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Effects of link-orientation methods on robustness against cascading failures in complex networks
Physica A 457 (2016) 1–7
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Effects of link-orientation methods on robustness against cascading failures in complex networks Zhong-Yuan Jiang a,b,∗ , Jian-Feng Ma a,b,c , Yu-Long Shen b,c , Yong Zeng a,b a
School of Cyber Engineering, Xidian University, Xi’an, Shaanxi 710071, China
b
Shaanxi Key Laboratory of Network and System Security, Xidian University, Xi’an, Shaanxi 710071, China
c
School of Computer Science and Technology, Xidian University, Xi’an, Shaanxi 710071, China
highlights • Three link orientation methods are discussed and employed in many classical network models. • Traffic capacity improvement and network safety are considered simultaneously. • Links direction-determining methods deteriorate the robustness against cascading failure.
article
info
Article history: Received 28 August 2015 Received in revised form 29 January 2016 Available online 5 April 2016 Keywords: Link orientation Cascade defense Robustness Complex network
abstract Unidirectional and bidirectional links may coexist in many realistic networked complex systems such as the city transportation networks. Even more, for some considerations, several bidirectional links are shifted to unidirectional ones. Many link-orientation strategies might be employed, including High-to-Low, Low-to-High and Random directiondetermining methods, abbreviated as HTLDD, LTHDD and RDD respectively. Traffic passing through a unidirectional link is restricted to one-side direction. In real complex systems, nodes are correlated with each other. The failure from an initial node may be propagated iteratively, resulting in a large scale of failures of other nodes, called cascade phenomenon which may damage the safety or security of the networked system. Assuming that traffic load on any failed node can be redistributed to its non-failed neighbors, in this work, we try to reveal the effects of unidirectional links on network robustness against cascades. Extensive simulations have been implemented on kinds of networks including Scale-Free networks, Small-World networks, and Erdös–Rényi random networks. The results showed that all of the above three direction-determining methods decrease the robustness of the original networks against cascading failure. This work can help network designers and managers understand the robustness of network well and efficiently prevent the safety events. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, complex networks are ubiquitous all over the world and provide us lots of basic requirements in working place or daily life, such as electricity generated in the power grid, information delivered on the Internet, vehicles passed
∗
Corresponding author at: School of Cyber Engineering, Xidian University, Xi’an, Shaanxi 710071, China. E-mail address: [email protected] (Z.-Y. Jiang).
http://dx.doi.org/10.1016/j.physa.2016.03.107 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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Z.-Y. Jiang et al. / Physica A 457 (2016) 1–7
through the transportation networks, planes navigated across the airline networks, new friends acquainted on social networks, and so on. However, when we enjoy the conveniences of these networked systems, the unexpected network safety issues may occur and strongly damage the normal network functionality. It brings immeasurable economical loss to us and has been the recent focus in many research areas. In realistic networks, the breakdown of a few nodes (or links) or even a single node (or link) caused by random failure or intentional attacks will trigger the propagation of traffic load from the failed nodes (or links) to non-failed ones iteratively until the collapse of the entire network. Such behavior is called cascade or avalanche, such as the largest blackout in US history in 2003 [1], the Western North American blackouts in 1996 [2], the Internet collapse [3,4] caused by traffic congestion, and the large-scale bankruptcy [5] during the recent global recession. Many literatures revealed the cascade phenomena and propagation process [6–19] to understand the dynamic evolutions of cascading failures in real networks. Methods to defend cascades have been studied widely in physical community, computer science and other interdisciplinary sciences, and include ‘‘soft’’ and ‘‘hard’’ strategies. ‘‘Soft’’ mechanisms mainly aim at designing efficient load distribution strategies [20–31] without changing network structure which is regarded too expensively and difficultly to be employed into real networks. ‘‘Hard’’ methods mainly aim at modifying the network underlying structures and can reveal the effects of regulating a part of link connections on robustness against cascades. It helps us understand cascade phenomenon from the network structural perspective. In ‘‘hard’’ strategies, Motter [32] studied a reactive cascade defense and control method in which shutting down many selected lowly loaded nodes or heavily loaded links can significantly restrict the cascade propagation after a single node or link failure. Zhao et al. [33] derived a theory to improve network robustness against attack-induced cascades via controlled removal of a small set of low-degree nodes. Wu et al. [34] proposed three link-removal strategies including flow-based removal, betweenness-based removal and mix-based removal to reduce the damage of cascade failures and delay the network breakdown time. Gao et al. [35] explored effects of link additions on network robustness against cascades, and three link-adding strategies were compared, including random linking, high-betweenness linking and low-polarization linking. To our best knowledge, all previous studies of cascades assumed all links of the network are undirected ones. However, in many real complex networks such as the urban traffic systems, a part of highly congested roads are generally limited to unidirectional ones to alleviate traffic jams. So far, there is a lack of study on investigating cascade propagation in networked complex systems when employing the direction shifting methods. From the perspective of network safety or security, can link orientation methods enhance the network robustness against cascading failures? To the best knowledge of ours, it is still an open problem and has not been investigated. In this work, we try to reveal the effects of edge-orientation strategies against cascading failures. This paper is organized as follows. In Section 2, three link-orientation methods and the cascading model are introduced. In Section 3, the simulation results on different network models are discussed. Finally, the conclusion is given in Section 4. 2. Link-orientations and cascading model 2.1. Link-orientation methods Many empirical studies [36,37] showed that the traffic load of a link lmn is proportional to the product (km × kn ), where km and kn are the link’s end-node degrees of node m and n respectively in the network. In real city transportation networks, a part of heavily loaded links are easily chosen to be directed according to government decisions or researchers’ advises to efficiently redistribute the overall traffic load and alleviate traffic jams on central nodes. For simplicity and without loss of generality, we assume the fraction of directed links is pre-defined and denoted as fr = Er /E, where Er is the number of directed links, and E is the number of total links in the original undirected network. The link-orientation process is implemented as follows. Based on the initial undirected network, firstly, all links are ranked according to the value of product (ki × kj ) from big to small. Er links are selected from the top rank list subsequently and directed according to one of the following direction-determining strategies.
