Percolation in self-similar networks

TL;DR

This paper proves that a broad class of self-similar networks have zero percolation threshold, indicating they percolate at arbitrarily small connection probabilities, regardless of clustering or metric structure.

Contribution

It provides a simple, general proof that self-similar networks have zero percolation threshold without assuming treelike structure, expanding understanding of percolation in complex networks.

Findings

Self-similar networks have zero percolation threshold.

The proof does not require treelike assumptions.

Hierarchical structure with increasing average degree is key.

Abstract

We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering, scale-free graphs with finite clustering and metric structure, growing scale-free networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that self-similar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks.