- Email: [email protected]
Abnormal cascading failure spreading on complex networks
Chaos, Solitons and Fractals 91 (2016) 695–701
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Abnormal cascading failure spreading on complex networks Jianwei Wang a, Enhui Sun a, Bo Xu a,∗, Peng Li b, Chengzhang Ni a a b
School of Business Administration, Northeastern University, Shenyang 110819, PR China Department of Mathematics, Zhongshan High School of Northeast, Shenyang 110819, PR China
a r t i c l e
i n f o
Article history: Received 10 November 2015 Revised 22 March 2016 Accepted 21 August 2016
Keywords: Cascading failure Node weight Betweenness Robustness
a b s t r a c t Applying the mechanism of the preferential selection of the flow destination, we develop a new method to quantify the initial load on an edge, of which the flow is transported along the path with the shortest edge weight between two nodes. Considering the node weight, we propose a cascading model on the edge and investigate cascading dynamics induced by the removal of the edge with the largest load. We perform simulated attacks on four types of constructed networks and two actual networks and observe an interesting and counterintuitive phenomenon of the cascading spreading, i.e., gradually improving the capacity of nodes does not lead to the monotonous increase in the robustness of these networks against cascading failures. The non monotonous behavior of cascading dynamics is well explained by the analysis on a simple graph. We additionally study the effect of the parameter of the node weight on cascading dynamics and evaluate the network robustness by a new metric. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Modern human societies are supported by various functional networks, such as power grids, the Internet, and traffic networks. The safety of these networks has been one of the classical research topics [1–9]. In particular, many researchers focus on the robustness of real networks against cascading failures [10–17]. Cascading failures are a sort of phenomena that a random failure or intentional attack on one or a few nodes triggers successive breakdowns and leads to serious damage to the whole network. Some typical real-world examples of cascading failures are the large-scale blackouts in some countries [10–15], e.g., the blackouts of America in 2003, Italy in 2003, London in 2003, and northern India in 2012. In addition, frequent traffic paralysis in some big cities and Internet collapse [16,17] are also caused by cascading failures. In order to mitigate and prevent various cascading-failure-induced disasters in the real world, many researchers investigated a number of important aspects of cascading failures, including the models for describing the cascade phenomena [18–24], the efficiency of random or targeted attacks [25–31], the cascade control and defense strategies [32–39], cascading failures in the multiplex networks [40–42], the percolation in the interdependent networks [43–52], and so on. In many infrastructure networks, some sort of flow is often required to realize its functionality and at the same time the flow plays a role of a load in the network, such as traffic flow in the ∗
Corresponding author. E-mail address: [email protected] (J. Wang).
http://dx.doi.org/10.1016/j.chaos.2016.08.007 0960-0779/© 2016 Elsevier Ltd. All rights reserved.
traffic network, electric current in a power grid, and data packet on the Internet [33]. Therefore, in previous studies on cascading failures, how to quantify the load on a node or an edge is the central issue. In earlier studies, the initial load on a node or an edge was generally given by the betweenness centrality of the node or the edge. Applying the betweenness centrality, the pioneering work by Motter et al. [53] discuss cascade-based attacks on complex networks and demonstrate that the initial removal of the highest degree (or highest load) node leads a large-scale cascade and scale-free networks are more fragile against cascading overload failures than homogeneous networks. Based on the strategy of the intentional removals of nodes and edges before the propagation of the cascade, Motter [54] propose a simple method to reduce the size of cascades of overload failures and show that the size of the cascade can be drastically reduced with the intentional removals of nodes having small load and/or edges having large excess of load. Defining the load on a node by the efficient paths, Crucitti et al. [55] propose a simple model for cascading failures and show how the breakdown of a single node is sufficient to collapse the entire system simply because of the dynamics of redistribution of flows on the network. Although the betweenness method can be widely applied to define the initial load on a node or an edge, it may be invalid for quantifying the flow of physical quantities in real networks. Since only one unit of the data packet between any two nodes is transported along with the shortest path, the betweenness method cannot thus approximate the traffic load in real networks, ignoring the differences among nodes, the weight of every edge, and the preferential characteristic of the destination
696
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
selection of the flow. Therefore, we propose an improved betweenness measurement and construct a simple cascading model. Considering the preferential mechanism of the flow transportation and the weights of nodes and edges, we analyze cascading dynamics on edges and develop a new method to assign the initial load of an edge. We propose a simple cascading model and study cascading failures on four artificial networks and two empirical networks. By disturbing the edge with the highest load, we observe an interesting and counterintuitive phenomenon in the cascading propagation. In these networks including the scale-free networks, the small-world network, the regular network, the random network, the traffic network, and the airport network, we find that, sometimes, improving the capacity of every edge inversely induces the robustness of these networks against cascading failures, evaluated by four metrics. We observe that the weights of nodes and edges have not affect the emergence of the abnormal cascading spreading in these networks. The non monotonous behavior of cascading dynamics is well explained by the analysis on a simple graph. Finally, we evaluate the network robustness by a new metric and give the correlation between the parameter of the node weight and the new metric.
2. The model In previous studies, the initial load on a node or an edge is given by the betweenness centrality of the node or the edge, i.e., the initial load of a node or an edge is naturally defined to the total number of shortest paths passing through it. However, in previous cascading models [53–55] based on the betweenness method, network weights have not been taken into consideration, regardless of the facts that real networks display a large heterogeneity in the weights which have a strong correlation with the network topology. For example, in the airport network, the number of the traveling passengers on every airport may be different, and these passengers do not randomly select the destinations. Similarly, in the Internet, the data packets generated by every router are also different, and they select the destinations according to some rules. Motivated by this fact, we propose a new method to define the initial load of an edge. Firstly, considering the heterogeneity in the capacity of nodes, we define the weight (or strength) of a node according to its local characteristics. Inspired by Refs. [56–60], we assume the weight wi of node i to kα , where α is a tunable weight parameter, i governing the strength of the node weight, and ki is the degree of node i. This assumption is supported by empirical evidence of real weighted networks [58,59], i.e., the bigger the node degree, the higher the node weight. Moreover, the ref [60] use this method to define the load or weight of a node. Based on the node weight, we study the preferential mechanism of the destination selection of flows. We use Fi → to denote the flow generated by node i. For simplicity, we assume Fi→ = wi . We use Fi → j to denote the flow transported from node i to node j. In Fi → , we assume that the flow transported from node i to node j (i = j) is proportional to the weight of node j, i.e.,
Fi→ j = Fi→
m∈N
wj , wm − wi
(1)
where N is the set of nodes in a network. Since the flow via the edge plays a role of a load, we focus on the effect of the flow transported between two nodes on the edges. We assume that the flow is transmitted along the shortest paths with the edge weight connecting two nodes. Without loss of generality, in later simulations, the weight of every edge is assigned by random numbers of the uniform distribution. If there is more than one shortest path with the edge weight connecting two
given nodes, the flow is divided evenly at each branching point (see Fig. 1). The initial load of an edge is defined as the amount of the flow between pairs of nodes that run through that edge. In Fig. 2, by the total amount of the flow passing through a given edge, we calculate the initial load Lij of edge ij, i.e.,
Li j =
Li(jm,n ) ,
(2)
m∈N,n∈N
where Li(jm,n ) denotes the load via edge ij among the flow transported between the ordered node pair m and n. In particular, when α = 0 and the weight of every edge is same, our method to define the initial load is as same as one of the betweenness centrality [53–55]. Following Refs. [13–15,18,19,26,27,53–55], we assume the capacity Cij of edge ij to be proportional to its initial load Lij .
