The Nonlinear Heat Equation on Dense Graphs and Graph Limits

Abstract

The continuum limit of coupled dynamical systems is an approximate procedure, by which the dynamical problem on a sequence of large graphs is replaced by an evolution integral equation on a continuous spatial domain. While this method has been widely used in the analysis of pattern formation in nonlocally coupled networks, its mathematical basis remained little understood. In this paper, 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 the continuum limit 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 r...