The Hamiltonian Mean Field model: effect of network structure on synchronization dynamics

TL;DR

This paper investigates how network structure influences synchronization in the Hamiltonian Mean Field model, revealing that the transition point depends on the network's largest eigenvalue and that synchronization robustness varies with coupling strength and network heterogeneity.

Contribution

It introduces a framework linking network eigenvalues to synchronization transition and analyzes the impact of network heterogeneity on synchronization dynamics.

Findings

Transition to synchrony depends on the inverse of the largest eigenvalue of the adjacency matrix.

Synchronization degree near the transition is sensitive to network heterogeneity.

For large coupling, synchronization becomes robust to network degree distribution variations.

Abstract

The Hamiltonian Mean Field (HMF) model of coupled inertial, Hamiltonian rotors is a prototype for conservative dynamics in systems with long-range interactions. We consider the case where the interactions between the rotors are governed by a network described by a weighted adjacency matrix. By studying the linear stability of the incoherent state, we find that the transition to synchrony occurs at a coupling constant $K$ inversely proportional to the largest eigenvalue of the adjacency matrix. We derive a closed system of equations for a set of local order parameters and use these equations to study the effect of network heterogeneity on the synchronization of the rotors. We find that for values of $K$ just beyond the transition to synchronization the degree of synchronization is highly dependent on the network's heterogeneity, but that for large values of $K$ the degree of synchronization is robust to changes in the heterogeneity of the network's degree distribution. Our results are illustrated with numerical simulations on Erd\"os-Renyi networks and networks with power-law degree distributions.