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

physical review. e

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