Bifurcations of mutually coupled equations in random graphs

TL;DR

This paper investigates how solutions of mutually coupled equations behave on heterogeneous random graphs, revealing that increased interaction strength causes a cascade of bifurcations and the emergence of stable solution subspaces.

Contribution

It provides a detailed analysis of bifurcation phenomena in coupled equations on heterogeneous networks, explicitly linking graph structure to solution stability.

Findings

Heterogeneity induces stable subspaces from initially unstable equations.

Increasing interaction strength causes a cascade of bifurcations.

Bifurcation scenarios are explicitly characterized by graph structure.

Abstract

We study the behavior of solutions of mutually coupled equations in heterogeneous random graphs. Heterogeneity means that some equations receive many inputs whereas most of the equations are given only with a few connections. Starting from a situation where the isolated equations are unstable, we prove that a heterogeneous interaction structure leads to the appearance of stable subspaces of solutions. Moreover, we show that, for certain classes of heterogeneous networks, increasing the strength of interaction leads to a cascade of bifurcations in which the dimension of the stable subspace of solutions increases. We explicitly determine the bifurcation scenario in terms of the graph structure.