• HTLDD (High-to-Low Direction-Determining). The direction of a unidirectional link is determined from the end-node with high degree to the other end with low degree. If the degrees of two nodes connecting to the link are equal, the direction is random. • LTHDD (Low-to-High Direction-Determining). The direction of a unidirectional link is determined from the end-node with low degree to the other one with high degree. If the degrees of two nodes connecting to the link are equal, the direction is random. • RDD (Random Direction-Determining). The direction of a unidirectional link is determined randomly. Links in the final network structure are with parts of bidirectional links and the other unidirectional ones as shown in Fig. 1(a). In fact, in real complex network structures, such cases as Fig. 1(b) existing nodes with no out-degrees and Fig. 1(c) existing nodes with no in-degrees cause these nodes partly isolated and are not allowed. Then in the link-orientation process, in order to guarantee each node’s arrivals at all other nodes, if determining a link’s direction may lead to at least one of its two end-nodes’ with no in-degrees or out-degrees, this link is not directed and go on to select the next one.
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Fig. 1. A simple example illustrates the link-orientation mechanism. (a) Network structure with a part of unidirectional links; (b) some node such as node i with no out-degrees; (c) some node such as node i with no in-degrees.
2.2. Cascading model Many previous studies have indicated that the traffic load of a node i is strongly related to its degree as [38,39] β
Li = ki ,
(1)
where β = 1.6 is observed and employed in many networked systems such as the communication networks, ki is the degree of node i in the initial undirected network, and Li is the initial traffic load on node i. The initial node capacity is defined as the maximum load that the node can handle per time step. Capacity of a node i is finite and denoted as [32] Ci = (1 + α)Li ,
(2)
where α (α ≥ 0) is a tolerance parameter and denotes the ability of nodes to handle extra load perturbations distributed from other nodes. Larger α represents higher network safety margin against cascading propagations. With increasing α , it exhibits a phase transition phenomenon [40] from full collapse of entire network to appearance of giant component at a critical value αc . When α < αc , the entire network becomes totally disconnected. When α goes beyond the critical value αc , the giant component appears, and it means the network is immune to global cascades. The critical value αc is a very important metric for evaluating cascades. The initial breakdown may occur at any node in the network under either random failures or intentional attacks. It is found that the eventual scale of collapse must be greater when a highly loaded node is broken [41]. Here we assume any node in the network can be used as the initial failure triggering node, and use local load distribution method to redistribute the load of the failed node to its nearest non-failed neighbors to avoid further overloads on other nodes. Fi (t ) denotes the dynamic load on node i at time t. Assume node i is the first failed node at the beginning, its traffic load Fi (t ) will be directed to its non-failed nearest neighbors along the unidirectional links and bidirectional links. Load cannot be directed to a nearest non-failed neighbor along the unidirectional link with direction from the non-failed node to the failed node. For instance, as in Fig. 1(b) load on node i cannot be directed to any node of its neighbors, while load on i can be redistributed to all of its neighbors in Fig. 1(c). Then the additional load 1Fj received by the neighboring node j is proportional to its weight given by Lj Lj
1Fj = Fi (t )
j∈Γi
Lj
,
(3)
where Γi is the set of non-failed reachable neighboring nodes of node i, not including the end-nodes as heads connecting to the failed node i with unidirectional links. After receiving the additional load, the total load on any neighboring node j is now Fj (t ) = Fj (t − 1) + 1Fj . If Fj (t ) > Cj on any node j, namely node j is overloaded due to its limited capacity, then it breaks down, and similar load distribution process is triggered repeatedly until the final stage of cascades. After the cascade propagation stops, a part of nodes may survive. For each cascade beginning from a node i, we denote the number of survived nodes as N ′ (i), the portion of survived nodes as g (i) = N ′ (i)/N. Here we denote the network robustness i=N G as G = i=1 g (i)/N. Larger G means higher network robustness against cascading failures. 3. Simulations Many large scale realistic networked systems appear to have scale-free phenomenon and small-world property, such as the Internet, World Wide Web (WWW), etc. and in this work, we investigate the robustness against cascades on the different network models, including well-known Barabási–Albert (BA) [42] scale-free network model, Watts–Strogatz (WS) [43] small-world model, and the classical Erdös–Rényi (ER) [44] random networks. Here we set network size N = 1000, and average degree ⟨k⟩ ≈ 10 for all the three models. Extensive simulations are implemented to reveal the effects of link-orientation methods on cascading failures in following parts.
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Fig. 2. (Color online) Simulation results on BA scale-free networks. (a) Evolution of G as a function of α and fr under the HTLDD method; (b) evolution of G as a function of α and fr under the LTHDD method; (c) evolution of G as a function of α and fr under the RDD method; (d) G vs. α with fr = 0.2; (e) G vs. fr with α = 0.2; (f) G vs. node degrees.
