The Cost of Representing Higher-Order Dynamics on Graphs: A Theoretical Framework. Leo Torres.

Can graphs represent the dynamics of higher-order (multi-body) interactions, or are hypergraphs necessary? We decompose network dynamics into three components: the combinatorial structure (graph or hypergraph), per-edge coupling functions, and per-node aggregation functions. Within this framework, we prove that graphs can always reproduce the trajectories of any hypergraph dynamics by absorbing the coupling logic into the aggregation. This generalizes a recent result by Peixoto et al., who showed the same under the assumption of flat dynamics. However, the graph representation pays a price in parsimony: when the coupling and aggregation functions are polynomials of degree at most D, the graph aggregation requires Θ(k^D/D!) parameters relative to the hypergraph alternative, where k is the interaction size. Without symmetry, the comparison can reverse. The framework also clarifies two other recent contributions by Moutuou and by Lacasa, and identifies overlooked modeling degrees of freedom including partial interaction symmetries and the role of non-additive aggregation.