Revealing the process of edge-based-attack cascading failures

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In this paper, concerned with the highest-load attack (HL) and random attack (RA) on one edge or one node, we model the cascading dynamics in scale-free networks (SF), WS small-world networks (WS) and ER random networks (ER). How long the cascade propagation in networks will persist and what features the avalanche edges or nodes at each time step will show in cascading failures are questions investigated. We find that, under both HL and RA attack, the persistent time of cascade propagation in SF networks is always longer than in ER and WS networks (except p=0.1). Under HL attack, there exists a threshold αc of tolerance parameter α making the cascading propagation in SF and WS networks longest-running. Moreover, under HL attack, the avalanche edges and nodes in WS and ER networks always reach a peak over a period of time, while the SF network shows similar characteristics only in some field of α. However, under RA attack, in the case of big α, most of the avalanche edges and nodes in SF, WS and ER networks always occur at the beginning of the cascading failures. Furthermore, under node-targeted attack, SF shows to be more vulnerable than under edge-targeted attack and most of avalanche nodes occur at the beginning of failures. The results remind us to grasp the rhythm of controlling disasters according to the features of cascading failures in different networks.