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.

💡 Research Summary

The paper tackles a growing misconception in the higher‑order network literature: that ordinary graph‑based models are fundamentally limited to pairwise interactions and therefore cannot capture the richer dependencies that hypergraphs are purported to represent. The authors set out to demonstrate, both theoretically and empirically, that graphs are in fact fully expressive for higher‑order (multibody) interactions, and that hypergraphs are merely a constrained subclass of a more general graph‑based formalism.

The argument begins by redefining what a “graph model” means in the context of interaction modeling. Instead of viewing an edge as a static binary relation, the paper treats each node i as being governed by a function f_i that depends on the state of i and on the collective state of its neighboring set N(i). Formally, the system energy (or loss) can be written as H(x)=∑{i∈V} f_i(x_i, x{N(i)}). Because f_i can be any multivariate function of the entire neighbor set, the model naturally incorporates interactions of arbitrary order: a term may involve three, four, or more nodes simultaneously, without any need to introduce a separate hyperedge object. This perspective aligns with modern message‑passing neural networks and factor‑graph formulations, where the “message” can be a function of an arbitrary subset of variables.

Next, the paper shows that hypergraphs can be mapped onto ordinary graphs by introducing auxiliary “hyper‑nodes” and constructing a bipartite incidence graph. While this conversion proves that any hypergraph can be represented as a graph, it also reveals the restrictive nature of the hypergraph formalism: all nodes incident to a given hyper‑node are forced to interact in an identical way, effectively imposing additional constraints on the interaction function. In contrast, a plain graph allows each original node to have its own bespoke function f_i, thereby offering a strictly larger expressive space. The authors therefore argue that hypergraphs are not an expansion of the modeling universe but a specialization that discards many possible interaction patterns.

To counter the claim that certain phenomena—most notably abrupt phase‑transition‑like behavior—are exclusive to hypergraph models, the authors provide a rigorous analysis of locally tree‑like random graphs (e.g., Erdős–Rényi and configuration models). By equipping these graphs with suitably chosen higher‑order interaction functions, they analytically derive the same critical points and scaling exponents that have been reported for hypergraph systems. Numerical simulations corroborate the theory, showing that abrupt transitions emerge under the same conditions of average degree and interaction strength, regardless of whether the underlying structure is a hypergraph or an ordinary graph with multivariate node functions. This demonstrates that the observed phenomenology is a property of the interaction function rather than of the hypergraph topology per se.

The paper then surveys a range of application domains—social contagion, biochemical reaction networks, transportation logistics—where hypergraphs have been championed as essential. In each case, the authors trace the modeling requirement back to a multivariate dependency (e.g., a joint probability distribution over a set of agents or a cost function involving several commodities). They show that such dependencies can be encoded directly in graph‑based models by defining appropriate f_i or by using higher‑order graph neural networks, without incurring the data‑collection overhead or algorithmic complexity associated with explicit hyperedges. Empirical evidence from several benchmark datasets indicates that graph‑only approaches achieve comparable predictive performance to hypergraph‑based counterparts, calling into question the asserted superiority of hypergraphs.

In the concluding discussion, the authors issue four practical recommendations for researchers: (1) treat the choice of interaction function as the primary modeling decision, not the choice between graph and hypergraph; (2) verify whether a hypergraph actually adds expressive power beyond a graph with multivariate node functions before adopting it; (3) leverage the extensive toolbox of graph theory, spectral methods, and graph neural networks to handle higher‑order effects efficiently; and (4) adopt a unified terminology that distinguishes “multivariate interactions parameterized by graphs” from “the specific functional forms that define them.” By doing so, the community can avoid unnecessary fragmentation, reduce model complexity, and build on the mature theoretical foundations of graph‑based analysis.

Overall, the paper makes a compelling case that graphs are maximally expressive for higher‑order interactions, that hypergraphs are a restrictive special case rather than a necessary extension, and that many of the claimed advantages of hypergraphs are either ill‑founded or can be replicated within a graph‑centric framework. This insight has significant implications for the design of future network models across physics, biology, sociology, and machine learning.