Subversion analyses of hierarchical networks based on (edge) neighbor connectivity

Abstract

The notion of neighbor connectivity was derived from the assessment of subversion in spy networks caused by the underground resistance movement. For a network G, the neighbor connectivity κNB(G) (resp. edge neighbor connectivity λNB(G)) is defined as the least number of vertices (resp. edges) such that if we remove the closed neighborhoods of them, the network will become disconnected, empty, or complete (resp. trivial). The two connectivities can also provide more accurate measures regarding the reliability and fault-tolerance of networks. Star graphs Sn and hypercubes Qn are the two most famous structures in the family of Cayley graphs. They are widely studied in the research of developing multiprocessor systems. In this paper, we investigate the neighbor connectivity and edge neighbor connectivity of two kinds of hierarchical networks, called hierarchical star network HSn and complete cubic network CCn, which take Sn and Qn as building blocks, respectively. Specifically, we obtain the following results: κNB(HSn)=n−1 and λNB(HSn)=n for n≥3, and κNB(CCn)=⌈n2⌉+1 and λNB(CCn)=n+1 for n≥2. In addition, to further determine the distribution of the number of subverted vertices and their occurrence status, we also carry out experiments to simulate the subversion at the vertices on these networks.