3.1. Results for BA networks BA scale-free model is generated by two general rules: the growth and preferential attachment. Starting from m0 fully connected nodes, at each time step a node with m (m ≤ m0 ) links is added into the existing network. The other end of each link is chosen preferentially among the old nodes and the selection probability is proportional to the degrees of the old nodes. The BA scale-free network exhibits a power-law degree distribution as P (k) ∼ k−3 . In Fig. 2, we show the simulation results on BA networks. Fig. 2(a)–(c) show the evolution of G as functions of α and fr under the HTLDD, LTHDD and RDD methods respectively. On the whole, with increasing α which means each node has increasing extra capacity to receive the redistributed load from its failed neighboring nodes, the network robustness G increases. With increasing fraction fr of directed links under the three methods, the robustness decreases. We can understand the reasons behind it. On the original network, when one node fails, the number of its available neighbors is the largest, and probability of preventing the further propagation of failure at early steps is the largest. Therefore, on the whole, under each link directed method, the probability of preventing cascade propagation at early steps is reduced. With larger fr , more nodes are influenced. In other words, the propagation of cascading failure is further and further with increasing fr . As showed in Fig. 3, the two-tuple for each node is composed of traffic load and extra capacity. For instance, in the original network in Fig. 3(a1 ), the traffic load and extra capacity on each node is 2 and 1 respectively. Then the capacity of each node is 3. In Fig. 3(b1 ), we assume that one link was directed. Assuming a random failure occurs on j1 in Fig. 3(a1 ), the load on this node is redistributed to its two neighbors. After the redistribution, the load on both nodes j2 and j3 is 3, not exceeding the capacity, and no further failure will happen. Meanwhile, similarly, assuming node j1 in Fig. 3(b1 ) fails first, the load is redistributed to its neighbor j3 in Fig. 3(b2 ). Then traffic load on j3 is 4, exceeding its capacity, resulting in j3 failed. The traffic load on j3 is redistributed to j4 , then j4 with load 6 larger than its capacity, and j4 fails. The cascading failure propagates iteratively, and finally all nodes in the network fail. With only one link directed in the original network, the results are remarkably different from the results on original network. In Fig. 2(d), with a link-orientation fraction fr = 0.2, one can see the RDD and LTHDD strategies have small effects on network robustness against cascades when compared with the original network without unidirectional links. The results under HTLDD method are the worst. In Fig. 2(e), With const α = 0.2, the evolution of comparisons under the three direction-determining methods, it can be seen that with increasing fr the network robustness G under HTLDD is the worst. In Fig. 2(f), we explore the evolution of G with node degrees. One can see that under the HTLDD method, if the initial node is selected as the one with the highest degree, the network robustness appears to have a little improvement. In our opinion, it is an interesting phenomenon and related to the network property of BA scale-free networks. On the original network, on the whole, nodes with larger degree often can deduce larger scale of failures, namely resulting in lower robustness. Because on the BA networks, the number of nodes with high degrees are very small. Meanwhile, nodes with small degrees are larger.
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Fig. 3. (Color online) The explanation of propagation of cascading failure. (a1 ) The original network; (a2 ) when node j1 fails, the load on this node is redistributed to its neighbors j2 and j3 . (b1 ) The network with one link directed; (b2 ) when node j1 fails, the load on this node is redistributed to its neighbor j3 , then load j3 exceeds its capacity and fails in (b3 ), then node j4 and j2 fail in (b4 ) and (b5 ) respectively.
Fig. 4. (Color online) Simulation results on ER networks. (a) Evolution of G as a function of α and fr under the HTLDD method; (b) evolution of G as a function of α and fr under the LTHDD method; (c) evolution of G as a function of α and fr under the RDD method; (d) G vs. α with fr = 0.2; (e) G vs. fr with α = 0.2; (f) G vs. node degrees.
We can see that under the HTLDD method, the nodes with lower degrees can result in higher scale of failed nodes. It is a very intersecting phenomenon that previous work [45] argued that with HTLDD method, the traffic capacity can be largely enhanced, while it reduces the average network robustness against cascading failures. 3.2. Results for ER networks ER random network is a very classical model in studying complex networks, denoted as G(N , p), where p is the probability of connecting a link between any two nodes. On ER networks, Fig. 4(a)–(c) show the evolution of G as functions of α and fr under the HTLDD, LTHDD and RDD methods respectively. Similar evolutions as in Fig. 2(a)–(c) can be observed. It can confirm that the link-orientation methods deteriorate the network robustness against cascading failures under the local load redistribution model. As shown in Fig. 4(d), with fr = 0.2, the evolution of G as a function of α is very similar under the three direction-determining methods, and the critical value of α denoted as αc can be obtained from the results. However, it is not the main focus of this work, so we will not discuss more details here.
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Fig. 5. (Color online) Simulation results on WS small-world networks. (a) Evolution of G as a function of α and fr under the HTLDD method; (b) evolution of G as a function of α and fr under the LTHDD method; (c) evolution of G as a function of α and fr under the RDD method; (d) G vs. α with fr = 0.2; (e) G vs. fr with α = 0.2; (f) G vs. node degrees.
In Fig. 4(e), one can see that on the whole all link-orientation strategies decrease the network robustness against cascading failures, even with few unidirectional links. The G for all the link orientation methods appears Original ≥ LTHDD ≥ RDD ≥ HTLDD. Combined with Fig. 4(f), similar to the above discussion in Fig. 2(f), the HTLDD method affects the failure scale which is the total number of failed nodes mostly, far away from the results of original network. On ER networks, the effects of direction-determining methods on G under the LTHDD and RDD are more obvious than under the HTLDD method. 3.3. Results for WS networks WS network model interpolates between regular and random networks. Starting from a ring lattice with N nodes and m links per vertex, each link is rewired randomly with probability q. Without loss of generality, here we set q = 0.05. On WS networks, Fig. 5(a)–(c) show the evolution of G as functions of α and fr under the HTLDD, LTHDD and RDD methods respectively. Similar evolution has been observed in Figs. 2(a)–(c) and 4(a)–(c). It can also further confirm that the link-orientation methods deteriorate the network robustness against cascading failures under the local load redistribution model. The results on the WS networks are similar to the results on the ER networks. In Fig. 5(e), the relations of G are still Original ≥ LTHDD ≥ RDD ≥ HTLDD. On the WS networks, under the three direction orientation methods, G decreases sharply with fr < 0.1. It is related to the network structures. As shown in Fig. 5(f), the distribution of G for different degrees is a bit different from the above BA and ER networks. Comparing simulation results of Fig. 2, Fig. 4 with Fig. 5, one can see that the three link-orientation methods lead to very similar results for different network models. It suggests us that all the link orientation strategies cannot enhance network robustness of BA, ER and WS scale-free networks in our simulations on the whole. Combined with the previous study, which indicated that HTLDD method on BA networks can enhance the network traffic capacity. However, cascading failures may happen on real networked systems such as city transportation networks. For instance, when one road is congested, the traffic cannot pass through the road timely and accumulates on its neighboring roads. The congestion may propagate iteratively through the observation of real traffic data. Though the link orientation methods can enhance the traffic capacity, it reduces the network robustness against cascading congestion or failures. Therefore, our work gives a good perspective to study the traffic dynamics of real networked systems. It gives suggestion that when the city managers try to use link orientation methods they should consider the positive and negative aspects. It should be noticed that enhancing traffic capacity and improving network robustness simultaneously is a great challenge. Inspired by the results in the work, we will aim to understand the relation between traffic capacity and network robustness against cascading failures more comprehensively in our future works.