Ci j = (1 + β )Li j ,
(3)
where the parameter β ≥ 0 is the tolerance parameter. This is a realistic assumption in real networks, since the capacity cannot be infinitely large because it is limited by the cost. With such a definition of capacity, initially the network is in a stationary state in which the load at each edge is not bigger than its capacity. The removal of an edge, changing the topological structure and the path lengths among some nodes, destroy the balance of the load and lead to a global redistribution of loads in the network. After updating the load of every remaining edge, some edges of overloads will be removed from the network, since their limited capacities are insufficient to handle the extra load. This leads to a new redistribution of loads and triggers a cascade of overload failures. This cascading process stops only when the capacity of every remaining edge is not smaller than its updated load. After the cascading propagation is over, we calculate the number G of nodes in the largest connected component, the number SE of failed edges, and the avalanche size SN of failed nodes. We additionally propose a new metric SC , i.e., the number of the connected component (there are at least two nodes in every connected component). SC can quantify the degree of fragmentation of the whole network. In later simulations, we use G, SE , SN , and SC to evaluate the network robustness against cascading failures. 3. The analysis of the cascading model Since the network structure plays an important role in the cascading propagation, we first select four types of classical artificial networks to study cascading dynamics on them. These networks are the BA scale-free networks [61], the Ring networks, the WS small-world networks [62], and the ER random networks. BA networks can be constructed as follows: starting from m0 fully connected nodes, a new node with m(m ≤ m0 ) edges is added to the existing network at each time step according to the preferential attachment, i.e., the probability of being connected to the existing node i is proportional to its ki . We set N = 200 and m0 = 2, m = 2, i.e., the average degree k is about 4. In the Ring network, the starting point is a N nodes ring, in which each node is symmetrically connected to its 2m nearest neighbors for a total of m × N edges. Here, we also set N = 200 and m = 2, i.e., the average degree k is about 4. WS networks can be constructed as follows: starting from a ring network with 200 nodes and 4 edges per node, we rewire each edge at random with the probability p. When p = 0.01, WS networks have both the small-world property and a high clustering coefficient. Therefore, in simulations, we set p = 0.01 in WS networks. To compare the effect of the network structure on cascading dynamics, we generate ER network with 200 nodes and k = 4. We propose a new method as follows: starting from a ring network with 200 nodes and 4 edges per node, we rewire each edge at random with probability p = 1.
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
697
Fig. 1. The effect of the flow transported between two nodes along their shortest weight paths on the edges.
Fig. 2. Calculation of the total amount of the flow passing through a given edge according to the transportation rule of the flow transported between two nodes.
According to the number of edges in every network, we generate one set of corresponding random numbers (values between 10 and 20) which obey the uniform distribution. We use these random numbers to assign the weight of every edge in this network. We calculate the initial load of every edge in a network and remove the edge with the highest load. By four measures (G, SE , SN , and SC ), in Fig. 3 we study the cascading dynamics in constructed networks when α = 0, α = 0.5, α = 1.0, and α = 1.5. Our main aim is to investigate whether there exists a positive relationship between the capacity parameter β and the robustness of a network. Therefore, each data point is obtained by only one realization on one generated network, applying one set of random numbers of the uniform distribution. As the value of β increases, we find an interesting and counterintuitive phenomenon, i.e., the abnormal spread of the cascading propagation after removing one edge with the highest load. In general, as the value of β increases, the capacity of each edge is stronger, i.e., the robustness of the whole network against cascading failures should be also stronger gradually. However, we difficultly obtain the correlation between the network robustness and the value of β . In BA scale-free networks, as β increases from 0 to 1, we find that values of G, SE , SN , and SC fluctuates dramatically. Too many irregular data points show that, sometime, investing the higher resources to protect a network may lead to more damage than investing the lower resources. Similarly, in the ring network, the WS smaller world network, and the ER random network, we observe the same phenomenon. In the ring network, because the degree of every node is same, several curves coincide with each other. In four typical types of constructed networks, we have observed the abnormal oscillation of cascading dynamics. We guess that
such counterintuitive phenomenon is universal in many networks. We further explore cascading dynamics in one traffic network and one airport network. We gather the data of the traffic network (TN) in Shenyang (the capital of Liaoning Province, China), of which the nodes and the edges are the main traffic junctions and the arterial roads in the third ring in Shenyang, respectively. The dataset of the network of airports in the United States was earlier used in Colizza et al. [63], including 500 nodes (the 500 busiest commercial airports in the United States) and 5960 edges (an edge exists between two airports if a flight was scheduled between them in 2002). In numerical simulations, each data point is the result of one realization on one generated network, applying one set of random numbers of the uniform distribution. Similar to Fig. 3, when α = 0, α = 0.5, α = 1.0, and α = 1.5, in Fig. 4 we also find the abnormal cascading propagation in two actual networks with the increase of the value of β from 0 to 1. In the traffic network, as the value of β increases gradually, all curves show the obvious and wide fluctuations, for example, when α = 0, as the value of β increases from 0.14 to 0.15, the corresponding G firstly and sharply increases from 41 to 65. And then, with the increase of the value of β from 0.15 to 0.19, the corresponding G quickly decreases from 65 to 37. When the value of β increases from 0.19 to 0.25, the corresponding G sharply increases from 37 to 74, and then G quickly decreases from 74 to 47, corresponding to the value of β from 0.25 to 0.28. These irregular data points show that we difficultly adopt some effective method to enhancing the robustness of these networks against cascading failures. In the airport network in US, we also observe some abnormal data points of the measure G. For example, when α = 0, the value of G does not change with the increase of the value of β from 0.24 to 0.34, or from 0.60 to 0.78, or from 0.80 to 0.90; when α = 0.5, the value of G does not change with the increase of the value of β from 0.12 to 0.28, or from 0.40 to 0.52, or from 0.54 to 0.72; when α = 1.0, the value of G does not change with the increase of the value of β from 0.12 to 0.24, or from 0.48 to 0.68, or from 0.76 to 0.86; when α = 1.5, the value of G does not change with the increase of the value of β from 0.34 to 0.52, or from 0.64 to 0.76, or from 0.84 to 0.90. The unchanging G with the increase of the value of β show that more resources invested do not improve the network robustness. Then, next a question arises: what factors can lead to the abnormal cascading spreading? Considering the evolving mechanisms of cascading dynamics, we speculate that some local network structure may lead to the oscillation phenomenon of cascading dynamics. In Fig. 5, we construct a simple network with four subnetworks and assume that the weight of every edge is same. The numbers of nodes in sub-network A, sub-network B, sub-network C, and sub-network D are represented by NA , NB , NC , and ND , respectively. Suppose NB = 3NA , NC = 2NA , and ND = 4NA . We use Lij , Lih , Lhj , Lik , and Lkj to denote the initial load on edges ij, ih, hj, ik, and kj, respectively. The flow quantity generated by every
698
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
Fig. 3. Abnormal phenomenon of the cascading propagation induced by removing an edge with the highest initial load, in four types of constructed networks. We observe the non monotonous behavior of cascading dynamics with the increase of β .
Fig. 4. Abnormal cascading spreading in two actual networks. We observe the non monotonous behavior of cascading dynamics with the increase of β .
node is given to N − 1 (here N = NA + NB + NC + ND ), which can ensure that the amount of the flow transported between any nodes is 1. Thus, by the calculated method to the initial load on each edge in cascading model, we can get Li j = NB × ND , Lih = NB × NC + 0.5NA × NC , Lh j = NC × ND + 0.5NA × NC , Lik =
NB × NA + 0.5NA × NC , Lk j = NA × ND + 0.5NA × NC , i.e., Li j = 12NA2 , Lih = 7NA2 , Lh j = 9NA2 , Lik = 4NA2 , and Lk j = 5NA2 . After edge ij fails, the flow transported between regions A and B will be transmitted ij along two new shortest paths: i↔h↔j and i↔k↔j. We use Lih to represent the recalculated load on edge ih after removing edge ij.
