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

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 Erdös-Renyi networks and networks with power-law degree distributions.

14 pages, 10 figures, 1 Appendix

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• Synchronization in complex networks
• Approximating the largest eigenvalue of network adjacency matrices
• Synchronization transition in scale-free networks: Clusters of synchrony
• Synchronizability determined by coupling strengths and topology on Complex Networks
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model

Cited by in corpus (3)

• Graph spectral characterisation of the XY model on complex networks
• Hamiltonian mean field model : effect of temporal perturbation in coupling matrix
• Networks of planar Hamiltonian systems