Graph partitions and cluster synchronization in networks of oscillators

TL;DR

This paper investigates how network structure influences cluster synchronization in oscillator networks, using graph theory to derive conditions for synchronized clusters and applying the framework to both linear and nonlinear models, including signed networks.

Contribution

It introduces a graph-theoretical framework based on External Equitable Partitions to analyze cluster synchronization in networks, extending to signed interactions and nonlinear dynamics.

Findings

Conditions for cluster synchronization derived from eigenvector localization.

Framework applicable to both linear and nonlinear oscillator models.

Extension of analysis to signed networks with positive and negative edges.

Abstract

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges, and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.