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
a
699
b
β <β <β <β
c
d
β <β <β <β <β
e
β ≤β <β <β <β
f
β <β ≤β <β <β
β <β ≤β <β <β
Fig. 5. A reasonable explanation of the abnormal phenomenon of the cascading propagation induced by removing one edge with the highest load. Given a network with four sub-networks. The numbers of nodes in sub-network A, sub-network B, sub-network C, and sub-network D are represented by NA , NB , NC , and ND , respectively. Suppose j denotes the recalculated load NB = 3NA , NC = 2NA , and ND = 4NA . We use Lij , Lih , Lhj , Lik , and Lkj to denote the initial load on edges ij, ih, hj, ik, and kj, respectively. Lihj,ih,ik,k j on edge hj after edges ij, ih, ik, and kj fail. As the value of β increases, we give the cascading propagation after edge ij fails.
ij
ij
ij
ij
We can get Lih = 13NA2 , Lh j = 15NA2 , Lik = 10NA2 , and Lk j = 11NA2 . We
βihi j
use to denote the minimum value of β required by edge ih to maintain its normal and efficient functioning, after removing edge ij ij ij ij ij. Thus, βih = 6/7, βh j = 6/9, βik = 6/4, and βih = 6/5. We get
βhi jj < βihi j < βki jj < βiki j . (c) We observe the change of the network structure with the increase of β . In our cascading model, the ij capacity of edge mn is Cmn = (1 + β )Lmn . Therefore, when β < βh j , ij G = ND , SE = 4, and SC = 4. (d) As the value of β , when βh j ≤ β < j i j,ih,ik,k j βihi j , edge hj revives and Lihj,ih,ik,k = 8NA2 , where Lh j denotes j
of the flow between two unconnected sub-networks, causing substantially more damage in the whole network. We further investigate how the parameter α of the node weight affect the network robustness against cascading failure. In our model, after the edge mn with the highest load fails, only if the capacity Lij of edge ij (among edges in the remaining network) is smaller than (1 + βimn )Li j , no cascading failure occurs and the sysj tem maintains its normal and efficient functioning. Therefore, we define the proportion Cβ of the minimum cost to
i j∈E,i j=mn
(1 + βimn )Li j j
βimn = 0, j s.t. Lmn −Li j βimn = i j Li j , j
Lmn ≤ Li j ij
the recalculated load on edge hj after edges ij, ih, ik, and kj fail. In this case, G = ND + NC , SE = 3, and SC = 3. (e) Once the value of β increases to the range of [βihi j , βki jj ), edge ih revives immediately
Cβ =
owing to the enough capacity to handle the load on it. Because of the connectivity of edge ih, a large number of flow transported between regions A and B will be transmitted along only one shorti j,ik,k j i j,ik,k j est path i↔h↔j. At this case, Lih = 18NA2 and Lh j = 20NA2 .
where E represents the set of edges in a network. The lower the value of Cβ , the stronger the robustness of the network against cascading failure. In Fig. 6, we plot the proportion Cβ of the minimum cost as a function of the parameter α in four types of constructed networks and two actual networks. Each data point is the result of averaging over 10 independent network realizations by generating 10 sets of numbers in the uniform distribution. We obtain three conclusions. a. In the ring network, the value of Cβ is constant, regardless of the value of the parameter α . Because every node in the ring network has the same degree, the different value of α has not affect the flow transported between any two
(f) Owing to the limited capacities of edges ih and hj, these two edges fail immediately. Thus, G = ND , SE = 4, and SC = 4. From the above analysis (c,d,e,f), we observe that, as the value of β , G first increases and then decreases, while SE and SC first decreases and then increases. Therefore, we conclude that the above abnormal phenomenon of cascading dynamics is mainly originated from the revivals of some edges, which facilitates the transportation
i j∈E,i j=mn Li j
Lmn > Li j , ij (4)
700
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
Fig. 6. Correlation between the minimum cost Cβ and the parameter α in four types of constructed networks and two actual networks.
nodes. Therefore, the value of Cβ is fixed. b. In the WS network and the traffic network (TN), the value of Cβ has almost a negative correlation with α . In these two networks, owing to the smaller difference of the degree in all nodes, the difference of the flow transported between any two nodes is also smaller. Because of the rapid increase of the value of the denominator in the expression (4), the value of the proportion Cβ generally decreases with the increase of α . c. In the BA network, the ER network, and the airport network in the US, the value of Cβ exhibits a minimum value with the increase of α . After an edge fails, assume that the recalculated load on only two edges in a network with the bigger size is larger than their initial load. When α increases in the range of the smaller value, owing to the weaker effect of the smaller α and the increase of the recalculated load on only two edges, the increase of the value of the denominator is generally faster than one of the numerator in the expression (4). Thus, the value of the proportion Cβ decreases with the increase of α . However, when the value of α , the increase of the recalculated load on only two edges may be faster than the one of the numerator in the expression (4). Thus, the value of the proportion Cβ increases with the increase of α . Therefore, in these three networks, the value of Cβ exhibits a minimum value with the increase of α . 4. Conclusion In this paper, by extending the utilization of a classic betweenness method, we propose a new method to calculate the initial load of an edge, taking into account the weights of nodes and edges. We construct a cascading model of the edge and give four metrics to quantify the network robustness against cascading failures. We focus on cascading dynamics of several types of networks induced by removing the edge with the highest load. We observe an interesting and counterintuitive phenomenon, gradually increasing the capacity of every edge inversely weakens the resilience of these networks against cascading failures. The abnormal spreading of cascading failures is independent of the weight parameter of nodes. By a simple network with four sub-networks, we step
by step analyze the cascading propagation by gradually increasing the capacity of every edge and explain the reason of the non monotonous behavior of cascading dynamics. Since the revivals of some edges in the cascading propagation facilitates the transportation of the flow between two unconnected sub-networks, causing substantially more damage in the whole network, we not only protect the edge with the higher load but also prevent the malfunctions of those revival edges. Finally, we investigate the effect of the parameter α of the node weight on the network robustness against cascading failure and give the correlation between the parameter α of the node weight and the proposed metric. It is worth emphasizing that our work gives us useful suggestions on the prevention and mitigation of cascading failures. By excavating the hidden revival edges, we can better protect them to avoid the deeper propagation of cascading overload failures. Furthermore, our findings can provides us a guidance to improve the network robustness under the premise of limited capacity resource.
Acknowledgment The research is supported by the National Natural Science Foundation of China under Grant No. 61673096 and 71501035, the Fundamental Research Funds for the Central Universities under Grant No. 140604005, the Program for New Century Excellent Talents in University under Grant No. NCET-12-0100, Social science project of the Ministry of Education No. 16YJC630118, the National Natural Science Foundation of Liaoning Province under Grant No. 2015020073, and Fundamental Research Funds of Northeastern University under grant No. 22114001.
References [1] Albert R, Jeong H, Barabási AL. Attack and error tolerance in complex networks. Nature 20 0 0;406:378–82.
