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Motivated by network resilience and insurance premiums in the context ofcyber security, we derive universal upper bounds for the first and secondmoments of the size of bond percolation clusters on finite regular graphs.Thinking of the clusters as dynamical objects coupled with branching processesgives a first set of bounds that are accurate when the probability of an edgebeing open is small. Estimating the number of isolated vertices, we also obtaina second set of bounds that are accurate when the probability of an edge beingclosed is small. As an illustration, we apply our results to the first threePlatonic solids.

First and second moments of the size distribution of bond percolation clusters on regular graphs