Hidden Symmetries in Real and Theoretical Networks

TL;DR

This paper introduces the concept of latent symmetry in networks, a generalized form of symmetry that exists at multiple scales, and demonstrates its prevalence and significance in real and modeled networks.

Contribution

The paper defines latent symmetry, shows its occurrence in real and model networks, and proves its relation to eigenvector centrality, extending the understanding of network symmetries.

Findings

Latent symmetries are more common in preferential attachment networks.

Latent symmetry vertices share the same eigenvector centrality.

Real networks contain latent symmetries at multiple scales.

Abstract

Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called \emph{latent symmetry}, which is an extension of the standard notion of symmetry. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a \emph{size}, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erd{\H o}s-R\'enyi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that \emph{latent symmetries} are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.