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

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.