A rigorous and efficient approach to finding and quantifying symmetries in complex networks

TL;DR

This paper introduces a new, efficient method using structural position vectors to identify and analyze symmetric nodes in large complex networks, significantly improving computational speed and understanding of network symmetries.

Contribution

The paper presents a novel SPV-based approach that finds all symmetric nodes in linear time and characterizes their influence, surpassing traditional algebraic methods in efficiency and interpretability.

Findings

All symmetric nodes can be identified in linear time.

SPVs effectively characterize node similarity and influence.

The method is validated on six real-world complex networks.

Abstract

Symmetries are fundamental to dynamical processes in complex networks such as cluster synchronization, which have attracted a great deal of current research. Finding symmetric nodes in large complex networks, however, has relied on automorphism groups in algebraic group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each and every node in the network. We prove mathematically that nodes with the identical SPV are symmetrical to each other. Utilizing six representative complex networks from the real world, we demonstrate that all symmetric nodes can be found in linear time, and the SPVs can not only characterize the similarity of nodes but also quantify the nodal influences in spreading dynamics on the network. Our SPV-based framework, in additional to being rigorously justified, provides a physically intuitive way to uncover, understand and exploit symmetric structures in complex networks.