Localized attack on networks with clustering

Gaogao Dong, Huifang Xiao, Fan Wang, Ruijin Du, Shuai Shao, Lixin Tian, H. Eugene Stanley, Shlomo Havlin

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12 Scopus citations


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

Original languageEnglish
Article number013014
JournalNew Journal of Physics
Issue number1
StatePublished - 18 Jan 2019

Bibliographical note

Funding Information:
SH acknowledges support of the Israel–Italian collaborative project NECST, Israel Science Foundation, ONR, Japan Science Foundation, BSF-NSF, the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime MinisterʼsOffice, and DTRA (Grant no. HDTRA-1-10-1-0014) for financial support. The Boston University Center for Polymer Studies is supported by NSF Grant PHY-1505000 and by DTRA Grant HDTRA1-14-1-0017. We also thank National Natural Science Foundation of China (Grant Nos. 61403171, 71403105, 71303095), the Jiangsu Postdoctoral Science Foundation (Grant No. 1501100B), the China Postdoctoral Science Foundation (Grant No. 2015M581738), the Senior talents Foundation of Jiangsu University (Grant Nos. 14JDG143, 14JDG144) and Scientific research project of Jiangsu University (Grant No. 17A293) for support.

Publisher Copyright:
© 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.


  • clustering network
  • localized attack
  • percolation
  • resilience
  • robustness


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