Generalized k-cores of networks under attack with limited knowledge

Network Robustness Analysis Based on Maximum Flow

“…Mariani et al [27] focused on one of the nonrandom structure patterns in networks-nestedness, and concentrated on their discussion on three main aspects: the existing methodologies to nestedness in networks, the key theoretical mechanisms to explain nestedness in ecological and socioeconomic networks, and implications of the nested topology of interactions for the stability and viability of a given interacting systems. Wuellner et al [28] analyzed the individual structures of the seven largest U.S. passenger carriers and found that networks with dense interconnectivity are extremely resilient to both targeted removal of airports (nodes) and random removal of flight paths (edges), and here, they measured the interconnectivity of the network using the k-core structure, which is a subgraph of the network constructed by iteratively pruning all vertices with a degree less than k. Shang Y-L [14] developed a mathematical framework for understanding the robustness of networks based on the number of nodes and edges in the Gk-core (a generalization of the ordinary k-core decomposition) under two general attacks with limited knowledge (min-n and max-n attacks), and it was found that knowing one more node (from n 1 to n 2) during attacks is most beneficial in terms of changing the robustness of the Gk-core. Therefore, research studies related to network robustness can help people understand the mechanisms and rules of network system failure or collapse and can identify better ways to prevent the failure of real network systems and build more robust systems, making real life more stable [29].…”

“…To ensure that the evaluation indicators can truly reflect the robustness of the complex network, measurability, sensitivity, and objectivity are required. Nowadays, robustness evaluation indicators generally include the network global effect, average path length, connectivity, relative size of the maximum connected subgraph, betweenness, circle rate, clustering coefficient [9], k-core structure [10,11], core [12], and generalized k-cores [13,14]. Among them, as the level of network damage caused by the attack increases, the average shortest path becomes larger and then smaller [9], and this trend of change is not a significant guide for practical applications; the betweenness index takes into account the changes of nodes and edges in the network but does not consider changes in the network size and structure as a whole [9]; the clustering coefficient reflects the tightness of connections between nodes in the network and is also an indicator of local change in the network; considering the maximum connected subgraph, the robustness of the complex network is defined as the size of the maximum connected subgraph in the network after randomly or deliberately removing a certain percentage of nodes from the network [15]; in single networks, k-core is defined as a maximal set of nodes that have at least k neighbors within the set [16], and the generalized k-core (Gk-core) is a core structure, which is obtained by implementing a k-leaf pruning procedure that progressively removes nodes with degree less than k alongside their nearest neighbors [14].…”

Percolation of interdependent networks with limited knowledge

“…A versatile mirror image process of the percolation under limited knowledge has been proposed in [22], where the targeted nodes for removal can be random or most/least connected with certain probabilities. The network is shown to undergo a hybrid phase transition for the k-core organization under such pruning processes [22,23]. An analytical framework for studying network immunization with limited knowledge is investigated in [24], where the immunity acquired may decline over time.…”