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Abstract 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 Evripidouet al(2022J. Phys. A: Math. Theor.55325201) and graph embedding by Paik and Griffin (2024Phys. Rev.E107L052202). 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. 

Abstract We introduce and study the spatial replicator equation with higher order interactions and both infinite (spatially homogeneous) populations and finite (spatially inhomogeneous) populations. We show that in the special case of three strategies (rock–paper–scissors) higher order interaction terms allow travelling waves to emerge in non-declining finite populations. We show that these travelling waves arise from diffusion stabilisation of an unstable interior equilibrium point that is present in the aspatial dynamics. Based on these observations and prior results, we offer two conjectures whose proofs would fully generalise our results to all odd cyclic games, both with and without higher order interactions, assuming a spatial replicator dynamic. Intriguingly, these generalisations for $N\text{⩾}5$strategies seem to require declining populations, as we show in our discussion. 

Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. Here, we integrate evolutionary game theory and behavioral ecology into a unified statistical mechanics framework, allowing all agents (modeled as nodes) and their bidirectional, signed, and weighted interactions at various orders (modeled as links or hyperlinks) to be coded into hypernetworks. Such hypernetworks can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurrence of active and passive HOI can drive complex systems to evolve at multiple time and space scales. We apply the model to reconstruct a hypernetwork of hexa-species microbial communities, and by dissecting the topological architecture of the hypernetwork using GLMY homology theory, we find distinct roles of pairwise interactions and HOI in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species. 

Abstract Recent work has shown that pairwise interactions may not be sufficient to fully model ecological dynamics in the wild. In this letter, we consider a replicator dynamic that takes both pairwise and triadic interactions into consideration using a rank-three tensor. We study these new nonlinear dynamics using a generalized rock-paper-scissors game whose dynamics are well understood in the standard replicator sense. We show that the addition of higher-order dynamics leads to the creation of a subcritical Hopf bifurcation and consequently an unstable limit cycle. It is known that this kind of behaviour cannot occur in the pairwise replicator in any three-strategy games, showing the effect higher-order interactions can have on the resulting dynamics of the system. We numerically characterize parameter regimes in which limit cycles exist and discuss possible ways to generalize this approach to studying higher-order interactions.