Higher orders need higher standards
Network science has a hypergraph problem
TL;DR — Contrary to widespread claims in recent years, it’s not true that graphs encode “pairwise” interactions. Graphs encode neighborhoods—sets of nodes incident to other nodes—on which multivariate interactions can be defined. This has been the standard way to model networks for decades. Hypergraphs are only special cases of this standard and more general graph-based framework. We demonstrate this in a new work [1].
Hype cycles in network science
Network science has always been a heterogeneous field, with many interesting problems and far-reaching questions, but it has also always been prone to uneven standards—presumably due to its interdisciplinary appeal and low technical barriers to entry. A certain laxity in rigor is not, in itself, a serious issue; one could even argue it is essential to deal with complex problems at the frontier of knowledge. It allows for intuition and creativity to play a bigger role in accelerating understanding, with increased rigor coming at a later stage as a mechanism of error correction and consolidation.
Under these circumstances, it has always been part of the game to have to sift through the sands of confusion and irrelevance to find the valuable nuggets of rare metals and gemstones that make this field worthwhile. This is likely true in most active areas of science.
Occasionally, however, the sands accumulate into dunes, and we are caught in a hype sandstorm that obscures everything else. This has happened many times before, and the current installment concerns the so-called “higher-order networks” (HONs).
In past hype cycles—from the early claims of power laws being everywhere, the excessive number of redundant heuristics for network clustering, to the gratuitous generalization to multilayer and temporal systems—the overindulgence has always been grounded in an element of truth: Networks do often exhibit broad degree distributions; network clustering is both useful and important; different interaction modalities matter; and certainly many networks do evolve in time. None of these are trivial aspects, and there’s a lot of depth required to tackle them properly. So one ignores the excesses, or spends some time correcting the most egregious ones, and then moves on to the actual work.
The situation with HONs, however, seems quite different. In my view, it is built largely on superficial misconceptions and quite a bit of smoke and mirrors that obscure relatively simple facts. Once these issues are cleared away, very little appears to remain—at least nothing especially distinctive.
“Higher-order networks”
The HON literature centers on the idea of models defined on hypergraphs, a mathematical generalization of graphs in which links may connect not just pairs of nodes but larger sets of three or more. The goal is to study processes such as epidemic spreading, synchronization, and other dynamical phenomena parameterized on hypergraphs rather than on conventional graphs.
Exploring the usefulness of hypergraph abstractions is, of course, entirely legitimate. Statistical physics has employed hypergraphs (calling them factor graphs) for decades, using them as a core concept in the study of spin glasses, constraint satisfaction, error correction, community detection, and nonequilibrium systems (see Ref. [2] for an introduction). Hypergraphs are likewise central in algebraic topology and topological data analysis. Their utility in appropriate contexts is therefore neither new nor controversial. One might reasonably argue that hypergraphs were understudied within network science and thus merited greater attention. I myself have worked on problems involving hypergraphs [3] and, I must admit, even contributed to a perspective article that helped fuel the very hype I criticize here (more on this below).
Much of the HON literature, however, is not just about exploring hypergraphs; it seeks to do something more ambitious. It presents itself as having uncovered a more fundamental representation of complex systems, explicitly casting alternative frameworks as incomplete or outdated in general. Almost invariably, the introduction of hundreds—perhaps thousands—of HON papers begin with slight variations of the following mantra:
“While networks provide important representations of real-world systems, they are inherently restricted to describing pairwise interactions. However, many real systems involve interactions among groups of more than two elements, requiring hypergraphs rather than conventional graphs for a more faithful modeling, leading to the understanding of entirely new phenomena.”
—About a quarter of all authors in network science the past five years (paraphrased)
This narrative is repeated with remarkable persistence and is often treated as self-evident and intuitive. Rhetorically, it sounds both obvious and progressive: pairwise interactions are “limited,” group interactions are “realistic,” and therefore hypergraphs are “necessary.” Once stated in this way, the conclusion appears to follow almost automatically. As a result, the vast majority of works built on this premise simply adopt it as a starting point, rarely pausing to articulate precisely what is meant, what assumptions are being made, or what—if anything—is genuinely precluded by conventional graph-based descriptions.