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701 [2] Holme P, Kim BJ, Yoon CN, Han SK. Attack vulnerability of complex networks. Phys Rev E 2002;65:056109. [3] Hong C, Cao XB, Du WB, Zhang J. The effect of attack cost on network robustness. Phys Scr 2013;87:055801. [4] M Jr R, Derzsy N, Szymanski BK, Korniss G. Building damage-resilient dominating sets in complex networks against random and targeted attacks. Sci Rep 2015;5:8321. [5] Callaway DS, Newman MEJ, Strogatz SH, Watts DJ. Network robustness and fragility: percolation on random graphs. Phys Rev Lett 20 0 0;85(25):5468–71. [6] Shang YL. Impact of self-healing capability on network robustness. Phys Rev E 2015;91:042804. [7] Sheinman M, Sharma A, Alvarado J, Koenderink GH, MacKintosh FC. Inherently unstable networks collapse to a critical point. Phys Rev E 2015;92:012710. [8] Wu J, Barahona M, Tan YJ, Deng HZ. Spectral measure of structural robustness in complex networks. IEEE Trans Syst Man Cybern Syst Humans 2011;41(6):1244–52. [9] Wang XG, Guan SG, Lai CH. Protecting infrastructure networks from cost-based attacks. New J Phys 20 09;11:0330 06. [10] Ricard VS, Martí RC, Bernat CM, Sergi V. Robustness of the european power grids under intentional attack. Phys Rev E 2008;77:026102. [11] Albert R, Albert I, Nakarado GL. Structural vulnerability of the north american power grid. Phys Rev E 2004;69:025103. [12] Tan F, Xia YX, Wei Z. Robust-yet-fragile nature of interdependent networks. Phys Rev E 2015;91:052809. [13] Wang JW, Rong LL. Cascade-based attack vulnerability on the U.S. power grid. Safety Sci 2009;47(10):1332–6. [14] Wang JW, Rong LL. Robustness of the western united states power grid under edge attack strategies due to cascading failures. Safety Sci 2011;49(6):807C812. [15] Wang JW. Robustness of complex networks with the local protection strategy against cascading failures. Safety Sci 2013;53:219–25. [16] Wu JJ, Sun HJ, Gao ZY. Cascading failures on weighted urban traffic equilibrium networks. Phys A 2007;386:407–13. [17] Goh KI, Hahng B, Kim D. Fluctuation-driven dynamics of the internet topology. Phys Rev Lett 2002;88(10):108701. [18] Wang WX, Chen GR. Universal robustness characteristic of weighted networks against cascading failure. Phys Rev E 2008;77:026101. [19] Wu ZX, Peng G, Wang WX, Chan S, Wong EEM. Cascading failure spreading on weighted heterogeneous networks. J Stat Mech 2008:P05013. [20] Lorenz J, Battiston S, Schweitzer F. Systemic risk in a unifying framework for cascading processes on networks. Eur Phys J B 2012;71(4):441–60. [21] Tessone CJ, Garas A, Guerra B, Schweitzer F. How big is too big? critical shocks for systemic failure cascades. J Stat Mech 2013;151(3):765–83. [22] Wang JW, Rong LL. Cascading failures on complex networks based on the local preferential redistribution rule of the load. Acta Phys Sin 2009;58:3714–21. [23] Sansavini G, Hajj MR, Puri IK, Zio E. A deterministic representation of cascade spreading in complex networks. Europhys Lett 20 09;87:480 04. [24] Liu CR, Li DQ. A modeling framework for system restoration from cascading failures. PLoS ONE 2014;9(12):e112363. [25] Shao S, Huang XQ, Stanley HE, Havlin S. Percolation of localized attack on complex networks. New J Phys 2015;17:023049. [26] Zhao L, Park K, Lai YC. Attack vulnerability of scale-free networks due to cascading breakdown. Phys Rev E 2004;70:035101(R). [27] Zhao L, Park K, Lai YC, Ye N. Tolerance of scale-free networks against attack-induced cascades. Phys Rev E 2005;72:025104. [28] Moreira AA, Andrade JS Jr, Herrmann HJ, Indekeu JO. How to make a fragile network robust and vice versa. Phys Rev Lett 2009;102:018701. [29] Li SD, Li LX, Yang YX, Luo Q. Revealing the process of edge-based-attack cascading failures. Nonlinear Dyn 2012;69:837–45. [30] Wang JW, Zhang C, Huang Y, Xin C. Attack robustness of cascading model with node weight. Nonlinear Dyn 2014;78:37–48. [31] Wang JW, Rong LL, Zhang L, Zhang ZZ. Attack vulnerability of scale-free networks due to cascading failures. Phys A 2008;387:6671–8. [32] Zhang SP, Huang ZG, Dong JQ, Eisenberg D, Seager TP, Lai YC. Optimization and resilience of complex supply-demand networks. New J Phys 2015;17:063029. [33] Mizutaka S, Yakubo K. Robustness of scale-free networks to cascading failures induced by fluctuating loads. Phys Rev E 2015;92:012814.
701
[34] Ash A, Newth D. Optimizing complex networks for resilience against cascading failure. Phys A 2007;380:673–83. [35] Wang B, Kim BJ. A high-robustness and low-cost model for cascading failures. Europhys Lett 20 07;78(4):480 01. [36] Wang JW. Mitigation of cascading failures on complex networks. Nonlinear Dyn 2012;70(3):1959–67. [37] Wang JW. Mitigation strategies on scale-free networks against cascading failures. Phys A 2013;392:2257–64. [38] Wang JW. Optimized scale-free networks against cascading failures. Int J Mod Phys C 2012;23(11):1250075. [39] Cao XB, Hong C, Du WB, Zhang J. Improving the network robustness against cascading failures by adding links. Chaos Soliton Fract 2013;57:35C40. [40] Azimi-Tafreshi N, nes JG-G, Dorogovtsev SN. k-core percolation on multiplex networks. Phys Rev E 2014;90:032816. [41] Lee KM, Brummitt CD, Goh KI. Threshold cascades with response heterogeneity in multiplex networks. Phys Rev E 2014;90:062816. [42] Brummitt CD, Kobayashi T. Cascades in multiplex financial networks with debts of different seniority. Phys Rev E 2015;91:062813. [43] Vespignani A. The fragility of interdependency. Nature 2010;464:984–5. [44] Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S. Catastrophic cascade of failures in interdependent networks. Nature 2010;464:1025–8. [45] Parshani R, Buldyrev SV, Havlin S. Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. Phys Rev Lett 2010;105(4):048701. [46] Gao J, Buldyrev SV, Havlin S, Stanley HE. Robustness of a network of networks. Phys Rev Lett 2011;107(19):195701. [47] Li W, Bashan A, Buldyrev SV, Stanley HE, Havlin S. Cascading failures in interdependent lattice networks: the critical role of the length of dependency links. Phys Rev Lett 2012;108:228702. [48] Zhou D, Bashan A, Cohen R, Berezin Y, Shnerb N, Havlin S. Simultaneous firstand second-order percolation transitions in interdependent networks. Phys Rev E 2014;90:012803. [49] Gao JX, Buldyrev SV, Havlin S, Stanley HE. Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Phys Rev E 2012;85:066134. [50] Wang JW, Jiang C, Qian JF. Robustness of interdependent networks with different link patterns against cascading failures. Phys A 2014;393:535–41. [51] Tian M, Wang XP, Dong ZC, Zhu GW, Long JH, Dai DD, et al. Cascading failures of interdependent modular scale-free networks with different coupling preferences. EPL 2015;111:18007. [52] Chen Z, Du WB, Cao XB, Zhou XL. Cascading failure of interdependent networks with different coupling preference under targeted attack. Chaos Solitons Fract 2015;80:7–12. [53] Motter AE, Lai YC. Cascade-based attacks on complex networks. Phys Rev E 2002;66:065102(R). [54] Motter AE. Cascade control and defense in complex networks. Phys Rev Lett 2004;93:098701. [55] Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks. Phys Rev E 2004;69:045104. [56] Boccaletti S, Latora V, Moreno Y, chavez M, Hwang DU. Complex networks: Structure and dynamics. Phys Rep 2006;424:175–308. [57] Barrat A, Barthélemy M, Pastor-Satorras R, Vespignani A. The architecture of complex weighted networks. Proc Natl Acad Sci USA 2004;101:3747. [58] Yook SH, Jeong H, Barabási A-L, Tu Y. Weighted evolving networks. Phys Rev Lett 2001;86:5835. [59] Macdonald PJ, Almaas E, Barabási A-L. Minimum spanning trees of weighted scale-free networks. Europhys Lett 2005;72:308. [60] Wang JW, Xu B, Wu YD. Ability paradox of cascading model based on betweenness. Sci Rep 2015;5:13939. [61] Barabási A-L, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. [62] Watts DJ, Strogatz SH. Collective dynamics of ’small-world’ networks. Nature 1998;393(6684):440–2. [63] Colizza V, Pastor-Satorras R, Vespignani A. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat Phys 2007;3:276–82.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Abnormal cascading failure spreading on complex networks Jianwei Wang a, Enhui Sun a, Bo Xu a,∗, Peng Li b, Chengzhang Ni a a b
School of Business Administration, Northeastern University, Shenyang 110819, PR China Department of Mathematics, Zhongshan High School of Northeast, Shenyang 110819, PR China
a r t i c l e
i n f o
Article history: Received 10 November 2015 Revised 22 March 2016 Accepted 21 August 2016
Keywords: Cascading failure Node weight Betweenness Robustness