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4. Conclusions The study of cascading phenomenon on complex networks has been a hot topic for many years, and numbers of interesting research results are emerging. However, the previous studies were mainly based on undirected network structures, and lacking study on cascades in many artificial networks with parts of unidirectional links and the other undirected ones. Inspired by many previous study that link-orientation methods can be used to enhance network traffic capacity [45], we considered similar link direction-determining methods under which we evaluated their effects on network robustness against cascading failures. We employed the average portion of survival nodes to represent the network robustness. Compared with simulation results on BA scale-free, WS small-world, and ER random networks, on the whole, it indicated that all employed link-orientations in this work reduced the network robustness against the cascading failures under the local load redistribution mechanism. This work gave out a great challenge that it is hard to improve the traffic capacity and robustness of many networks simultaneously by employing link direction-determining methods. Moreover, observed from the results, under the HTLDD method, on BA scale-free networks, with a tiny fraction of unidirectional links, if the initial failed node was selected as the node with the highest degree, the network robustness appears to have a small improvement. It might be related to the network property of BA networks, and needs further studies in the future works. Acknowledgments The authors are grateful to the anonymous reviewers for their valuable comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (No. 61502375), the Key Program of NSFC—Guangdong Union Foundation (No. U1135002), the Fundamental Research Funds for the Central Universities (No. 20101156150), the China 111 Project (No. B16037), and the National High Technology Research and Development Program (863 Program) (No. 2015AA011704). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
J. Glanz, R. Perez-Pena, 90 seconds that left tens of millions of people in the dark, New York Times, August 26, 2003. M.L. Sachtjen, B.A. Carreras, V.E. Lynch, Phys. Rev. E 61 (2000) 4877. R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. Lett. 87 (2001) 258701. K.I. Goh, B. Hahng, D. Kim, Phys. Rev. Lett. 88 (2002) 108701. W.X. Wang, R. Yang, Y.C. Lai, Phys. Rev. E 81 (2010) 035102(R). Z.X. Wu, G. Peng, W.X. Wang, S. Chan, E.E.M. Wong, J. Stat. Mech. 2008 (2008) P05013. W.X. Wang, G.R. Chen, Phys. Rev. E 77 (2008) 026101. Z.J. Bao, Y.J. Cao, L.J. Ding, Z.X. Han, G.Z. Wang, Phys. Lett. A 372 (2008) 5778. P. Li, B.H. Wang, H. Sun, P. Gao, T. Zhou, Eur. Phys. J. B 62 (2008) 1. W.X. Wang, Y.C. Lai, A. Dieter, Chaos 21 (2011) 033112. P. Crucitti, V. Latora, M. Marchiori, Phys. Rev. E 69 (2004) 045104. B. Mirzasoleiman, M. Babaei, M. Jalili, M. Safari, Phys. Rev. E 84 (2011) 046114. J.W. Wang, L.L. Rong, L. Zhang, Z.Z. Zhang, Physica A 387 (2008) 6671. J. Lehmann, J. Bernasconi, Phys. Rev. E 81 (2010) 031129. X.F. Wang, J. Xu, Phys. Rev. E 70 (2004) 056113. J. Wang, C. Zhang, Y. Huang, C. Xin, Nonlinear Dynam. 78 (2014) 37. S. Sahasrabudhe, A.E. Motter, Nature Commun. 2 (2011) 170. W. Chen, M. Schröder, R.M. D’Souza, D. Sornette, J. Nagler, Phys. Rev. Lett. 112 (2014) 155701. C.D. Brummitt, G. Barnett, R.M. D’Souza, Coupled catastrophes: sudden shifts cascade and hop among interdependent systems, arXiv:1410.4175 [physics.soc-ph]. M. Schäfer, J. Scholz, M. Greiner, Phys. Rev. Lett. 96 (2006) 108701. A. Ash, D. Newth, Physica A 380 (2007) 673. R. Yang, W.X. Wang, Y.C. Lai, G.R. Chen, Phys. Rev. E 79 (2009) 026112. I. Simonsen, L. Buzna, K. Peters, S. Bornholdt, D. Helbing, Phys. Rev. Lett. 100 (2008) 218701. L. Buzna, K. Peters, H. Ammoser, C. Kühnert, D. Helbing, Phys. Rev. E 75 (2007) 056107. K. Peters, L. Buzna, D. Helbing, Int. J. Crit. Infrastruct. 4 (2008) 46. B. Wang, B.J. Kim, Europhys. Lett. 78 (2007) 48001. J. Wang, Physica A 392 (2013) 2257. J. Wang, Saf. Sci. 53 (2013) 219. Y. Qiu, Eur. Phys. J. B 86 (2013) 329. S. Nie, X. Wang, H. Zhang, Q. Li, B. Wang, PLoS One 9 (2014) e89066. C.D. Brummitt, R.M. D’Souza, E.A. Leicht, Proc. Natl. Acad. Sci. 109 (2012) E680–E689. A.E. Motter, Phys. Rev. Lett. 93 (2004) 098701. L. Zhao, K. Park, Y.-C. Lai, N. Ye, Phys. Rev. E 72 (2005) 025104(R). J.J. Wu, Z.Y. Gao, H.J. Sun, Physica A 378 (2007) 505. X.-B. Cao, C. Hong, W.-B. Du, J. Zhang, Chaos Solitons Fractals 57 (2013) 35. P.J. Macdonald, E. Almaas, A.-L. Barabási, Europhys. Lett. 72 (2005) 308. Z. Liu, M.-B. Hu, R. Jiang, W.-X. Wang, Q.-S. Wu, Phys. Rev. E 76 (2007) 037101. K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. 87 (2001) 278701. K. Park, Y.-C. Lai, N. Ye, Phys. Rev. E 70 (2004) 026109. L. Zhao, K. Park, Y.-C. Lai, Phys. Rev. E 70 (2004) 035101(R). A.E. Motter, Y.-C. Lai, Phys. Rev. E 66 (2002) 065102(R). A.-L. Barabási, R. Albert, Science 286 (1999) 509. D.J. Watts, S.H. Strogatz, Nature 393 (1998) 440. P. Erdös, A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17. K. Hu, C. Liu, T. Hu, Y. Tang, J. Phys. A: Math. Theor. 43 (2010) 175101.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Effects of link-orientation methods on robustness against cascading failures in complex networks Zhong-Yuan Jiang a,b,∗ , Jian-Feng Ma a,b,c , Yu-Long Shen b,c , Yong Zeng a,b a
School of Cyber Engineering, Xidian University, Xi’an, Shaanxi 710071, China
b
Shaanxi Key Laboratory of Network and System Security, Xidian University, Xi’an, Shaanxi 710071, China
c
School of Computer Science and Technology, Xidian University, Xi’an, Shaanxi 710071, China
highlights • Three link orientation methods are discussed and employed in many classical network models. • Traffic capacity improvement and network safety are considered simultaneously. • Links direction-determining methods deteriorate the robustness against cascading failure.
