How Symmetric Are Real-World Graphs? A Large-Scale Study

Fabian Ball,Andreas Geyer-Schulz

doi:10.3390/sym10010029

Abstract

The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the “network redundancy” measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.

Highlights

The analysis and understanding of the effects of symmetry has been a guiding principle in physics since the beginning of the 20th century [1]
This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent
In chemistry, the recognition of graph symmetry plays a role, as the structure of molecules can be modeled with graphs and the distinction of molecular structures (i.e., checking for isomorphism) is of great interest [2]

Summary

Introduction

The analysis and understanding of the effects of symmetry has been a guiding principle in physics since the beginning of the 20th century [1]. In chemistry, the recognition of graph symmetry plays a role, as the structure of molecules can be modeled with graphs and the distinction of molecular structures (i.e., checking for (non-) isomorphism) is of great interest [2]. This has led to entropy-based measures, applicable in chemistry and in the fields of biology, sociology and psychology [3,4]. Both problems (graph isomorphism—GI, and graph automorphism—GA) are not in P and not easy to solve in general. For GI it is even unclear whether it is in N P C

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