A global synchronization theorem for oscillators on a random graph

Abstract

Consider [Formula: see text] identical Kuramoto oscillators on a random graph. Specifically, consider Erdős–Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability [Formula: see text]. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for [Formula: see text] above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, [Formula: see text] for [Formula: see text]. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if [Formula: see text], then Erdős–Rényi networks of Kuramoto oscillators are globally synchronizing with high probability as [Formula: see text]. Here, we improve that result by showing that [Formula: see text] suffices. Our estimates are explicit: for example, we can say that there is more than a [Formula: see text] chance that a random network with [Formula: see text] and [Formula: see text] is globally synchronizing.