The claim thus functions less as a carefully examined hypothesis and more as a ritual demarcation. It establishes a sense of conceptual rupture—the assertion that we have long been looking at complex systems through an impoverished lens, and that only now, through the language of hypergraphs, can we finally see clearly.
Taking the hype out of hypergraphs
In a new work with Leto Peel, Thilo Gross, and Manlio de Domenico1 [1], we scrutinize the premises underlying the HON literature and the conclusions drawn from them, showing that they all disintegrate under close examination.
Our analysis goes beyond typical gripes one could have with this literature.
For example, a natural first reaction to claims about the supposed uniqueness of hypergraphs is to recall that they admit an equivalent representation as bipartite graphs: each hyperedge can be mapped to a factor node connected to the nodes it contains. From this perspective, assertions of hypergraph uniqueness relative to graph-based representations seem to evaporate rather quickly. Indeed, many papers ostensibly about hypergraphs read like papers about conventional bipartite graphs once one runs the text through sed s/hypergraph/bipartite graph/.
This is not a trivial observation. It implies that hypergraphs are intrinsically compatible with standard tools of network analysis and that, whether one likes this fact or not, any generative model for a hypergraph is simultaneously a model for a bipartite graph—and conversely. Frankly, for the same reason, I always found that the development of stand-alone software frameworks dedicated to hypergraphs to be ill-conceived—they all internally use bipartite graph representations, since these are the most efficient, and hence could have been developed on top of established graph-based frameworks without any loss at all; on the contrary, it would have avoided fragmentation, incompatibility, and code duplication.2
2 The specification of standard file formats for hypergraphs seem also equally redundant, and in fact quite counter-productive. Since hypergraphs are indistinguisable from bipartite graphs, the most obvious and meaningful choice would have been to use established graph formats, not create a new one for this special case alone.
On top of this, I never fully understood the appeal of using such cumbersome formalisms as simplicial complexes, which seem far removed from any reasonable modeling of real world systems that are the typical subject of study in network science, such as socio-economical and biological networks. The raison d’être of complexity science—in fact of science in general—is to find simple mechanisms that explain seemingly complex behavior. If the best one can do to model reality is to represent it as, say, a multilayer, time-varying, simplicial complex, I think this counts mostly as a failure of this program.
But, the above considerations should be put in the background if nature demands it. Indeed, this is how the HON literature frames things, insisting on the following very specific set of claims, often stated explicitly and repeatedly:
- Graphs encode only “pairwise interactions,”
- 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.
- “Group interactions” give rise to new phenomenology, not explainable by graph-based models.
The central observation we make is that this set of claims rests on a rather elementary conflation between structure and function: A graph does not define interactions; it merely constrains them.
Beyond the edge: two edges
Suppose we model a system with a graph and then ask someone to simulate the dynamics it represents. They would immediately face an obvious question: which dynamics? An epidemic spreading? Coupled oscillators? A diffusion process? And if so, which specific formulation?
To specify the dynamics, we have to actually define the interactions, which are based on, but are absent from, the graph description. In other words, for a node \(i\), all a graph does is define the neighborhood set \[ \partial i = \{ j \mid (i,j) \in \mathcal{E} \}, \] where \(\mathcal{E}\) is the edge set. This doesn’t yet define any interaction. It tells us who interacts with node \(i\), but not how they interact. To do that we have to give extra information: what are the interaction values and functions. For example, if our system is defined on continuous scalar variables \(x_i\), that evolve in time via a system of ordinary differential equations, in general we can write \[ \dot x_i = f_i(x_i, \boldsymbol x_{\partial i}), \tag{1}\] where \(f_i\) is a multivariate function that depends simultaneously on the values of all neighbors, i.e. \[ \boldsymbol x_{\partial i} = \{x_j \mid j \in \partial i\}. \] Once we acknowledge this elementary fact, we notice immediately that there is nothing “pairwise” about this picture. Even though the graph is defined on node pairs, the interaction functions can be in general multivariate. This is the classic separation between “structure” and “function” in network science [4]. One would imagine that this would be common knowledge in the field, but apparently not everyone got the memo.
