Let 𝐺={𝐺(ℓ)}ℓ∈ℒ be a hypergraph with 𝐿 unique (but not necessarily consecutive) layers 𝐺(ℓ) indexed by ℓ, each layer 𝐺(ℓ) containing all hyperedges of size ℓ from 𝐺. Let ℒ={ℓ1,…,ℓ𝐿} denote the set of 𝐿 unique layer indices. For example, a hypergraph 𝐺 with only layers ℓ=2 and ℓ =5 would have 𝐺={𝐺(2),𝐺(5)} and ℒ={2,5}. We consider 𝐺(ℓ) as a set of undirected, sorted tuples of size ℓ with no repeated entries, and let 𝐸(ℓ)=|𝐺(ℓ)| be the number of hyperedges in 𝐺(ℓ) (in other words, 𝐺 is a simple, undirected hypergraph). There are (𝑁ℓ) possible undirected, sorted tuples of size ℓ so 𝐸(ℓ)≤(𝑁ℓ). We also let 𝐺(𝑘→ℓ) be the projection of layer 𝑘 >ℓ onto the order of ℓ, which extracts all unique ℓ-tuples nested within the 𝑘-tuples in 𝐺(𝑘). For example, if 𝐺(3)={(0,1,2),(0,2,4)}, we would have 𝐺(3→2)={(0,1),(0,2),(1,2),(0,4),(2,4)}. We define 𝐸(𝑘→ℓ)=| | |𝐺(𝑘→ℓ)| | |, similarly to the unprojected layers, and use the convention 𝐺(ℓ→ℓ)=𝐺(ℓ).

The structural reducibility of a hypergraph can be defined based on the overlap among its pairs of layers , where overlap is defined based on the projection of each layer to the lower order of the two. This convention is required because higher-order hyperedges have unique projections onto lower-order interactions (as defined above), but one cannot conversely determine higher-order structure uniquely from lower-order structure alone [1]. Higher overlap among the layers indicates higher structural redundancy among different orders of hyperedges, suggesting a higher structural reducibility for the hypergraph . Formally, we can define the overlap of the layers indexed by as , which is the number of hyperedges the two layers share when they are projected to the lower order of the two layers.

Our proposed reducibility measure reflects the extent to which a hypergraph can be compressed in an information-theoretic sense when we exploit the structural redundancy among its hyperedge layers of different orders (i.e., their layer overlaps). To formalize this concept mathematically, we consider transmitting the hypergraph to a receiver using two different schemes. In the first (naïve) scheme, we transmit each of ’s layers individually. In the second scheme, we first transmit a set of “representative” layers which capture most of the heterogeneity in the hyperedge structure in , and then we transmit each remaining layer as a noisy copy of a representative layer . A similar concept has been employed to compress multilayer network structures [30] and sets of network partitions [31] using intermediate representative structures.

We assume that the receiver knows the orders of the layers and the number of hyperedges for each —specifying these counts incurs a comparatively negligible information cost anyway. Since each layer has possible configurations of its hyperedges when is known, we need to send a bit string of length approximately equal to bits to fully specify which configuration corresponds to . The naïve transmission of as individual layers, therefore, requires an information content of

bits, using the convention for brevity.

A better way to transmit is to exploit the overlaps among layers of hyperedges of different sizes to save information. To do this, we first transmit a representative subset of layers indexed by , which incurs a cost of bits. Then we transmit each remaining layer by (a) transmitting a representative layer of a higher order, costing us as there are layers to choose from; (b) transmitting the overlap among the layer and its representative , costing us as ; and (c) transmitting layer given the constraints imposed by the overlap value . If of the possible edges in are present in , and of the possible edges absent from are present in , then we require bits to specify given knowledge of and . Steps (a) and (b) incur negligible information costs compared to step (c) and can be ignored. The total information content of this scheme is then

bits. For a minimal information cost, the representative layer for each layer can be assigned as

The information cost of this transmission scheme depends on which layers are selected as representatives—the better the layers capture the heterogeneity in the hyperedge structure in , the lower the information cost . Thus, to maximize compression from layer overlap, we must find the optimal set of representative layers according to

We will describe shortly how to solve this optimization problem.

