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

Abstract

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.

Revised version, to appear in Phys. Rev. E

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• Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks
• Epidemics on random intersection graphs
• Deciphering the global organization of clustering in real complex networks
• Dynamics on Modular Networks with Heterogeneous Correlations
• Network cloning unfolds the effect of clustering on dynamical processes
• Endemic infections are always possible on regular networks
• Hierarchical scale-free network is fragile against random failure
• Exact epidemic dynamics for generally clustered, complex networks
• Improving the accuracy of the k-shell method by removing redundant links-from a perspective of spreading dynamics