a b s t r a c t Applying the mechanism of the preferential selection of the flow destination, we develop a new method to quantify the initial load on an edge, of which the flow is transported along the path with the shortest edge weight between two nodes. Considering the node weight, we propose a cascading model on the edge and investigate cascading dynamics induced by the removal of the edge with the largest load. We perform simulated attacks on four types of constructed networks and two actual networks and observe an interesting and counterintuitive phenomenon of the cascading spreading, i.e., gradually improving the capacity of nodes does not lead to the monotonous increase in the robustness of these networks against cascading failures. The non monotonous behavior of cascading dynamics is well explained by the analysis on a simple graph. We additionally study the effect of the parameter of the node weight on cascading dynamics and evaluate the network robustness by a new metric. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Modern human societies are supported by various functional networks, such as power grids, the Internet, and traffic networks. The safety of these networks has been one of the classical research topics [1–9]. In particular, many researchers focus on the robustness of real networks against cascading failures [10–17]. Cascading failures are a sort of phenomena that a random failure or intentional attack on one or a few nodes triggers successive breakdowns and leads to serious damage to the whole network. Some typical real-world examples of cascading failures are the large-scale blackouts in some countries [10–15], e.g., the blackouts of America in 2003, Italy in 2003, London in 2003, and northern India in 2012. In addition, frequent traffic paralysis in some big cities and Internet collapse [16,17] are also caused by cascading failures. In order to mitigate and prevent various cascading-failure-induced disasters in the real world, many researchers investigated a number of important aspects of cascading failures, including the models for describing the cascade phenomena [18–24], the efficiency of random or targeted attacks [25–31], the cascade control and defense strategies [32–39], cascading failures in the multiplex networks [40–42], the percolation in the interdependent networks [43–52], and so on. In many infrastructure networks, some sort of flow is often required to realize its functionality and at the same time the flow plays a role of a load in the network, such as traffic flow in the ∗
Corresponding author. E-mail address: [email protected] (J. Wang).
http://dx.doi.org/10.1016/j.chaos.2016.08.007 0960-0779/© 2016 Elsevier Ltd. All rights reserved.
traffic network, electric current in a power grid, and data packet on the Internet [33]. Therefore, in previous studies on cascading failures, how to quantify the load on a node or an edge is the central issue. In earlier studies, the initial load on a node or an edge was generally given by the betweenness centrality of the node or the edge. Applying the betweenness centrality, the pioneering work by Motter et al. [53] discuss cascade-based attacks on complex networks and demonstrate that the initial removal of the highest degree (or highest load) node leads a large-scale cascade and scale-free networks are more fragile against cascading overload failures than homogeneous networks. Based on the strategy of the intentional removals of nodes and edges before the propagation of the cascade, Motter [54] propose a simple method to reduce the size of cascades of overload failures and show that the size of the cascade can be drastically reduced with the intentional removals of nodes having small load and/or edges having large excess of load. Defining the load on a node by the efficient paths, Crucitti et al. [55] propose a simple model for cascading failures and show how the breakdown of a single node is sufficient to collapse the entire system simply because of the dynamics of redistribution of flows on the network. Although the betweenness method can be widely applied to define the initial load on a node or an edge, it may be invalid for quantifying the flow of physical quantities in real networks. Since only one unit of the data packet between any two nodes is transported along with the shortest path, the betweenness method cannot thus approximate the traffic load in real networks, ignoring the differences among nodes, the weight of every edge, and the preferential characteristic of the destination
696
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
selection of the flow. Therefore, we propose an improved betweenness measurement and construct a simple cascading model. Considering the preferential mechanism of the flow transportation and the weights of nodes and edges, we analyze cascading dynamics on edges and develop a new method to assign the initial load of an edge. We propose a simple cascading model and study cascading failures on four artificial networks and two empirical networks. By disturbing the edge with the highest load, we observe an interesting and counterintuitive phenomenon in the cascading propagation. In these networks including the scale-free networks, the small-world network, the regular network, the random network, the traffic network, and the airport network, we find that, sometimes, improving the capacity of every edge inversely induces the robustness of these networks against cascading failures, evaluated by four metrics. We observe that the weights of nodes and edges have not affect the emergence of the abnormal cascading spreading in these networks. The non monotonous behavior of cascading dynamics is well explained by the analysis on a simple graph. Finally, we evaluate the network robustness by a new metric and give the correlation between the parameter of the node weight and the new metric.
2. The model In previous studies, the initial load on a node or an edge is given by the betweenness centrality of the node or the edge, i.e., the initial load of a node or an edge is naturally defined to the total number of shortest paths passing through it. However, in previous cascading models [53–55] based on the betweenness method, network weights have not been taken into consideration, regardless of the facts that real networks display a large heterogeneity in the weights which have a strong correlation with the network topology. For example, in the airport network, the number of the traveling passengers on every airport may be different, and these passengers do not randomly select the destinations. Similarly, in the Internet, the data packets generated by every router are also different, and they select the destinations according to some rules. Motivated by this fact, we propose a new method to define the initial load of an edge. Firstly, considering the heterogeneity in the capacity of nodes, we define the weight (or strength) of a node according to its local characteristics. Inspired by Refs. [56–60], we assume the weight wi of node i to kα , where α is a tunable weight parameter, i governing the strength of the node weight, and ki is the degree of node i. This assumption is supported by empirical evidence of real weighted networks [58,59], i.e., the bigger the node degree, the higher the node weight. Moreover, the ref [60] use this method to define the load or weight of a node. Based on the node weight, we study the preferential mechanism of the destination selection of flows. We use Fi → to denote the flow generated by node i. For simplicity, we assume Fi→ = wi . We use Fi → j to denote the flow transported from node i to node j. In Fi → , we assume that the flow transported from node i to node j (i = j) is proportional to the weight of node j, i.e.,
Fi→ j = Fi→
m∈N
wj , wm − wi
(1)
where N is the set of nodes in a network. Since the flow via the edge plays a role of a load, we focus on the effect of the flow transported between two nodes on the edges. We assume that the flow is transmitted along the shortest paths with the edge weight connecting two nodes. Without loss of generality, in later simulations, the weight of every edge is assigned by random numbers of the uniform distribution. If there is more than one shortest path with the edge weight connecting two
given nodes, the flow is divided evenly at each branching point (see Fig. 1). The initial load of an edge is defined as the amount of the flow between pairs of nodes that run through that edge. In Fig. 2, by the total amount of the flow passing through a given edge, we calculate the initial load Lij of edge ij, i.e.,
Li j =
Li(jm,n ) ,
(2)
m∈N,n∈N
where Li(jm,n ) denotes the load via edge ij among the flow transported between the ordered node pair m and n. In particular, when α = 0 and the weight of every edge is same, our method to define the initial load is as same as one of the betweenness centrality [53–55]. Following Refs. [13–15,18,19,26,27,53–55], we assume the capacity Cij of edge ij to be proportional to its initial load Lij .