Therefore, the claim that graph-based models can only encode “pairwise” interactions is not only untrue, but also ahistorical, since there are many network models defined in the literature that take advantage of this kind of generality, from Kauffman’s boolean networks, to threshold dynamics, and many others.
This simple fact about graph-based models has important consequences. One of them is that graph-based models generalize hypergraph-based ones, not the other way around!
Graph models generalize hypergraph models
This is simple to explain: since graphs only constrain the interactions by specifying their domain, and as they contain more information, hypergraphs can only add further constraints to the interaction functions. Indeed, hypergraph parameterizations typically inform interaction functions that act on all members of a hyperedge in a symmetric and mutually coherent manner. Graphs leave completely open the kinds of interactions possible, which means they can accommodate exactly the same functions one can define on hypergraphs, but in fact many others that hypergraphs cannot.
In addition, we also show in the paper that, from a purely parametric perspective, ignoring the interaction functions, multilayer networks also generalize hypergraphs, further undermining the central premises of the HON literature.
No network phenomena are “invisible” to graph models
The above generalization hierarchy already completely disables the claim that phenomenology observed with hypergraph models is “invisible” to graph-based models [5]. However, if one inspects the alleged unique phenomena more closely, such as the onset of abrupt transitions in synchronization [6] or contagion [7], one sees that they rely on homogeneous mean-field calculations that completely disregard the hypergraph structure. The abrupt transitions described occur in exactly the same way on locally tree-like graphs, which do not possess any cliques, let alone hyperedges. Therefore, these works actually prove the opposite of what they claim: the hypergraph structure is neither necessary nor relevant to explain the observed phenomena. Furthermore, as we show, these abrupt transitions belong to the same class as bootstrap (\(k\)-core) percolation and interdependent percolation—two classic and well-known graph processes, which somehow are never referenced in the HON literature. As it turns out, it seems that it is the correct interpretation of mean-field calculations and the broader literature of cooperative phenomena in networks that can be invisible to some.
Absence of empirical evidence
Networks and hypergraphs are not literal features of the world but abstract tools we use to model complex systems. A single system can often be represented in multiple mathematical ways, and choosing among them involves balancing interpretability, versatility, and empirical support.
Network models have historically offered a powerful balance of simplicity and analytical depth, supported by a rich ecosystem of mature mathematical tools. While hypergraphs provide a more structurally specific language for representing particular kinds of group interactions, their analytical framework is still comparatively limited. In the absence of clear mathematical advantages, we contend that networks—including multilayer extensions—currently offer a more practical and versatile foundation for modeling many complex systems.
That said, empirical performance should be the final arbiter: if hypergraphs yield more parsimonious or predictive descriptions in specific contexts, they should be preferred.
However, as we argue, the HON literature frequently makes clearly excessive claims about hypegraphs being “ubiquitous” and “essential” to model real world systems while giving virtually no real evidence to support them.
We examine several common arguments presented as evidence for the ubiquity of hypergraphs and find them lacking. Toy models formulated with hypergraphs may reproduce interesting phenomena, but they do not demonstrate that real systems require hypergraph structure, especially when the exact same behaviors can often be generated by graph-based models. Likewise, reinterpreting bipartite data as hypergraphs frequently amounts to a relabeling exercise rather than a substantive empirical advance. Attempts to impute hypergraphs from graph data or reconstruct them from time series typically rely on heuristics, suffer from non-identifiability, and lack rigorous model comparison. Without principled statistical frameworks [8] that test hypergraph models against graph-based alternatives, these approaches cannot establish that the particular complexity of hypergraph parametrizations is warranted by the data.
