Concentric Network Symmetry

TL;DR

This paper introduces a new framework for locally quantifying node symmetries in complex networks using connectivity patterns, with applications to various models and real-world networks, revealing unique symmetry properties.

Contribution

The authors develop two topological transformations to characterize local symmetries, extending previous global symmetry measures to a local context in networks.

Findings

Erdős-Rényi networks exhibit high symmetry.

Proposed measures show low correlation with traditional metrics.

Principal component analysis indicates potential for global network characterization.

Abstract

Quantification of symmetries in complex networks is typically done globally in terms of automorphisms. Extending previous methods to locally assess the symmetry of nodes is not straightforward. Here we present a new framework to quantify the symmetries around nodes, which we call connectivity patterns. We develop two topological transformations that allow a concise characterization of the different types of symmetry appearing on networks and apply these concepts to six network models, namely the Erd\H{o}s-R\'enyi, Barab\'asi-Albert, random geometric graph, Waxman, Voronoi and rewired Voronoi. Real-world networks, namely the scientific areas of Wikipedia, the world-wide airport network and the street networks of Oldenburg and San Joaquin, are also analyzed in terms of the proposed symmetry measurements. Several interesting results emerge from this analysis, including the high symmetry exhibited by the Erd\H{o}s-R\'enyi model. Additionally, we found that the proposed measurements present low correlation with other traditional metrics, such as node degree and betweenness centrality. Principal component analysis is used to combine all the results, revealing that the concepts presented here have substantial potential to also characterize networks at a global scale.