Ci j = (1 + β )Li j ,
(3)
where the parameter β ≥ 0 is the tolerance parameter. This is a realistic assumption in real networks, since the capacity cannot be infinitely large because it is limited by the cost. With such a definition of capacity, initially the network is in a stationary state in which the load at each edge is not bigger than its capacity. The removal of an edge, changing the topological structure and the path lengths among some nodes, destroy the balance of the load and lead to a global redistribution of loads in the network. After updating the load of every remaining edge, some edges of overloads will be removed from the network, since their limited capacities are insufficient to handle the extra load. This leads to a new redistribution of loads and triggers a cascade of overload failures. This cascading process stops only when the capacity of every remaining edge is not smaller than its updated load. After the cascading propagation is over, we calculate the number G of nodes in the largest connected component, the number SE of failed edges, and the avalanche size SN of failed nodes. We additionally propose a new metric SC , i.e., the number of the connected component (there are at least two nodes in every connected component). SC can quantify the degree of fragmentation of the whole network. In later simulations, we use G, SE , SN , and SC to evaluate the network robustness against cascading failures. 3. The analysis of the cascading model Since the network structure plays an important role in the cascading propagation, we first select four types of classical artificial networks to study cascading dynamics on them. These networks are the BA scale-free networks [61], the Ring networks, the WS small-world networks [62], and the ER random networks. BA networks can be constructed as follows: starting from m0 fully connected nodes, a new node with m(m ≤ m0 ) edges is added to the existing network at each time step according to the preferential attachment, i.e., the probability of being connected to the existing node i is proportional to its ki . We set N = 200 and m0 = 2, m = 2, i.e., the average degree k is about 4. In the Ring network, the starting point is a N nodes ring, in which each node is symmetrically connected to its 2m nearest neighbors for a total of m × N edges. Here, we also set N = 200 and m = 2, i.e., the average degree k is about 4. WS networks can be constructed as follows: starting from a ring network with 200 nodes and 4 edges per node, we rewire each edge at random with the probability p. When p = 0.01, WS networks have both the small-world property and a high clustering coefficient. Therefore, in simulations, we set p = 0.01 in WS networks. To compare the effect of the network structure on cascading dynamics, we generate ER network with 200 nodes and k = 4. We propose a new method as follows: starting from a ring network with 200 nodes and 4 edges per node, we rewire each edge at random with probability p = 1.
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
697
Fig. 1. The effect of the flow transported between two nodes along their shortest weight paths on the edges.
Fig. 2. Calculation of the total amount of the flow passing through a given edge according to the transportation rule of the flow transported between two nodes.
According to the number of edges in every network, we generate one set of corresponding random numbers (values between 10 and 20) which obey the uniform distribution. We use these random numbers to assign the weight of every edge in this network. We calculate the initial load of every edge in a network and remove the edge with the highest load. By four measures (G, SE , SN , and SC ), in Fig. 3 we study the cascading dynamics in constructed networks when α = 0, α = 0.5, α = 1.0, and α = 1.5. Our main aim is to investigate whether there exists a positive relationship between the capacity parameter β and the robustness of a network. Therefore, each data point is obtained by only one realization on one generated network, applying one set of random numbers of the uniform distribution. As the value of β increases, we find an interesting and counterintuitive phenomenon, i.e., the abnormal spread of the cascading propagation after removing one edge with the highest load. In general, as the value of β increases, the capacity of each edge is stronger, i.e., the robustness of the whole network against cascading failures should be also stronger gradually. However, we difficultly obtain the correlation between the network robustness and the value of β . In BA scale-free networks, as β increases from 0 to 1, we find that values of G, SE , SN , and SC fluctuates dramatically. Too many irregular data points show that, sometime, investing the higher resources to protect a network may lead to more damage than investing the lower resources. Similarly, in the ring network, the WS smaller world network, and the ER random network, we observe the same phenomenon. In the ring network, because the degree of every node is same, several curves coincide with each other. In four typical types of constructed networks, we have observed the abnormal oscillation of cascading dynamics. We guess that
such counterintuitive phenomenon is universal in many networks. We further explore cascading dynamics in one traffic network and one airport network. We gather the data of the traffic network (TN) in Shenyang (the capital of Liaoning Province, China), of which the nodes and the edges are the main traffic junctions and the arterial roads in the third ring in Shenyang, respectively. The dataset of the network of airports in the United States was earlier used in Colizza et al. [63], including 500 nodes (the 500 busiest commercial airports in the United States) and 5960 edges (an edge exists between two airports if a flight was scheduled between them in 2002). In numerical simulations, each data point is the result of one realization on one generated network, applying one set of random numbers of the uniform distribution. Similar to Fig. 3, when α = 0, α = 0.5, α = 1.0, and α = 1.5, in Fig. 4 we also find the abnormal cascading propagation in two actual networks with the increase of the value of β from 0 to 1. In the traffic network, as the value of β increases gradually, all curves show the obvious and wide fluctuations, for example, when α = 0, as the value of β increases from 0.14 to 0.15, the corresponding G firstly and sharply increases from 41 to 65. And then, with the increase of the value of β from 0.15 to 0.19, the corresponding G quickly decreases from 65 to 37. When the value of β increases from 0.19 to 0.25, the corresponding G sharply increases from 37 to 74, and then G quickly decreases from 74 to 47, corresponding to the value of β from 0.25 to 0.28. These irregular data points show that we difficultly adopt some effective method to enhancing the robustness of these networks against cascading failures. In the airport network in US, we also observe some abnormal data points of the measure G. For example, when α = 0, the value of G does not change with the increase of the value of β from 0.24 to 0.34, or from 0.60 to 0.78, or from 0.80 to 0.90; when α = 0.5, the value of G does not change with the increase of the value of β from 0.12 to 0.28, or from 0.40 to 0.52, or from 0.54 to 0.72; when α = 1.0, the value of G does not change with the increase of the value of β from 0.12 to 0.24, or from 0.48 to 0.68, or from 0.76 to 0.86; when α = 1.5, the value of G does not change with the increase of the value of β from 0.34 to 0.52, or from 0.64 to 0.76, or from 0.84 to 0.90. The unchanging G with the increase of the value of β show that more resources invested do not improve the network robustness. Then, next a question arises: what factors can lead to the abnormal cascading spreading? Considering the evolving mechanisms of cascading dynamics, we speculate that some local network structure may lead to the oscillation phenomenon of cascading dynamics. In Fig. 5, we construct a simple network with four subnetworks and assume that the weight of every edge is same. The numbers of nodes in sub-network A, sub-network B, sub-network C, and sub-network D are represented by NA , NB , NC , and ND , respectively. Suppose NB = 3NA , NC = 2NA , and ND = 4NA . We use Lij , Lih , Lhj , Lik , and Lkj to denote the initial load on edges ij, ih, hj, ik, and kj, respectively. The flow quantity generated by every
698
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
Fig. 3. Abnormal phenomenon of the cascading propagation induced by removing an edge with the highest initial load, in four types of constructed networks. We observe the non monotonous behavior of cascading dynamics with the increase of β .
Fig. 4. Abnormal cascading spreading in two actual networks. We observe the non monotonous behavior of cascading dynamics with the increase of β .
node is given to N − 1 (here N = NA + NB + NC + ND ), which can ensure that the amount of the flow transported between any nodes is 1. Thus, by the calculated method to the initial load on each edge in cascading model, we can get Li j = NB × ND , Lih = NB × NC + 0.5NA × NC , Lh j = NC × ND + 0.5NA × NC , Lik =
NB × NA + 0.5NA × NC , Lk j = NA × ND + 0.5NA × NC , i.e., Li j = 12NA2 , Lih = 7NA2 , Lh j = 9NA2 , Lik = 4NA2 , and Lk j = 5NA2 . After edge ij fails, the flow transported between regions A and B will be transmitted ij along two new shortest paths: i↔h↔j and i↔k↔j. We use Lih to represent the recalculated load on edge ih after removing edge ij.