article
info
Article history: Received 28 August 2015 Received in revised form 29 January 2016 Available online 5 April 2016 Keywords: Link orientation Cascade defense Robustness Complex network
abstract Unidirectional and bidirectional links may coexist in many realistic networked complex systems such as the city transportation networks. Even more, for some considerations, several bidirectional links are shifted to unidirectional ones. Many link-orientation strategies might be employed, including High-to-Low, Low-to-High and Random directiondetermining methods, abbreviated as HTLDD, LTHDD and RDD respectively. Traffic passing through a unidirectional link is restricted to one-side direction. In real complex systems, nodes are correlated with each other. The failure from an initial node may be propagated iteratively, resulting in a large scale of failures of other nodes, called cascade phenomenon which may damage the safety or security of the networked system. Assuming that traffic load on any failed node can be redistributed to its non-failed neighbors, in this work, we try to reveal the effects of unidirectional links on network robustness against cascades. Extensive simulations have been implemented on kinds of networks including Scale-Free networks, Small-World networks, and Erdös–Rényi random networks. The results showed that all of the above three direction-determining methods decrease the robustness of the original networks against cascading failure. This work can help network designers and managers understand the robustness of network well and efficiently prevent the safety events. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, complex networks are ubiquitous all over the world and provide us lots of basic requirements in working place or daily life, such as electricity generated in the power grid, information delivered on the Internet, vehicles passed
∗
Corresponding author at: School of Cyber Engineering, Xidian University, Xi’an, Shaanxi 710071, China. E-mail address: [email protected] (Z.-Y. Jiang).
http://dx.doi.org/10.1016/j.physa.2016.03.107 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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through the transportation networks, planes navigated across the airline networks, new friends acquainted on social networks, and so on. However, when we enjoy the conveniences of these networked systems, the unexpected network safety issues may occur and strongly damage the normal network functionality. It brings immeasurable economical loss to us and has been the recent focus in many research areas. In realistic networks, the breakdown of a few nodes (or links) or even a single node (or link) caused by random failure or intentional attacks will trigger the propagation of traffic load from the failed nodes (or links) to non-failed ones iteratively until the collapse of the entire network. Such behavior is called cascade or avalanche, such as the largest blackout in US history in 2003 [1], the Western North American blackouts in 1996 [2], the Internet collapse [3,4] caused by traffic congestion, and the large-scale bankruptcy [5] during the recent global recession. Many literatures revealed the cascade phenomena and propagation process [6–19] to understand the dynamic evolutions of cascading failures in real networks. Methods to defend cascades have been studied widely in physical community, computer science and other interdisciplinary sciences, and include ‘‘soft’’ and ‘‘hard’’ strategies. ‘‘Soft’’ mechanisms mainly aim at designing efficient load distribution strategies [20–31] without changing network structure which is regarded too expensively and difficultly to be employed into real networks. ‘‘Hard’’ methods mainly aim at modifying the network underlying structures and can reveal the effects of regulating a part of link connections on robustness against cascades. It helps us understand cascade phenomenon from the network structural perspective. In ‘‘hard’’ strategies, Motter [32] studied a reactive cascade defense and control method in which shutting down many selected lowly loaded nodes or heavily loaded links can significantly restrict the cascade propagation after a single node or link failure. Zhao et al. [33] derived a theory to improve network robustness against attack-induced cascades via controlled removal of a small set of low-degree nodes. Wu et al. [34] proposed three link-removal strategies including flow-based removal, betweenness-based removal and mix-based removal to reduce the damage of cascade failures and delay the network breakdown time. Gao et al. [35] explored effects of link additions on network robustness against cascades, and three link-adding strategies were compared, including random linking, high-betweenness linking and low-polarization linking. To our best knowledge, all previous studies of cascades assumed all links of the network are undirected ones. However, in many real complex networks such as the urban traffic systems, a part of highly congested roads are generally limited to unidirectional ones to alleviate traffic jams. So far, there is a lack of study on investigating cascade propagation in networked complex systems when employing the direction shifting methods. From the perspective of network safety or security, can link orientation methods enhance the network robustness against cascading failures? To the best knowledge of ours, it is still an open problem and has not been investigated. In this work, we try to reveal the effects of edge-orientation strategies against cascading failures. This paper is organized as follows. In Section 2, three link-orientation methods and the cascading model are introduced. In Section 3, the simulation results on different network models are discussed. Finally, the conclusion is given in Section 4. 2. Link-orientations and cascading model 2.1. Link-orientation methods Many empirical studies [36,37] showed that the traffic load of a link lmn is proportional to the product (km × kn ), where km and kn are the link’s end-node degrees of node m and n respectively in the network. In real city transportation networks, a part of heavily loaded links are easily chosen to be directed according to government decisions or researchers’ advises to efficiently redistribute the overall traffic load and alleviate traffic jams on central nodes. For simplicity and without loss of generality, we assume the fraction of directed links is pre-defined and denoted as fr = Er /E, where Er is the number of directed links, and E is the number of total links in the original undirected network. The link-orientation process is implemented as follows. Based on the initial undirected network, firstly, all links are ranked according to the value of product (ki × kj ) from big to small. Er links are selected from the top rank list subsequently and directed according to one of the following direction-determining strategies.
