Exploiting symmetry in network analysis

TL;DR

This paper investigates how symmetry in real-world networks affects various measures and demonstrates methods to exploit this symmetry for computational efficiency and redundancy reduction, with practical applications to large networks.

Contribution

It provides a systematic study of network symmetry effects on measures and introduces practical techniques to leverage symmetry for analysis and computation.

Findings

Symmetry impacts network measures like the graph Laplacian.

Exploiting symmetry reduces computational complexity.

Efficient algorithms for detecting network symmetry in large graphs.

Abstract

Virtually all network analyses involve structural measures between pairs of vertices, or of the vertices themselves, and the large amount of symmetry present in real-world complex networks is inherited by such measures. This has practical consequences which have not yet been explored in full generality, nor systematically exploited by network practitioners. Here we study the effect of network symmetry on arbitrary network measures, and show how this can be exploited in practice in a number of ways, from redundancy compression, to computational reduction. We also uncover the spectral signatures of symmetry for an arbitrary network measure such as the graph Laplacian. Computing network symmetries is very efficient in practice, and we test real-world examples up to several million nodes. Since network models are ubiquitous in the Applied Sciences, and typically contain a large degree of structural redundancy, our results are not only significant, but widely applicable.