Balanced Colorings and Bifurcations in Rivalry and Opinion Networks

TL;DR

This paper investigates balanced colorings in symmetric networks, focusing on their role in synchronization patterns and bifurcations, especially in Wilson and opinion networks, revealing conditions for exotic colorings and their impact on stability.

Contribution

It characterizes exotic balanced colorings in Wilson and opinion networks and proves their absence in simple Wilson networks with limited learned patterns.

Findings

Exotic colorings exist in certain network types.

Wilson networks with up to two learned patterns lack exotic colorings.

Exotic colorings influence bifurcation structures and stability.

Abstract

Balanced colorings of networks classify robust synchrony patterns -- those that are defined by subspaces that are flow-invariant for all admissible ODEs. In symmetric networks the obvious balanced colorings are orbit colorings, where colors correspond to orbits of a subgroup of the symmetry group. All other balanced colorings are said to be exotic. We analyze balanced colorings for two closely related types of network encountered in applications: trained Wilson networks, which occur in models of binocular rivalry, and opinion networks, which occur in models of decision making. We give two examples of exotic colorings which apply to both types of network, and prove that Wilson networks with at most two learned patterns have no exotic colorings. We discuss how exotic colorings affect the existence and stability of branches for bifurcations of the corresponding model ODEs.