This persistent lack of decisive evidence is not accidental but reflects a deeper epistemic problem. Any system representable as a hypergraph can also be represented as a graph; the real differences lie in assumptions about hidden interaction rules, which are rarely observed directly in empirical data. For instance, a protein–protein interaction database typically indicates only whether the interaction is believed to be positive or negative; a survey reporting friendship ties between students does not specify how gossip or infection would propagate; and a neuronal map does not reveal firing dynamics. This pattern holds for virtually all network data. Because these rules are typically latent and measurements are incomplete, it may be impossible to obtain direct proof that one representation is uniquely correct. Instead, as in all of science, models must be evaluated comparatively, with preference given to those that offer the best balance of parsimony, predictive accuracy, and explanatory power. In this light, hypergraphs should be viewed as one option among many—not as universally necessary, but as tools whose value must be demonstrated case by case.
Falling from the edge: post mortem
The observations we make are not particularly deep, and one can’t help but wonder how such misconceptions went on for so long as they did unchecked.
I believe the rapid rise of publications on HONs is not difficult to understand. These works do not require a great deal of creativity. The usual protocol is just to take any old idea and apply it to a hypergraph. If something superficially different is seen, call that a novel phenomenon, throw in an unjustified assertion that this is “invisible” on graph-based representations, collect citations, lather, rinse, repeat. Bonus points if you combine it with generalizations such as multilayer and temporal. If you write a paper on temporal multilayer simplicial complexes, you win the HON bingo. Careers can and have been made based on such triteness.
I’m not doing justice to all papers on hypergraph models, since there are surely many interesting ones—I’m talking about the sand, not the gemstones.
Given the widespread hype, it’s unlikely many will snap out of the “hypergraphs go brrr” mentality very soon, but for those that actually wish to understand something meaningful about the world, value quality over quantity and substance over hype, the above considerations are relevant.
Hopefully our work will help at least to cut down the grandiose, self-serving, and misleading rhetoric that permeates these papers, which can get a bit embarrassing.
More constructively, the distinction between multivariate interactions and hypergraphs that we elucidate could motivate more refined models and analyses of network phenomena. What other ways do we have of formulating multivariate interactions that break away from the restrictive hypergraph framework but are also analytically convenient in some cases? How can we develop network reconstruction methods that can wield the full expressive power of graph-based formulations? How do we develop principled inference methods to select between different network abstractions? Answering these questions, and others, requires us to be actually creative and venture into new ground—not reachable by only recycling old ideas.
Mea culpa
I have to admit that I share some responsibility for the current state of the HON literature. In 2021, I co-authored a perspective article [9] that, in retrospect, helped fuel a lot of the hype. The paper had fourteen authors, and it was hardly an “indivisible group interaction.” I was invited to contribute to a specific section on statistical inference—especially how to reconstruct hypergraphs from indirect data—building on recent work at the time [3], which I thought was an important and worthwhile direction.
My reading of the paper back then was that it called for richer models of multibody interactions, not for replacing graphs wholesale with hypergraphs. That idea was already in the text, but at the time I considered it innocuous—specially given the counterpoint I was helping provide with a call for principled statistical methods. I didn’t anticipate how influential that particular framing would become. It was only later, once the topic started to go viral, that I realized how literally it was being interpreted.
To my surprise and mortification, the paper now has over a thousand citations and is still being cited in support of claims I don’t necessarily agree with. As more recent publications indicate [5], the hype has not quite given way to more critical reflection. That association is something I’m not especially proud of.
This is a lesson in academic promiscuity and carelessness that I won’t soon forget. In fact, part of my motivation behind the current work of Ref. [1] is to revisit some of these issues and clarify what I think the right perspective should have been all along—even if the original effects were unintended.
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