Universality in percolation of arbitrary Uncorrelated Nested Subgraphs

TL;DR

This paper analytically demonstrates that in uncorrelated nested subgraphs, the cluster size distribution at criticality universally follows a power law with exponent 3/2, regardless of specific graph processes.

Contribution

It provides the first analytical derivation of the universal power-law exponent for cluster sizes in uncorrelated nested subgraphs at criticality.

Findings

Cluster size distribution follows a power law with exponent 3/2 at criticality.

The universality applies to a wide class of nesting processes.

The results generalize percolation theory to nested subgraphs.

Abstract

The study of percolation in so-called {\em nested subgraphs} implies a generalization of the concept of percolation since the results are not linked to specific graph process. Here the behavior of such graphs at criticallity is studied for the case where the nesting operation is performed in an uncorrelated way. Specifically, I provide an analyitic derivation for the percolation inequality showing that the cluster size distribution under a generalized process of uncorrelated nesting at criticality follows a power law with universal exponent $\gamma=3/2$. The relevance of the result comes from the wide variety of processes responsible for the emergence of the giant component that fall within the category of nesting operations, whose outcome is a family of nested subgraphs.