Geiger, Philipp M.; Knebel, Johannes; Frey, Erwin
(2018):
Topologically robust zerosum games and Pfaffian orientation: How network topology determines the longtime dynamics of the antisymmetric LotkaVolterra equation.
In: Physical Review E, Vol. 98, No. 6, 62316

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Abstract
To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric LotkaVolterra equation (ALVE). The ALVE is the replicator equation of zerosum 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 zerosum games, such as the rockpaperscissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zerosum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the longtime dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graphtheoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio phi = 1.6180 ..., cycles with complete subnetworks, and nonHamiltonian networks. In graphtheoretical terms, we extend the concept of a Pfaffian orientation from evensized to oddsized networks. Our results show that the topology of interaction networks alone can determine the longtime behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.