Complex Networks and Symmetry I: A Review

TL;DR

This review explores the relationship between complex networks and symmetry, emphasizing the importance of stochastic symmetry in understanding real-world noisy networks and highlighting its topological significance.

Contribution

It introduces a stochastic symmetry concept for real networks, connecting symmetry analysis with network topology in noisy and imperfect conditions.

Findings

Stochastic symmetry captures essential topological features of real networks.

Traditional symmetry concepts are limited in analyzing noisy, real-world networks.

Stochastic symmetry provides insights inaccessible to exact symmetry methods.

Abstract

In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques.