Graphs are maximally expressive for higher-order interactions

We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on "higher-order networks" that graph-based representations are fundamentally limited to "pairwise" interactions, requiring hypergraph formulations to capture richer dependencies. We clarify this issue by emphasizing two frequently overlooked facts. First, graph-based models are not restricted to pairwise interactions, as they naturally accommodate interactions that depend simultaneously on multiple adjacent nodes. Second, hypergraph formulations are strict special cases of more general graph-based representations, as they impose additional constraints on the allowable interactions between adjacent elements rather than expanding the space of possibilities. We show that key phenomenology commonly attributed to hypergraphs -- such as abrupt transitions -- can, in general, be recovered exactly using graph models, even locally tree-like ones, and thus do not constitute a class of phenomena that is inherently contingent on hypergraphs models. Finally, we argue that the broad relevance of hypergraphs for applications that is sometimes claimed in the literature is not supported by evidence. Instead it is likely grounded in misconceptions that network models cannot accommodate multibody interactions or that certain phenomena can only be captured with hypergraphs. We argue that clearly distinguishing between multivariate interactions, parametrized by graphs, and the functions that define them enables a more unified and flexible foundation for modeling interacting systems.