Local symmetry in random graphs

TL;DR

This paper introduces the concept of local symmetry in graphs, analyzing how local structural equivalences emerge or disappear in Erdős-Rényi random graphs as the average degree varies.

Contribution

It defines local symmetry based on egonet similarity and studies its emergence in Erdős-Rényi graphs, revealing regimes where local symmetry persists or breaks down.

Findings

Local symmetry persists up to average degree of n^{1/3}.

Local asymmetry emerges at average degree not greater than n^{1/2}.

Local symmetry regimes are larger than those for global symmetry.

Abstract

Quite often real-world networks can be thought of as being symmetric, in the abstract sense that vertices can be found to have similar or equivalent structural roles. However, traditional measures of symmetry in graphs are based on their automorphism groups, which do not account for the similarity of local structures. We introduce the concept of local symmetry, which reflects the structural equivalence of the vertices' egonets. We study the emergence of asymmetry in the Erd\H{o}s-R\'enyi random graph model and identify regimes of both asymptotic local symmetry and asymptotic local asymmetry. We find that local symmetry persists at least to an average degree of $n^{1/3}$ and local asymmetry emerges at an average degree not greater than $n^{1/2}$, which are regimes of much larger average degree than for traditional, global asymmetry.