Keeping the interaction structure explicit: comment on"Graphs are maximally expressive for higher-order interactions"

Abstract

Peixoto et al. (arXiv:2602.16937v2) stress that graph-based formulations can express any interaction model, a point the literature on higher-order networks (HONs) has often overlooked. We agree with much of their critique. We argue that expressiveness, however, might not be the main reason hypergraphs are often preferred. A network is read from a model rather than assumed, and to properly study the role of structure-one of the most basic questions in complex systems research-one needs a representation that keeps that structure separate from the functional form of the interactions. We show that such representation is in general a directed hypergraph (equivalently, a directed factor graph), with one hyperarc per interaction term, recovering a (directed) graph for pairwise interactions, and an undirected hypergraph for symmetric, higher-order ones. Most of the HONs literature has focused so far on symmetric interactions, which, in light of what established here, might explain why there (undirected) hypergraphs are regularly presented as the natural representation for systems with higher-order interactions.