Geometric unfolding of synchronization dynamics on networks

TL;DR

This paper introduces a geometric series approach to analyze the synchronized state in network-coupled oscillators, revealing how local and global network structures influence synchronization.

Contribution

It presents a novel geometric expansion method for the steady-state solution of oscillator synchronization, applicable to various network types and distributions.

Findings

The geometric series converges for arbitrary frequency distributions.

Error growth in truncation is related to the second largest eigenvalue.

A local approximation using the first neighborhood term is derived.

Abstract

We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Namely, each addition term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Lastly, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.