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
a
699
b
β <β <β <β
c
d
β <β <β <β <β
e
β ≤β <β <β <β
f
β <β ≤β <β <β
β <β ≤β <β <β
Fig. 5. A reasonable explanation of the abnormal phenomenon of the cascading propagation induced by removing one edge with the highest load. Given a network with four sub-networks. The numbers of nodes in sub-network A, sub-network B, sub-network C, and sub-network D are represented by NA , NB , NC , and ND , respectively. Suppose j denotes the recalculated load NB = 3NA , NC = 2NA , and ND = 4NA . We use Lij , Lih , Lhj , Lik , and Lkj to denote the initial load on edges ij, ih, hj, ik, and kj, respectively. Lihj,ih,ik,k j on edge hj after edges ij, ih, ik, and kj fail. As the value of β increases, we give the cascading propagation after edge ij fails.
ij
ij
ij
ij
We can get Lih = 13NA2 , Lh j = 15NA2 , Lik = 10NA2 , and Lk j = 11NA2 . We
βihi j
use to denote the minimum value of β required by edge ih to maintain its normal and efficient functioning, after removing edge ij ij ij ij ij. Thus, βih = 6/7, βh j = 6/9, βik = 6/4, and βih = 6/5. We get
βhi jj < βihi j < βki jj < βiki j . (c) We observe the change of the network structure with the increase of β . In our cascading model, the ij capacity of edge mn is Cmn = (1 + β )Lmn . Therefore, when β < βh j , ij G = ND , SE = 4, and SC = 4. (d) As the value of β , when βh j ≤ β < j i j,ih,ik,k j βihi j , edge hj revives and Lihj,ih,ik,k = 8NA2 , where Lh j denotes j
of the flow between two unconnected sub-networks, causing substantially more damage in the whole network. We further investigate how the parameter α of the node weight affect the network robustness against cascading failure. In our model, after the edge mn with the highest load fails, only if the capacity Lij of edge ij (among edges in the remaining network) is smaller than (1 + βimn )Li j , no cascading failure occurs and the sysj tem maintains its normal and efficient functioning. Therefore, we define the proportion Cβ of the minimum cost to
i j∈E,i j=mn
(1 + βimn )Li j j
βimn = 0, j s.t. Lmn −Li j βimn = i j Li j , j
Lmn ≤ Li j ij
the recalculated load on edge hj after edges ij, ih, ik, and kj fail. In this case, G = ND + NC , SE = 3, and SC = 3. (e) Once the value of β increases to the range of [βihi j , βki jj ), edge ih revives immediately
Cβ =
owing to the enough capacity to handle the load on it. Because of the connectivity of edge ih, a large number of flow transported between regions A and B will be transmitted along only one shorti j,ik,k j i j,ik,k j est path i↔h↔j. At this case, Lih = 18NA2 and Lh j = 20NA2 .
where E represents the set of edges in a network. The lower the value of Cβ , the stronger the robustness of the network against cascading failure. In Fig. 6, we plot the proportion Cβ of the minimum cost as a function of the parameter α in four types of constructed networks and two actual networks. Each data point is the result of averaging over 10 independent network realizations by generating 10 sets of numbers in the uniform distribution. We obtain three conclusions. a. In the ring network, the value of Cβ is constant, regardless of the value of the parameter α . Because every node in the ring network has the same degree, the different value of α has not affect the flow transported between any two
(f) Owing to the limited capacities of edges ih and hj, these two edges fail immediately. Thus, G = ND , SE = 4, and SC = 4. From the above analysis (c,d,e,f), we observe that, as the value of β , G first increases and then decreases, while SE and SC first decreases and then increases. Therefore, we conclude that the above abnormal phenomenon of cascading dynamics is mainly originated from the revivals of some edges, which facilitates the transportation
i j∈E,i j=mn Li j
Lmn > Li j , ij (4)
700
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701
Fig. 6. Correlation between the minimum cost Cβ and the parameter α in four types of constructed networks and two actual networks.
nodes. Therefore, the value of Cβ is fixed. b. In the WS network and the traffic network (TN), the value of Cβ has almost a negative correlation with α . In these two networks, owing to the smaller difference of the degree in all nodes, the difference of the flow transported between any two nodes is also smaller. Because of the rapid increase of the value of the denominator in the expression (4), the value of the proportion Cβ generally decreases with the increase of α . c. In the BA network, the ER network, and the airport network in the US, the value of Cβ exhibits a minimum value with the increase of α . After an edge fails, assume that the recalculated load on only two edges in a network with the bigger size is larger than their initial load. When α increases in the range of the smaller value, owing to the weaker effect of the smaller α and the increase of the recalculated load on only two edges, the increase of the value of the denominator is generally faster than one of the numerator in the expression (4). Thus, the value of the proportion Cβ decreases with the increase of α . However, when the value of α , the increase of the recalculated load on only two edges may be faster than the one of the numerator in the expression (4). Thus, the value of the proportion Cβ increases with the increase of α . Therefore, in these three networks, the value of Cβ exhibits a minimum value with the increase of α . 4. Conclusion In this paper, by extending the utilization of a classic betweenness method, we propose a new method to calculate the initial load of an edge, taking into account the weights of nodes and edges. We construct a cascading model of the edge and give four metrics to quantify the network robustness against cascading failures. We focus on cascading dynamics of several types of networks induced by removing the edge with the highest load. We observe an interesting and counterintuitive phenomenon, gradually increasing the capacity of every edge inversely weakens the resilience of these networks against cascading failures. The abnormal spreading of cascading failures is independent of the weight parameter of nodes. By a simple network with four sub-networks, we step
by step analyze the cascading propagation by gradually increasing the capacity of every edge and explain the reason of the non monotonous behavior of cascading dynamics. Since the revivals of some edges in the cascading propagation facilitates the transportation of the flow between two unconnected sub-networks, causing substantially more damage in the whole network, we not only protect the edge with the higher load but also prevent the malfunctions of those revival edges. Finally, we investigate the effect of the parameter α of the node weight on the network robustness against cascading failure and give the correlation between the parameter α of the node weight and the proposed metric. It is worth emphasizing that our work gives us useful suggestions on the prevention and mitigation of cascading failures. By excavating the hidden revival edges, we can better protect them to avoid the deeper propagation of cascading overload failures. Furthermore, our findings can provides us a guidance to improve the network robustness under the premise of limited capacity resource.
Acknowledgment The research is supported by the National Natural Science Foundation of China under Grant No. 61673096 and 71501035, the Fundamental Research Funds for the Central Universities under Grant No. 140604005, the Program for New Century Excellent Talents in University under Grant No. NCET-12-0100, Social science project of the Ministry of Education No. 16YJC630118, the National Natural Science Foundation of Liaoning Province under Grant No. 2015020073, and Fundamental Research Funds of Northeastern University under grant No. 22114001.
References [1] Albert R, Jeong H, Barabási AL. Attack and error tolerance in complex networks. Nature 20 0 0;406:378–82.
