Complex networks: when random walk dynamics equals synchronization

Abstract

Synchrony prevalently emerges from the interactions of coupled dynamical units. For simple systems such as networks of phase oscillators, the asymptotic synchronization process is assumed to be equivalent to a Markov process that models standard diffusion or random walks on the same network topology. In this paper, we analytically derive the conditions for such equivalence for networks of pulse-coupled oscillators, which serve as models for neurons and pacemaker cells interacting by exchanging electric pulses or fireflies interacting via light flashes. We find that the pulse synchronization process is less simple, but there are classes of, e.g., network topologies that ensure equivalence. In particular, local dynamical operators are required to be doubly stochastic. These results provide a natural link between stochastic processes and deterministic synchronization on networks. Tools for analyzing diffusion (or, more generally, Markov processes) may now be transferred to pin down features of synchronization in networks of pulse-coupled units such as neural circuits.