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THE CONTINUUM LIMIT OF THE KURAMOTO MODEL ON SPARSE RANDOM GRAPHS
Journal article   Open access

THE CONTINUUM LIMIT OF THE KURAMOTO MODEL ON SPARSE RANDOM GRAPHS

Georgi S. Medvedev
Communications in mathematical sciences, v 17(4), pp 883-898
01 Jan 2019
DOI: https://doi.org/10.4310/CMS.2019.v17.n4.a1
url
http://arxiv.org/abs/1802.03787View
SubmittedOpen Access (License Unspecified),  Open

Abstract

Mathematics Mathematics, Applied Physical Sciences Science & Technology
In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Medvedev, The nonlinear heat equation on W-random graphs, Arch. Rational Mech. Anal., 212(3): 781-803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems o n sparse graphs and yields almost sure convergence on time intervals of order logn, where n is the number of vertices. Finally, we prove convergence of the averaged model to the continuum limit. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erdos-Renyi, small-world, and power law graphs.

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Mathematics, Applied
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