Generalized k-cores of networks under attack with limited knowledge

Network theory has been used as an effective approach for understanding and controlling many real world large-scale systems. A significant aspect of network operation is its robustness against failures and attacks. Here, we develop a theoretical framework for two classes of network attack with limited knowledge, namely, min - n and max - n attacks, where only n nodes are observed and a node with smallest or largest degree is removed at a time until a fraction 1 - p of nodes are attacked. We study the effect of these attacks on the generalized k-core ( Gk-core) of the network, which is obtained by implementing a k-leaf pruning process, removing progressively nodes with degree smaller than k alongside their nearest neighbors. This removal process can be viewed as a generation of the ordinary k-core decomposition. It is found that the G 2-core undergoes a continuous phase transition with respect to p while Gk-core shows a first-order percolation transition for k >= 3 under both types of attacks for all n . We reveal that knowing one more node during attacks, improving from n = 1 to n = 2 , turns out to be most beneficial in terms of changing the robustness of Gk-core in both directions. Moreover, it is shown that degree heterogeneity plays a role in robustness as prioritizing attack on small-degree nodes in heterogeneous networks may help consolidate the Gk-core, but also in stability where hub nodes act like anchors stabilizing the Gk core structure. Our results offer insight into the design of resilient complex systems and evaluation of network robustness and stability. (c) 2021 Elsevier Ltd. All rights reserved.