Pattern Formation in Random Networks Using Graphons

TL;DR

This paper investigates Turing bifurcations in large random ring networks using graphon theory, providing a continuum approximation and analyzing how finite network bifurcations relate to the graphon limit.

Contribution

It introduces a method to approximate bifurcations in large random networks via graphons, extending classical PDE bifurcation analysis to network structures.

Findings

Bifurcations in large networks are well approximated by the graphon limit.

Eigenvalue estimates relate finite graph Laplacian to graphon Laplacian.

The spectral gap influences the accuracy of bifurcation approximation.

Abstract

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