The nonlinear heat equation on dense graphs and graph limits

TL;DR

This paper establishes a rigorous mathematical framework for deriving continuum limits of dynamical networks on convergent graph sequences, linking graph theory, nonlinear PDEs, and numerical methods, with applications to complex network dynamics.

Contribution

It provides the first rigorous proof of continuum limits for certain complex networks and connects these limits to classical numerical scheme convergence analysis.

Findings

Proves convergence of discrete network solutions to continuous equations.

Shows convergence rate depends on the fractal dimension of graphon boundaries.

Links continuum limits to classical numerical methods like Galerkin and Monte Carlo.

Abstract

We use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking continuum limit for certain nonlocally coupled networks and to extend this method to cover many complex networks, for which it has not been applied before. Specifically, for dynamical networks on convergent sequences of simple and weighted graphs, we prove convergence of solutions of the initial-value problems for discrete models to those of the limiting continuous equations. In addition, for sequences of simple graphs converging to {0, 1}-valued graphons, it is shown that the convergence rate depends on the fractal dimension of the boundary of the support of the graph limit. These results are then used to study the regions of continuity of chimera states and the attractors of the nonlocal Kuramoto equation on certain multipartite graphs. Furthermore, the analytical tools developed in this work are used in the rigorous justification of the continuum limit for networks on random graphs that we undertake in a companion paper (Medvedev, 2013). As a by-product of the analysis of the continuum limit on deterministic and random graphs, we identify the link between this problem and the convergence analysis of several classical numerical schemes: the collocation, Galerkin, and Monte-Carlo methods. Therefore, our results can be used to characterize convergence of these approximate methods of solving initial-value problems for nonlinear evolution equations with nonlocal interactions.