Continuum limit of the nonlocal p-Laplacian evolution problem on random inhomogeneous graphs
Provides theoretical guarantees for approximating PDEs on random graphs, relevant for network-based modeling and analysis.
The paper proves continuum limits and convergence rates for the nonlocal p-Laplacian evolution problem on inhomogeneous random graphs, showing that discrete solutions converge to continuum counterparts as graph size increases.
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.