Robustness paradox of attack behavior induced by cascading failures: The larger the attack, the more vulnerable the network?

The robustness of complex networks responding to attacks has long been the focus of network science researching. Nonetheless, the precious studies mostly focus on network performance when facing malicious attacks and random failures while rarely pay attention to the influences of scales of attacking. It is wondering if it is an actual fact that the network is more fragile when attacking scale is exacerbated. In this paper, we are committed to exploring the influences related to the very factor of attacking scale from the perspective of cascading failure problem of dynamic network theory. We construct the model with a regular ranking edge deletion method by simulating attacking scale with Na and a is denoted as attacked edge number. To be specific, we rank the edges according to initial distributed loads and delete edges in the ranked list, and subsequently observe the changes of robustness in the networks, including BA scale-free network, WS small-world network and several real traffic networks. During the process, an unusual counterintuitive phenomenon captures our attention that the network damages caused by attacks do not always grow with the increase of attacked edges number. We specifically demonstrate and analyze this abnormal cascading propagation phenomenon, ascribing this paradox to the dynamics of the load and the connections of the network structure. Our work may offer a new angle on better controlling the spread of cascading failure and remind the importance of effectively protecting networks from underlying dangers.