The lower bound of the network connectivity guaranteeing in-phase synchronization

Abstract

In-phase synchronization is a stable state of identical Kuramoto oscillators coupled on a network with identical positive connections, regardless of network topology. However, this fact does not mean that the networks always synchronize in-phase because other attractors besides the stable state may exist. The critical connectivity μc is defined as the network connectivity above which only the in-phase state is stable for all the networks. In other words, below μc, one can find at least one network that has a stable state besides the in-phase sync. The best known evaluation of the value so far is 0.6828…≤μc≤0.7889. In this paper, focusing on the twisted states of the circulant networks, we provide a method to systematically analyze the linear stability of all possible twisted states on all possible circulant networks. This method using integer programming enables us to find the densest circulant network having a stable twisted state besides the in-phase sync, which breaks a record of the lower bound of the μc from 0.6828… to 0.6838…. We confirm the validity of the theory by numerical simulations of the networks not converging to the in-phase state.

From weighted to unweighted graphs in Synchronizing Graph Theory

“…In our case this formula turn out to be The first k such that the strong density µ k is greater than the best lower bound known so far 7 , i.e., 0.6838, is k = 34, that correspond to a graph with 272 vertices, an order far away from exhaustive numerical experiments. The first k such that the strong density µ k is greater than than 0.6874, is k = 1250, that corresponds to a graph with 10000 vertices.…”

The Kuramoto model on dynamic random graphs

“…Furthermore, Canale and Monzón [13,14] constructed a sequence of networks from the lexicographic product of a WSG network together with complete graphs to exhibit networks that are highly connected (µ = (15/22) − ) but do not globally synchronise. This was improved in [57] to 0.6838 . .…”

The Kuramoto model on dynamic random graphs

“…Furthermore, Canale and Monzón [12,13] constructed a sequence of networks from the lexicographic product of a WSG network together with complete graphs to exhibit networks that are highly connected (µ = (15/22) − ) but do not globally synchronize. This was improved in [55] to 0.6838 . .…”