Integrability of generalised skew-symmetric replicator equations via graph embeddings

It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville-Arnold integrable, building on prior work in graph decloning by Evripidou et al (2022 J. Phys. A: Math. Theor. 55 325201) and graph embedding by Paik and Griffin (2024 Phys. Rev. E 107 L052202). Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results, along with numerical methods, to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs.