Algebraic bounds for heterogeneous site percolation on directed and undirected graphs

TL;DR

This paper develops algebraic bounds for heterogeneous site percolation on directed and undirected graphs, providing insights into percolation thresholds and giant component formation using weighted matrices.

Contribution

It introduces new algebraic bounds for percolation and connectivity in heterogeneous graphs, linking spectral properties to phase transition phenomena.

Findings

Bounds on cluster susceptibilities and connectivity functions

Lower bounds for percolation and giant component formation

Spectral criteria related to percolation thresholds

Abstract

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.