Continuum limit of the nonlocal p-Laplacian evolution problem on random inhomogeneous graphs

TL;DR

This paper investigates the continuum limit of the nonlocal p-Laplacian evolution problem on inhomogeneous random graphs, establishing convergence rates and analyzing parameter effects.

Contribution

It provides the first rigorous analysis of the continuum limit and convergence rates for the nonlocal p-Laplacian on inhomogeneous random graphs.

Findings

Established continuum limits for the nonlocal p-Laplacian evolution problem.

Derived convergence rates depending on graph parameters and data regularity.

Analyzed the influence of p and data geometry on solution behavior.

Abstract

In this paper we study numerical approximations of the evolution problem for the nonlocal $p$-Laplacian operator with homogeneous Neumann boundary conditions on inhomogeneous random convergent graph sequences. More precisely, for networks on convergent inhomogeneous random graph sequences (generated first by deterministic and then random node sequences), we establish their continuum limits and provide rate of convergence of solutions for the discrete models to their continuum counterparts as the number of vertices grows. Our bounds reveals the role of the different parameters, and in particular that of $p$ and the geometry/regularity of the data.