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Algebraic bounds for heterogeneous site percolation on directed and undirected graphs

We analyze site percolation on directed and undirected graphs with site-dependent open-site probabilities. We construct upper bounds on cluster susceptibilities, vertex connectivity functions, and the expected number of simple open cycles through a chosen arc; separate bounds are given on finite and infinite (di)graphs. These produce lower bounds for percolation and uniqueness transitions in infinite (di)graphs, and for the formation of a giant component in finite (di)graphs. The bounds are formulated in terms of appropriately weighted adjacency and non-backtracking (Hashimoto) matrices. It turns out to be the uniqueness criterion that is most closely associated with an asymptotically vanishing probability of forming a giant strongly-connected component on a large finite (di)graph.

preprint2016arXivOpen access
Kathleen E. HamiltonLeonid P. Pryadko
math-phmath.MPcond-mat.dis-nn
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Kathleen E. HamiltonLeonid P. Pryadko

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