Beyond expressiveness in pairwise and higher-order models

Abstract

The debate over pairwise and higher-order models is often cast as a contest of expressive power, but this framing is misleading. A graph with arbitrary multivariate node functions can emulate the node-level dynamics of many hypergraph models, yet this does not erase the grouping information encoded by the hypergraph: it may simply be shifted from structure to dynamics. We distinguish four notions that are often conflated: structural projection, functional representability, statistical identifiability, and mechanistic adequacy. We show that the interaction order of a finite-state map is an invariant of the map itself, so an exact change of representation cannot reduce the underlying order of dependence. We then recast the comparison in terms of description length. At the unrestricted algorithmic level, a fixed compiler can move information between structure and rule with only constant overhead, so expressiveness alone does not privilege graphs or hypergraphs. Preferences arise only relative to explicit model classes, coding schemes, regularity assumptions, and data. This motivates an operational minimum-description-length criterion combining structural cost, rule cost conditional on structure, and model fit. For $M$ disjoint groups of size $k$, we show that the clique projection requires an edge list asymptotically $k-1$ times longer than the corresponding hyperedge list, making projection a more expensive encoding of the same grouping. Examples spanning diffusion, Boolean dynamics, ecological interactions, and ambiguous projections illustrate graph-preferred, hypergraph-preferred, and unresolved cases. The resulting position is symmetric: higher-order structure should not be inferred from phenomenology alone, but neither should graph-based emulation be taken as evidence that a graph is the most parsimonious or scientifically adequate description.