The optimal information cost of this transmission scheme is bounded in the interval , where

The lower bound follows from always minimally needing to transmit the top layer of as a representative layer, at a cost , and the upper bound follows from being in the solution space over which we minimize to find . Therefore, exploiting layer overlap always provides compression relative to the naïve transmission of layers independently. The reducibility of the hypergraph can then be computed based on the extent to which can be compressed relative to the baseline cost of .

Using these bounds, we can construct a properly normalized structural reducibility measure for a hypergraph as

which satisfies . If is maximally compressible—i.e., is a nested hypergraph where all layers with are given by —then we have and and, thus, . On the other hand, we have an information cost when is highly incompressible (i.e., has little to no structural overlap among its layers), as there is little shared information that can be exploited to improve on the naïve information cost of ; thus, .

In Supplemental Material [35], we discuss extending our reducibility concept to understand the structural redundancy of multiscale coarse-grainings of hypergraphs (Sec. I), as well as individual hypergraph layers (Sec. V) and individual hyperedges (Sec. VI).

By solving Eq. (4) to maximize compression of , we can also obtain a compressed hypergraph representation for given by , which captures the critical higher-order structure of while removing structurally redundant layers. In Fig. 1(a), we show an example hypergraph with layers and its corresponding reduced representation of layers , giving a reducibility value of .

FIG. 1.

To identify the optimal representative layers , when —a value satisfied by most real hypergraph datasets [32]—we can use a simple brute-force search. Letting be the largest hyperedge size in , we must have , since there is no higher layer from which to transmit . Therefore, we have possible subsets of representatives that we must search through. In our method we must first compute , , and for all pairs of layers with . For layer pairs with small values , we can directly compute and using the projection and the lower layer . For larger values, however, we cannot compute the projection directly. In this case, we can compute by iterating through the edges in a fixed order, for each edge checking its overlaps with all previously checked edges. Then, we can compute the number of new projected tuples that contributes to as , where is the number of unique subtuples of size within the set of overlapping tuples , which can be computed recursively using the same approach. When computing the projection is infeasible, we also must compute in a more efficient way. We do this by iterating over hyperedges and incrementing for each edge that fully overlaps with , removing from afterward.

We can then compute a matrix such that

for layer pairs . The computation of certain conditional entropies is often the computational bottleneck in practice and takes roughly run-time for large where projection is intractable and run-time when using direct projection. We also compute a vector with entries

storing the individual layer information costs for all layers . Then, for each valid subset , we can compute

where . We then select as the representative subset that achieves the minimum value of . There are subsets to check, and each takes in the worst case operations to compute for each layer . The iteration over thus has a time complexity of roughly , which in practice is tractable for most real-world datasets with (see Table I).

For systems with many layers , optimizing over subsets through direct enumeration is unfeasible. In such cases, we use an approximate greedy method for identifying to compute the structural reducibility. Starting with , we can iteratively add the best layer to following the rule until . Then, we choose among the explored solution candidates to find the representative layer set giving the lowest information cost . In practice, this approximation always obtains accurate results (see Sec. IV in Supplemental Material [35]), significantly speeding up the computation and making our method available to otherwise intractable datasets.

To validate our approach, we investigate our method on synthetic hypergraphs with tunable structure. First, we consider nested hypergraphs, where all lower-order interactions are fully encapsulated into those of higher order. Noisy nested hypergraphs are nested hypergraph where a noise parameter determines the fraction of its hyperedges to be rewired, replacing each selected hyperedge with a hyperedge of the same order drawn uniformly at random. In Fig. 1(b), we plot the results of this test, averaged over ten realizations of the randomness for each value of . Standard errors (vanishingly small) are shown as error bars. We can observe that indicates complete reducibility when for each fully nested hypergraph and that decreases smoothly as the hypergraph becomes noisier, eventually bottoming out at for purely random hypergraphs ( ). As the system gets larger ( increases), we observe greater reducibility values, since a larger fraction of the layers are structurally redundant due to being nested within the top layer .

