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 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 [31].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.

Dynamics and control for multi-agent networked systems: A finite-difference approach

“…Recall that (1.1), (2.1) can be seen as a set of N coupled equations on a graph. To analyze the limit when N → +∞, we adopt the graph limit method presented in [57], where the author combines techniques from the theory of evolution equations and the recent theory of graph limits ( [11,53]) to rigorously justify the possibility of taking the continuum limit for a large class of dynamical models on deterministic graphs.…”

“…On the one hand, one can employ the so-called graph limit method ( [57]) to describe the limit of (1.1) as N → +∞ by means of non-local diffusive models in the form ∂ t x(s, t) = I W (s, s * )(x(s * , t) − x(s, t)) ds * .…”

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

“…Thus, one might expect that in the limit as the size of the network goes to infinity, the dynamics of all three models are approximated by the same continuum model. In fact, for the Kuramoto models on K n and G(n, p) such limit was established in [28] and [30] respectively. It was shown that the solutions of the IVPs for these models for large n are approximated by the solutions of the IVP for the continuum equation ∂ ∂x u(x, t) = (−1) α I sin(u(y, t) − u(x, t))dy.…”

“…However, even for the Kuramoto models on K n and G(n, p), the results in [28,30] establish the proximity of solutions of the IVPs for discrete and continuum models only on finite time intervals, which is not sufficient to guarantee that the solutions of the discrete and continuum models have the same asymptotic behavior.…”

Continuum Limits of Nonlocal $p$-Laplacian Variational Problems on Graphs

“…Relying on what we did in [30], the rate of convergence of the solution of the discrete problem to the solution of the limiting problem depends on the regularity of the boundary bd(supp(K)) of the closure of the support. Following [37], we recall the upper box-counting (or Minkowski-Bouligand) dimension of bd(supp(K)) as a subset of R 2 :…”