Topologically robust zero-sum games and Pfaffian orientation -- How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra equation

TL;DR

This paper investigates how the topology of interaction networks influences the long-term dynamics of the antisymmetric Lotka-Volterra equation, revealing topologically robust zero-sum games called coexistence networks that ensure strategy coexistence.

Contribution

It introduces the concept of coexistence networks in zero-sum games, extending Pfaffian orientation to odd-sized networks, and provides graph-theoretical rules for constructing such robust networks.

Findings

Existence of topologically robust zero-sum games with all strategies coexisting.

Construction of coexistence networks with arbitrary strategies.

Extension of Pfaffian orientation concept to odd-sized networks.

Abstract

To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is the replicator equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio $\varphi = 1.6180...$, cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.