Identifying symmetries and predicting cluster synchronization in complex networks

TL;DR

This paper introduces a computationally efficient method for identifying network symmetries and predicting cluster synchronization in complex networks, linking eigen-vector centrality to cluster formation, and enabling analysis of large graphs.

Contribution

It presents a novel, linear-cost framework connecting eigen-vector centrality with network clusters, simplifying symmetry analysis and synchronization prediction in large graphs.

Findings

Eigen-vector centrality elements correlate with network clusters.

The method accurately predicts cluster synchronization sequences.

Analysis aligns with traditional symmetry-based approaches.

Abstract

Symmetries in a network connectivity regulate how the graph's functioning organizes into clustered states. Classical methods for tracing the symmetry group of a network require very high computational costs, and therefore they are of hard, or even impossible, execution for large sized graphs. We here unveil that there is a direct connection between the elements of the eigen-vector centrality and the clusters of a network. This gives a fresh framework for cluster analysis in undirected and connected graphs, whose computational cost is linear in $N$. We show that the cluster identification is in perfect agreement with symmetry based analyses, and it allows predicting the sequence of synchronized clusters which form before the eventual occurrence of global synchronization.