Pattern Formation in Random Networks Using Graphons

Abstract

We study Turing bifurcations on one-dimensional random ring networks where\nthe probability of a connection between two nodes depends on the distance\nbetween the two nodes. Our approach uses the theory of graphons to approximate\nthe graph Laplacian in the limit as the number of nodes tends to infinity by a\nnonlocal operator -- the graphon Laplacian. For the ring networks considered\nhere, we employ center manifold theory to characterize Turing bifurcations in\nthe continuum limit in a manner similar to the classical partial differential\nequation case and classify these bifurcations as sub/super/trans-critical. We\nderive estimates that relate the eigenvalues and eigenvectors of the finite\ngraph Laplacian to those of the graphon Laplacian. We are then able to show\nthat, for a sufficiently large realization of the network, with high\nprobability the bifurcations that occur in the finite graph are well\napproximated by those in the graphon limit. The number of nodes required\ndepends on the spectral gap between the critical eigenvalue and the remaining\nones, with the smaller this gap the more nodes that are required to guarantee\nthat the graphon and graph bifurcations are similar. We demonstrate that if\nthis condition is not satisfied then the bifurcations that occur in the finite\nnetwork can differ significantly from those in the graphon limit.\n

Topics & Concepts