Local symmetries in complex networks

TL;DR

This paper introduces measures of local symmetry in complex networks based on path permutations, providing tools to identify symmetric vertices and their surroundings in various network types.

Contribution

It proposes new continuous symmetry measures for vertices in networks, focusing on path permutation invariance to better analyze complex network structures.

Findings

Symmetry coefficients can identify local centers of symmetry in protein interaction networks.

Proposed measures are stable against randomness in complex networks.

Different definitions of symmetry coefficients are applicable to real-world networks.

Abstract

Symmetry -- invariance to certain operators -- is a fundamental concept in many branches of physics. We propose ways to measure symmetric properties of vertices, and their surroundings, in networks. To be stable to the randomness inherent in many complex networks, we consider measures that are continuous rather than dichotomous. The main operator we suggest is permutations of the paths of a certain length leading out from a vertex. If these paths are more similar (in some sense) than expected, the vertex is a local center of symmetry in networks. We discuss different precise definitions based on this idea and give examples how different symmetry coefficients can be applied to protein interaction networks.