• HTLDD (High-to-Low Direction-Determining). The direction of a unidirectional link is determined from the end-node with high degree to the other end with low degree. If the degrees of two nodes connecting to the link are equal, the direction is random. • LTHDD (Low-to-High Direction-Determining). The direction of a unidirectional link is determined from the end-node with low degree to the other one with high degree. If the degrees of two nodes connecting to the link are equal, the direction is random. • RDD (Random Direction-Determining). The direction of a unidirectional link is determined randomly. Links in the final network structure are with parts of bidirectional links and the other unidirectional ones as shown in Fig. 1(a). In fact, in real complex network structures, such cases as Fig. 1(b) existing nodes with no out-degrees and Fig. 1(c) existing nodes with no in-degrees cause these nodes partly isolated and are not allowed. Then in the link-orientation process, in order to guarantee each node’s arrivals at all other nodes, if determining a link’s direction may lead to at least one of its two end-nodes’ with no in-degrees or out-degrees, this link is not directed and go on to select the next one.
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Fig. 1. A simple example illustrates the link-orientation mechanism. (a) Network structure with a part of unidirectional links; (b) some node such as node i with no out-degrees; (c) some node such as node i with no in-degrees.
2.2. Cascading model Many previous studies have indicated that the traffic load of a node i is strongly related to its degree as [38,39] β
Li = ki ,
(1)
where β = 1.6 is observed and employed in many networked systems such as the communication networks, ki is the degree of node i in the initial undirected network, and Li is the initial traffic load on node i. The initial node capacity is defined as the maximum load that the node can handle per time step. Capacity of a node i is finite and denoted as [32] Ci = (1 + α)Li ,
(2)
where α (α ≥ 0) is a tolerance parameter and denotes the ability of nodes to handle extra load perturbations distributed from other nodes. Larger α represents higher network safety margin against cascading propagations. With increasing α , it exhibits a phase transition phenomenon [40] from full collapse of entire network to appearance of giant component at a critical value αc . When α < αc , the entire network becomes totally disconnected. When α goes beyond the critical value αc , the giant component appears, and it means the network is immune to global cascades. The critical value αc is a very important metric for evaluating cascades. The initial breakdown may occur at any node in the network under either random failures or intentional attacks. It is found that the eventual scale of collapse must be greater when a highly loaded node is broken [41]. Here we assume any node in the network can be used as the initial failure triggering node, and use local load distribution method to redistribute the load of the failed node to its nearest non-failed neighbors to avoid further overloads on other nodes. Fi (t ) denotes the dynamic load on node i at time t. Assume node i is the first failed node at the beginning, its traffic load Fi (t ) will be directed to its non-failed nearest neighbors along the unidirectional links and bidirectional links. Load cannot be directed to a nearest non-failed neighbor along the unidirectional link with direction from the non-failed node to the failed node. For instance, as in Fig. 1(b) load on node i cannot be directed to any node of its neighbors, while load on i can be redistributed to all of its neighbors in Fig. 1(c). Then the additional load 1Fj received by the neighboring node j is proportional to its weight given by Lj Lj
1Fj = Fi (t )
j∈Γi
Lj
,
(3)
where Γi is the set of non-failed reachable neighboring nodes of node i, not including the end-nodes as heads connecting to the failed node i with unidirectional links. After receiving the additional load, the total load on any neighboring node j is now Fj (t ) = Fj (t − 1) + 1Fj . If Fj (t ) > Cj on any node j, namely node j is overloaded due to its limited capacity, then it breaks down, and similar load distribution process is triggered repeatedly until the final stage of cascades. After the cascade propagation stops, a part of nodes may survive. For each cascade beginning from a node i, we denote the number of survived nodes as N ′ (i), the portion of survived nodes as g (i) = N ′ (i)/N. Here we denote the network robustness i=N G as G = i=1 g (i)/N. Larger G means higher network robustness against cascading failures. 3. Simulations Many large scale realistic networked systems appear to have scale-free phenomenon and small-world property, such as the Internet, World Wide Web (WWW), etc. and in this work, we investigate the robustness against cascades on the different network models, including well-known Barabási–Albert (BA) [42] scale-free network model, Watts–Strogatz (WS) [43] small-world model, and the classical Erdös–Rényi (ER) [44] random networks. Here we set network size N = 1000, and average degree ⟨k⟩ ≈ 10 for all the three models. Extensive simulations are implemented to reveal the effects of link-orientation methods on cascading failures in following parts.
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Fig. 2. (Color online) Simulation results on BA scale-free networks. (a) Evolution of G as a function of α and fr under the HTLDD method; (b) evolution of G as a function of α and fr under the LTHDD method; (c) evolution of G as a function of α and fr under the RDD method; (d) G vs. α with fr = 0.2; (e) G vs. fr with α = 0.2; (f) G vs. node degrees.
3.1. Results for BA networks BA scale-free model is generated by two general rules: the growth and preferential attachment. Starting from m0 fully connected nodes, at each time step a node with m (m ≤ m0 ) links is added into the existing network. The other end of each link is chosen preferentially among the old nodes and the selection probability is proportional to the degrees of the old nodes. The BA scale-free network exhibits a power-law degree distribution as P (k) ∼ k−3 . In Fig. 2, we show the simulation results on BA networks. Fig. 2(a)–(c) show the evolution of G as functions of α and fr under the HTLDD, LTHDD and RDD methods respectively. On the whole, with increasing α which means each node has increasing extra capacity to receive the redistributed load from its failed neighboring nodes, the network robustness G increases. With increasing fraction fr of directed links under the three methods, the robustness decreases. We can understand the reasons behind it. On the original network, when one node fails, the number of its available neighbors is the largest, and probability of preventing the further propagation of failure at early steps is the largest. Therefore, on the whole, under each link directed method, the probability of preventing cascade propagation at early steps is reduced. With larger fr , more nodes are influenced. In other words, the propagation of cascading failure is further and further with increasing fr . As showed in Fig. 3, the two-tuple for each node is composed of traffic load and extra capacity. For instance, in the original network in Fig. 3(a1 ), the traffic load and extra capacity on each node is 2 and 1 respectively. Then the capacity of each node is 3. In Fig. 3(b1 ), we assume that one link was directed. Assuming a random failure occurs on j1 in Fig. 3(a1 ), the load on this node is redistributed to its two neighbors. After the redistribution, the load on both nodes j2 and j3 is 3, not exceeding the capacity, and no further failure will happen. Meanwhile, similarly, assuming node j1 in Fig. 3(b1 ) fails first, the load is redistributed to its neighbor j3 in Fig. 3(b2 ). Then traffic load on j3 is 4, exceeding its capacity, resulting in j3 failed. The traffic load on j3 is redistributed to j4 , then j4 with load 6 larger than its capacity, and j4 fails. The cascading failure propagates iteratively, and finally all nodes in the network fail. With only one link directed in the original network, the results are remarkably different from the results on original network. In Fig. 2(d), with a link-orientation fraction fr = 0.2, one can see the RDD and LTHDD strategies have small effects on network robustness against cascades when compared with the original network without unidirectional links. The results under HTLDD method are the worst. In Fig. 2(e), With const α = 0.2, the evolution of comparisons under the three direction-determining methods, it can be seen that with increasing fr the network robustness G under HTLDD is the worst. In Fig. 2(f), we explore the evolution of G with node degrees. One can see that under the HTLDD method, if the initial node is selected as the one with the highest degree, the network robustness appears to have a little improvement. In our opinion, it is an interesting phenomenon and related to the network property of BA scale-free networks. On the original network, on the whole, nodes with larger degree often can deduce larger scale of failures, namely resulting in lower robustness. Because on the BA networks, the number of nodes with high degrees are very small. Meanwhile, nodes with small degrees are larger.