J. Wang et al. / Chaos, Solitons and Fractals 91 (2016) 695–701 [2] Holme P, Kim BJ, Yoon CN, Han SK. Attack vulnerability of complex networks. Phys Rev E 2002;65:056109. [3] Hong C, Cao XB, Du WB, Zhang J. The effect of attack cost on network robustness. Phys Scr 2013;87:055801. [4] M Jr R, Derzsy N, Szymanski BK, Korniss G. Building damage-resilient dominating sets in complex networks against random and targeted attacks. Sci Rep 2015;5:8321. [5] Callaway DS, Newman MEJ, Strogatz SH, Watts DJ. Network robustness and fragility: percolation on random graphs. Phys Rev Lett 20 0 0;85(25):5468–71. [6] Shang YL. Impact of self-healing capability on network robustness. Phys Rev E 2015;91:042804. [7] Sheinman M, Sharma A, Alvarado J, Koenderink GH, MacKintosh FC. Inherently unstable networks collapse to a critical point. Phys Rev E 2015;92:012710. [8] Wu J, Barahona M, Tan YJ, Deng HZ. Spectral measure of structural robustness in complex networks. IEEE Trans Syst Man Cybern Syst Humans 2011;41(6):1244–52. [9] Wang XG, Guan SG, Lai CH. Protecting infrastructure networks from cost-based attacks. New J Phys 20 09;11:0330 06. [10] Ricard VS, Martí RC, Bernat CM, Sergi V. Robustness of the european power grids under intentional attack. Phys Rev E 2008;77:026102. [11] Albert R, Albert I, Nakarado GL. Structural vulnerability of the north american power grid. Phys Rev E 2004;69:025103. [12] Tan F, Xia YX, Wei Z. Robust-yet-fragile nature of interdependent networks. Phys Rev E 2015;91:052809. [13] Wang JW, Rong LL. Cascade-based attack vulnerability on the U.S. power grid. Safety Sci 2009;47(10):1332–6. [14] Wang JW, Rong LL. Robustness of the western united states power grid under edge attack strategies due to cascading failures. Safety Sci 2011;49(6):807C812. [15] Wang JW. Robustness of complex networks with the local protection strategy against cascading failures. Safety Sci 2013;53:219–25. [16] Wu JJ, Sun HJ, Gao ZY. Cascading failures on weighted urban traffic equilibrium networks. Phys A 2007;386:407–13. [17] Goh KI, Hahng B, Kim D. Fluctuation-driven dynamics of the internet topology. Phys Rev Lett 2002;88(10):108701. [18] Wang WX, Chen GR. Universal robustness characteristic of weighted networks against cascading failure. Phys Rev E 2008;77:026101. [19] Wu ZX, Peng G, Wang WX, Chan S, Wong EEM. Cascading failure spreading on weighted heterogeneous networks. J Stat Mech 2008:P05013. [20] Lorenz J, Battiston S, Schweitzer F. Systemic risk in a unifying framework for cascading processes on networks. Eur Phys J B 2012;71(4):441–60. [21] Tessone CJ, Garas A, Guerra B, Schweitzer F. How big is too big? critical shocks for systemic failure cascades. J Stat Mech 2013;151(3):765–83. [22] Wang JW, Rong LL. Cascading failures on complex networks based on the local preferential redistribution rule of the load. Acta Phys Sin 2009;58:3714–21. [23] Sansavini G, Hajj MR, Puri IK, Zio E. A deterministic representation of cascade spreading in complex networks. Europhys Lett 20 09;87:480 04. [24] Liu CR, Li DQ. A modeling framework for system restoration from cascading failures. PLoS ONE 2014;9(12):e112363. [25] Shao S, Huang XQ, Stanley HE, Havlin S. Percolation of localized attack on complex networks. New J Phys 2015;17:023049. [26] Zhao L, Park K, Lai YC. Attack vulnerability of scale-free networks due to cascading breakdown. Phys Rev E 2004;70:035101(R). [27] Zhao L, Park K, Lai YC, Ye N. Tolerance of scale-free networks against attack-induced cascades. Phys Rev E 2005;72:025104. [28] Moreira AA, Andrade JS Jr, Herrmann HJ, Indekeu JO. How to make a fragile network robust and vice versa. Phys Rev Lett 2009;102:018701. [29] Li SD, Li LX, Yang YX, Luo Q. Revealing the process of edge-based-attack cascading failures. Nonlinear Dyn 2012;69:837–45. [30] Wang JW, Zhang C, Huang Y, Xin C. Attack robustness of cascading model with node weight. Nonlinear Dyn 2014;78:37–48. [31] Wang JW, Rong LL, Zhang L, Zhang ZZ. Attack vulnerability of scale-free networks due to cascading failures. Phys A 2008;387:6671–8. [32] Zhang SP, Huang ZG, Dong JQ, Eisenberg D, Seager TP, Lai YC. Optimization and resilience of complex supply-demand networks. New J Phys 2015;17:063029. [33] Mizutaka S, Yakubo K. Robustness of scale-free networks to cascading failures induced by fluctuating loads. Phys Rev E 2015;92:012814.
701
[34] Ash A, Newth D. Optimizing complex networks for resilience against cascading failure. Phys A 2007;380:673–83. [35] Wang B, Kim BJ. A high-robustness and low-cost model for cascading failures. Europhys Lett 20 07;78(4):480 01. [36] Wang JW. Mitigation of cascading failures on complex networks. Nonlinear Dyn 2012;70(3):1959–67. [37] Wang JW. Mitigation strategies on scale-free networks against cascading failures. Phys A 2013;392:2257–64. [38] Wang JW. Optimized scale-free networks against cascading failures. Int J Mod Phys C 2012;23(11):1250075. [39] Cao XB, Hong C, Du WB, Zhang J. Improving the network robustness against cascading failures by adding links. Chaos Soliton Fract 2013;57:35C40. [40] Azimi-Tafreshi N, nes JG-G, Dorogovtsev SN. k-core percolation on multiplex networks. Phys Rev E 2014;90:032816. [41] Lee KM, Brummitt CD, Goh KI. Threshold cascades with response heterogeneity in multiplex networks. Phys Rev E 2014;90:062816. [42] Brummitt CD, Kobayashi T. Cascades in multiplex financial networks with debts of different seniority. Phys Rev E 2015;91:062813. [43] Vespignani A. The fragility of interdependency. Nature 2010;464:984–5. [44] Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S. Catastrophic cascade of failures in interdependent networks. Nature 2010;464:1025–8. [45] Parshani R, Buldyrev SV, Havlin S. Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. Phys Rev Lett 2010;105(4):048701. [46] Gao J, Buldyrev SV, Havlin S, Stanley HE. Robustness of a network of networks. Phys Rev Lett 2011;107(19):195701. [47] Li W, Bashan A, Buldyrev SV, Stanley HE, Havlin S. Cascading failures in interdependent lattice networks: the critical role of the length of dependency links. Phys Rev Lett 2012;108:228702. [48] Zhou D, Bashan A, Cohen R, Berezin Y, Shnerb N, Havlin S. Simultaneous firstand second-order percolation transitions in interdependent networks. Phys Rev E 2014;90:012803. [49] Gao JX, Buldyrev SV, Havlin S, Stanley HE. Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Phys Rev E 2012;85:066134. [50] Wang JW, Jiang C, Qian JF. Robustness of interdependent networks with different link patterns against cascading failures. Phys A 2014;393:535–41. [51] Tian M, Wang XP, Dong ZC, Zhu GW, Long JH, Dai DD, et al. Cascading failures of interdependent modular scale-free networks with different coupling preferences. EPL 2015;111:18007. [52] Chen Z, Du WB, Cao XB, Zhou XL. Cascading failure of interdependent networks with different coupling preference under targeted attack. Chaos Solitons Fract 2015;80:7–12. [53] Motter AE, Lai YC. Cascade-based attacks on complex networks. Phys Rev E 2002;66:065102(R). [54] Motter AE. Cascade control and defense in complex networks. Phys Rev Lett 2004;93:098701. [55] Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks. Phys Rev E 2004;69:045104. [56] Boccaletti S, Latora V, Moreno Y, chavez M, Hwang DU. Complex networks: Structure and dynamics. Phys Rep 2006;424:175–308. [57] Barrat A, Barthélemy M, Pastor-Satorras R, Vespignani A. The architecture of complex weighted networks. Proc Natl Acad Sci USA 2004;101:3747. [58] Yook SH, Jeong H, Barabási A-L, Tu Y. Weighted evolving networks. Phys Rev Lett 2001;86:5835. [59] Macdonald PJ, Almaas E, Barabási A-L. Minimum spanning trees of weighted scale-free networks. Europhys Lett 2005;72:308. [60] Wang JW, Xu B, Wu YD. Ability paradox of cascading model based on betweenness. Sci Rep 2015;5:13939. [61] Barabási A-L, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. [62] Watts DJ, Strogatz SH. Collective dynamics of ’small-world’ networks. Nature 1998;393(6684):440–2. [63] Colizza V, Pastor-Satorras R, Vespignani A. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat Phys 2007;3:276–82.