Analytical results for bond percolation and k-core sizes on clustered networks

James P. Gleeson, Sergey Melnik

Published 2008-11-27, updated 2009-09-22Version 2

An analytical approach to calculating bond percolation thresholds, sizes of $k$-cores, and sizes of giant connected components on structured random networks with non-zero clustering is presented. The networks are generated using a generalization of Trapman's [P. Trapman, Theor. Pop. Biol. {\bf 71}, 160 (2007)] model of cliques embedded in tree-like random graphs. The resulting networks have arbitrary degree distributions and tunable degree-dependent clustering. The effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature. For very high levels of clustering the percolation threshold in these generalized Trapman networks is increased above the value it takes in a randomly-wired (unclustered) network of the same degree distribution. In assortative scale-free networks, where the variance of the degree distribution is infinite, this clustering effect can lead to a non-zero percolation (epidemic) threshold.