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Fig. 3. (Color online) The explanation of propagation of cascading failure. (a1 ) The original network; (a2 ) when node j1 fails, the load on this node is redistributed to its neighbors j2 and j3 . (b1 ) The network with one link directed; (b2 ) when node j1 fails, the load on this node is redistributed to its neighbor j3 , then load j3 exceeds its capacity and fails in (b3 ), then node j4 and j2 fail in (b4 ) and (b5 ) respectively.
Fig. 4. (Color online) Simulation results on ER networks. (a) Evolution of G as a function of α and fr under the HTLDD method; (b) evolution of G as a function of α and fr under the LTHDD method; (c) evolution of G as a function of α and fr under the RDD method; (d) G vs. α with fr = 0.2; (e) G vs. fr with α = 0.2; (f) G vs. node degrees.
We can see that under the HTLDD method, the nodes with lower degrees can result in higher scale of failed nodes. It is a very intersecting phenomenon that previous work [45] argued that with HTLDD method, the traffic capacity can be largely enhanced, while it reduces the average network robustness against cascading failures. 3.2. Results for ER networks ER random network is a very classical model in studying complex networks, denoted as G(N , p), where p is the probability of connecting a link between any two nodes. On ER networks, Fig. 4(a)–(c) show the evolution of G as functions of α and fr under the HTLDD, LTHDD and RDD methods respectively. Similar evolutions as in Fig. 2(a)–(c) can be observed. It can confirm that the link-orientation methods deteriorate the network robustness against cascading failures under the local load redistribution model. As shown in Fig. 4(d), with fr = 0.2, the evolution of G as a function of α is very similar under the three direction-determining methods, and the critical value of α denoted as αc can be obtained from the results. However, it is not the main focus of this work, so we will not discuss more details here.
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Fig. 5. (Color online) Simulation results on WS small-world networks. (a) Evolution of G as a function of α and fr under the HTLDD method; (b) evolution of G as a function of α and fr under the LTHDD method; (c) evolution of G as a function of α and fr under the RDD method; (d) G vs. α with fr = 0.2; (e) G vs. fr with α = 0.2; (f) G vs. node degrees.
In Fig. 4(e), one can see that on the whole all link-orientation strategies decrease the network robustness against cascading failures, even with few unidirectional links. The G for all the link orientation methods appears Original ≥ LTHDD ≥ RDD ≥ HTLDD. Combined with Fig. 4(f), similar to the above discussion in Fig. 2(f), the HTLDD method affects the failure scale which is the total number of failed nodes mostly, far away from the results of original network. On ER networks, the effects of direction-determining methods on G under the LTHDD and RDD are more obvious than under the HTLDD method. 3.3. Results for WS networks WS network model interpolates between regular and random networks. Starting from a ring lattice with N nodes and m links per vertex, each link is rewired randomly with probability q. Without loss of generality, here we set q = 0.05. On WS networks, Fig. 5(a)–(c) show the evolution of G as functions of α and fr under the HTLDD, LTHDD and RDD methods respectively. Similar evolution has been observed in Figs. 2(a)–(c) and 4(a)–(c). It can also further confirm that the link-orientation methods deteriorate the network robustness against cascading failures under the local load redistribution model. The results on the WS networks are similar to the results on the ER networks. In Fig. 5(e), the relations of G are still Original ≥ LTHDD ≥ RDD ≥ HTLDD. On the WS networks, under the three direction orientation methods, G decreases sharply with fr < 0.1. It is related to the network structures. As shown in Fig. 5(f), the distribution of G for different degrees is a bit different from the above BA and ER networks. Comparing simulation results of Fig. 2, Fig. 4 with Fig. 5, one can see that the three link-orientation methods lead to very similar results for different network models. It suggests us that all the link orientation strategies cannot enhance network robustness of BA, ER and WS scale-free networks in our simulations on the whole. Combined with the previous study, which indicated that HTLDD method on BA networks can enhance the network traffic capacity. However, cascading failures may happen on real networked systems such as city transportation networks. For instance, when one road is congested, the traffic cannot pass through the road timely and accumulates on its neighboring roads. The congestion may propagate iteratively through the observation of real traffic data. Though the link orientation methods can enhance the traffic capacity, it reduces the network robustness against cascading congestion or failures. Therefore, our work gives a good perspective to study the traffic dynamics of real networked systems. It gives suggestion that when the city managers try to use link orientation methods they should consider the positive and negative aspects. It should be noticed that enhancing traffic capacity and improving network robustness simultaneously is a great challenge. Inspired by the results in the work, we will aim to understand the relation between traffic capacity and network robustness against cascading failures more comprehensively in our future works.
