Localized attack on networks with clustering

Network systems with clustering have been given much attention due to their wide occurrence in the real world. One focus of these studies has been on robustness of single clustered networks and interdependent clustered networks under random attack (RA) or hub-targeted attack. However, infrastructure networks could suffer from a damage that is localized, i.e. a group of neighboring nodes attacked or fail, a topic that was not studied earlier on clustered networks. In this paper, we analytically and via simulations study the robustness under localized attack (LA) of single Erdos-Renyi 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.