Generalized Hamiltonian Dynamics and Chaos in Evolutionary Games on Networks

TL;DR

This paper explores the dynamics of evolutionary games on networks, revealing conditions for fixed points, the emergence of chaos in certain graph structures, and linking dynamical properties to graph theory.

Contribution

It introduces a generalized Hamiltonian framework for network replicator dynamics and characterizes when chaos can occur in these systems.

Findings

Complex behavior cannot emerge in 2x2 games on networks.

Chaos appears in RPS on 3-cycle graphs.

Network replicator with RPS on larger graphs is a generalized Hamiltonian system.

Abstract

We study the network replicator equation and characterize its fixed points on arbitrary graph structures for $2 \times 2$ symmetric games. We show a relationship between the asymptotic behavior of the network replicator and the existence of an independent vertex set in the graph and also show that complex behavior cannot emerge in $2 \times 2$ games. This links a property of the dynamical system with a combinatorial graph property. We contrast this by showing that ordinary rock-paper-scissors (RPS) exhibits chaos on the 3-cycle and that on general graphs with $\geq 3$ vertices the network replicator with RPS is a generalized Hamiltonian system. This stands in stark contrast to the established fact that RPS does not exhibit chaos in the standard replicator dynamics or the bimatrix replicator dynamics, which is equivalent to the network replicator on a graph with one edge and two vertices ($K_2$).