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4. Conclusions The study of cascading phenomenon on complex networks has been a hot topic for many years, and numbers of interesting research results are emerging. However, the previous studies were mainly based on undirected network structures, and lacking study on cascades in many artificial networks with parts of unidirectional links and the other undirected ones. Inspired by many previous study that link-orientation methods can be used to enhance network traffic capacity [45], we considered similar link direction-determining methods under which we evaluated their effects on network robustness against cascading failures. We employed the average portion of survival nodes to represent the network robustness. Compared with simulation results on BA scale-free, WS small-world, and ER random networks, on the whole, it indicated that all employed link-orientations in this work reduced the network robustness against the cascading failures under the local load redistribution mechanism. This work gave out a great challenge that it is hard to improve the traffic capacity and robustness of many networks simultaneously by employing link direction-determining methods. Moreover, observed from the results, under the HTLDD method, on BA scale-free networks, with a tiny fraction of unidirectional links, if the initial failed node was selected as the node with the highest degree, the network robustness appears to have a small improvement. It might be related to the network property of BA networks, and needs further studies in the future works. Acknowledgments The authors are grateful to the anonymous reviewers for their valuable comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (No. 61502375), the Key Program of NSFC—Guangdong Union Foundation (No. U1135002), the Fundamental Research Funds for the Central Universities (No. 20101156150), the China 111 Project (No. B16037), and the National High Technology Research and Development Program (863 Program) (No. 2015AA011704). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
J. Glanz, R. Perez-Pena, 90 seconds that left tens of millions of people in the dark, New York Times, August 26, 2003. M.L. Sachtjen, B.A. Carreras, V.E. Lynch, Phys. Rev. E 61 (2000) 4877. R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. Lett. 87 (2001) 258701. K.I. Goh, B. Hahng, D. Kim, Phys. Rev. Lett. 88 (2002) 108701. W.X. Wang, R. Yang, Y.C. Lai, Phys. Rev. E 81 (2010) 035102(R). Z.X. Wu, G. Peng, W.X. Wang, S. Chan, E.E.M. Wong, J. Stat. Mech. 2008 (2008) P05013. W.X. Wang, G.R. Chen, Phys. Rev. E 77 (2008) 026101. Z.J. Bao, Y.J. Cao, L.J. Ding, Z.X. Han, G.Z. Wang, Phys. Lett. A 372 (2008) 5778. P. Li, B.H. Wang, H. Sun, P. Gao, T. Zhou, Eur. Phys. J. B 62 (2008) 1. W.X. Wang, Y.C. Lai, A. Dieter, Chaos 21 (2011) 033112. P. Crucitti, V. Latora, M. Marchiori, Phys. Rev. E 69 (2004) 045104. B. Mirzasoleiman, M. Babaei, M. Jalili, M. Safari, Phys. Rev. E 84 (2011) 046114. J.W. Wang, L.L. Rong, L. Zhang, Z.Z. Zhang, Physica A 387 (2008) 6671. J. Lehmann, J. Bernasconi, Phys. Rev. E 81 (2010) 031129. X.F. Wang, J. Xu, Phys. Rev. E 70 (2004) 056113. J. Wang, C. Zhang, Y. Huang, C. Xin, Nonlinear Dynam. 78 (2014) 37. S. Sahasrabudhe, A.E. Motter, Nature Commun. 2 (2011) 170. W. Chen, M. Schröder, R.M. D’Souza, D. Sornette, J. Nagler, Phys. Rev. Lett. 112 (2014) 155701. C.D. Brummitt, G. Barnett, R.M. D’Souza, Coupled catastrophes: sudden shifts cascade and hop among interdependent systems, arXiv:1410.4175 [physics.soc-ph]. M. Schäfer, J. Scholz, M. Greiner, Phys. Rev. Lett. 96 (2006) 108701. A. Ash, D. Newth, Physica A 380 (2007) 673. R. Yang, W.X. Wang, Y.C. Lai, G.R. Chen, Phys. Rev. E 79 (2009) 026112. I. Simonsen, L. Buzna, K. Peters, S. Bornholdt, D. Helbing, Phys. Rev. Lett. 100 (2008) 218701. L. Buzna, K. Peters, H. Ammoser, C. Kühnert, D. Helbing, Phys. Rev. E 75 (2007) 056107. K. Peters, L. Buzna, D. Helbing, Int. J. Crit. Infrastruct. 4 (2008) 46. B. Wang, B.J. Kim, Europhys. Lett. 78 (2007) 48001. J. Wang, Physica A 392 (2013) 2257. J. Wang, Saf. Sci. 53 (2013) 219. Y. Qiu, Eur. Phys. J. B 86 (2013) 329. S. Nie, X. Wang, H. Zhang, Q. Li, B. Wang, PLoS One 9 (2014) e89066. C.D. Brummitt, R.M. D’Souza, E.A. Leicht, Proc. Natl. Acad. Sci. 109 (2012) E680–E689. A.E. Motter, Phys. Rev. Lett. 93 (2004) 098701. L. Zhao, K. Park, Y.-C. Lai, N. Ye, Phys. Rev. E 72 (2005) 025104(R). J.J. Wu, Z.Y. Gao, H.J. Sun, Physica A 378 (2007) 505. X.-B. Cao, C. Hong, W.-B. Du, J. Zhang, Chaos Solitons Fractals 57 (2013) 35. P.J. Macdonald, E. Almaas, A.-L. Barabási, Europhys. Lett. 72 (2005) 308. Z. Liu, M.-B. Hu, R. Jiang, W.-X. Wang, Q.-S. Wu, Phys. Rev. E 76 (2007) 037101. K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. 87 (2001) 278701. K. Park, Y.-C. Lai, N. Ye, Phys. Rev. E 70 (2004) 026109. L. Zhao, K. Park, Y.-C. Lai, Phys. Rev. E 70 (2004) 035101(R). A.E. Motter, Y.-C. Lai, Phys. Rev. E 66 (2002) 065102(R). A.-L. Barabási, R. Albert, Science 286 (1999) 509. D.J. Watts, S.H. Strogatz, Nature 393 (1998) 440. P. Erdös, A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17. K. Hu, C. Liu, T. Hu, Y. Tang, J. Phys. A: Math. Theor. 43 (2010) 175101.