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.
In recent years there has been a surge of interest in modeling interacting systems via so-called "higher-order networks" (HONs), characterized by hypergraph parametrizations of interactions involving more than two elements [1][2][3][4][5][6][7][8]. This literature proposes hypergraphs as the foundation of a general theory of complex systems, claiming a level of explanatory power unattainable by graph-based formulations. On this basis, some authors advocate that graphs should be universally supplanted by hypergraphs as the most elementary representational object [1,7,8], relegating graph-based models to special cases. The alleged strict superiority of hypergraphs rests on the following claims:
Graphs encode only “pairwise interactions.” 2. Hypergraphs encode “group interactions,” indivisible interaction units with more than two elements that cannot be represented by graphs.
Many systems are better modeled with “group interactions,” and hence hypergraphs. 4. “Group interactions” give rise to new phenomenology, not explainable by graph-based models.
These claims follow a different line of reasoning and do not engage substantially with the long tradition in statistical physics of employing hypergraphs as bipartite factor graphs [9,10]-a framework central to the theory of constraint satisfaction [9,11], error correction [12,13], spin glasses [14], statistical inference [15], community detection [16], algorithmic hardness [17], and non-equilibrium disordered systems [18]. The notion of “pairwise interactions” in the HON literature conflates the structure of a graph-where edges connect pairs of nodes-with the functional form of interactions defined on those edges. This conflation suggests that if a system is represented by a graph, then the interactions of a node with its neighbors must decompose into independent or additive pairwise terms. However, graphs define neighborhoods, i.e. the set of nodes adjacent to a given node, not the interactions themselves. The functions defined on these neighborhoods can be arbitrarily complex and multivariate, depending on all adjacent nodes simultaneously in nonlinear ways. In this work, we show that once this conflation is resolved, graph-based formulations are revealed to be not inferior in expressive power, but in fact more general than hypergraphbased ones.
Specifically, we demonstrate that a hyperedge implies that a certain interaction plays out between a fixed set of nodes, in a manner coherently experienced by every member of the set. As a consequence, interactions across different hyperedges compose at most additively or in an otherwise simple manner. Hypergraphs therefore impose structure on the interactions that graph-based models leave open, making hypergraph models the more restrictive class. It follows that every phenomenon observable in a hypergraph model must also be observable in a graph-based model.
We examine prominent phenomenological claims attributed uniquely to hypergraph models, including abrupt transitions in synchronization [19], population dynamics [20][21][22][23], epidemic spreading [24,25], and equilibrium spin models [26,27], and show that identical behavior can be obtained with graphbased formulations that are asymptotically locally tree-like. We also show that hypergraph frameworks for dynamical systems [28][29][30] can be equivalently formulated using multilayer networks [31,32], obviating the need for hypergraphs. Beyond the theoretical arguments, we highlight the lack of empirical evidence supporting the utility of hypergraph-based models: graph data almost never contain information beyond neighborhoods, and rarely encode interaction rules explicitly. Such information is typically latent and must therefore be inferred statistically. Furthermore, as the size of interaction groups increases, these interactions are increasingly likely to exhibit sparse internal structure that is naturally represented by standard graph-based models. Hypergraph formulations therefore require explicit justification, rather than being assumed as a default.
This paper is organized as follows. In Sec. II we demonstrate that graphs are not confined to “pairwise interactions.” In Sec. III we show that hypergraph formulations are special cases of graph-based models, and in Sec. IV we show how every specific hypergraph formulation can also be converted into a multilayer network, but not vice versa. In Sec. V we turn to phenomena that have been attributed to hypergraph structure, and show they can be reproduced identically by locally tree-like graphs. In Sec. VI we discuss the lack of empirical evidence for the alleged universal relevance of hypergraph models, and in Sec. VII we discuss general aspects of the reducibility of multivariate interactions. We conclude in Sec. VIII with a final discussion.
The HON literature frequently asserts that graphs encode only “pairwise interactions,” but does not precisely define this concept. Instead, edges in a graph are of
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