First and second moments of the size distribution of bond percolation clusters on rings, paths and stars

The main objective of this paper is to study the first and second moments of the size of a typical cluster of bond percolation on ring, path, and star graphs. These graphs are essential building blocks of graphs that represent hybrid local area networks (LANs). In our setting, the edges are independently open with probability  p , and the aim is to find the exact expressions for the first and second moments of the number of vertices in the cluster of open edges containing a vertex chosen uniformly at random. This work is motivated by a network science question of resilience as well as pricing of cyber risk in hybrid LANs.