Spectral Characteristics of Network Redundancy

TL;DR

This paper explores how the symmetry properties of networks, described by automorphism groups, influence their spectral signatures, linking network motifs to specific eigenvalues and eigenvectors.

Contribution

It provides a group-theoretic framework to connect network automorphisms with spectral characteristics, offering a complete description of redundancy signatures in undirected networks.

Findings

Automorphism groups relate to specific eigenvalues.

Eigenvectors can be associated with network motifs.

Spectral signatures reveal structural redundancy.

Abstract

Many real-world complex networks contain a significant amount of structural redundancy, in which multiple vertices play identical topological roles. Such redundancy arises naturally from the simple growth processes which form and shape many real-world systems. Since structurally redundant elements may be permuted without altering network structure, redundancy may be formally investigated by examining network automorphism (symmetry) groups. Here, we use a group-theoretic approach to give a complete description of spectral signatures of redundancy in undirected networks. In particular, we describe how a network's automorphism group may be used to directly associate specific eigenvalues and eigenvectors with specific network motifs.