networks models, the effectiveness of this weighting strategy in improving the robustness of the networks was more pronounced in scale-free networks as compared to small-world and random ones. The results obtained in this paper have immediate applica- 554 tions in designing networks where the capacity of the links can 555 be designed apriori(having the topology of the connections 556 fixed). This is the case for many engineering networks 557 such as those used in water distribution, transportation, and 558 communication networks. Our findings revealed that taking the 559 capacity of the links as the multiplication of the betweenness 560 of the end nodes is better than taking the multiplication of 56 the degrees or the betweenness of the links. The weights 562 based on the node betweenness measures are in favor of 563 robustness against cascaded failures. 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