In a followup experiment, we examine the reducibility of more general synthetic hypergraphs with nested structure. We start by generating three planted representative layers , , and on nodes with , , , 1050, and 50 hyperedges, respectively, drawn uniformly at random without replacement. We then generate the layers , , and as noisy versions of the projected layers , , and , respectively, by selecting a fraction of the hyperedges in each projected layer randomly and replacing these hyperedges with those of the same size drawn uniformly at random. In Fig. 2, we plot layer-layer similarity matrices showing the network normalized mutual information [33] (over arbitrary tuple sizes) for pairs of layers in the generated hypergraphs, illustrating the effect of on the nested structure. As increases, we see a smooth decrease in the structural reducibility , with our method able to infer the planted set of representative layers . We compare our results with dynamical reducibility [32], which reduces structure based on the collective behavior supported by the hypergraph rather than topological overlap and which remains close to zero without detecting any change in this simple but nuanced hypergraph structure. All results are averaged over 20 realizations of the randomness at each . Section II in Supplemental Material [35] further investigates the reducibility of hypergraphs with tunable nested interactions.

FIG. 2.

We also examine our proposed multiscale reducibility measure in a similar experimental setting. For an node system, we synthesize each random hypergraph using a planted community partition by generating each of the hyperedges in layer as follows: (i) choose a random node to start a hyperedge ; (ii) add nodes to , drawing each node (without replacement) from the same community as with probability and from a different community with probability . The result is a hypergraph that is more tightly clustered under the planted node partition as and uncorrelated with as . All results are averaged over ten realizations of the randomness at each , and separate experiments are run for equally sized communities in . We observe that the standard reducibility of Eq. (6) cannot detect any changes in the mesoscale nested structure, while the multiscale reducibility of Eq. (S3) in Supplemental Material [35] exhibits a smooth descent as increases. For larger , we see that the communities in become smaller and the hypergraph becomes less reducible under the coarse-graining , approaching the standard structural reducibility value.

Finally, we apply our reducibility method to a range of real higher-order networks [34], as shown in Table I. We find a great variety in the reducibility of these systems, with many systems most parsimoniously represented by only a small subset of their layers. In Supplemental Material [35], we further investigate the nested organization of real-world systems in Sec. III, while in Sec. IV we compare run times of the exact and greedy optimization methods for finding , showing that the greedy scheme is considerably faster especially for larger hypergraphs. Finally, we explore the structural and dynamical properties of all reduced empirical hypergraphs in Sec. VII. We find that the reduced hypergraph representations consistently preserve the global, mesoscale, and local connectivity of the complete empirical hypergraphs—as quantified by the effective number of connected components, community structure, and degree ordering, respectively. We also find that these reduced systems preserve the consensus times in higher-order voter model dynamics [36]. In general, we observe that such properties are better preserved as the reducibility increases, due to improved compressibility of the original hypergraph structure.

Reducing the dimensionality of higher-order systems allows for more efficient analyses, with simpler interpretations and visualization. Here, we have developed a principled, efficient, and interpretable information-theoretic framework for assessing the structural reducibility of hypergraphs and removing structural redundancies to construct compressed hypergraph representations retaining the critical higher-order structure of complex networked systems. There are a number of ways in which this framework can be extended in future work to directed, weighted, temporal, or multilayer hypergraphs. This would allow the method to be applied to representations that capture additional nuances of the relational structure in a wider variety of systems. Our Letter sheds new light on the organizational principles of higher-order networks, distinguishing the extent to which lower-order information is redundant in the presence of higher-order information.

A. K. acknowledges support from the National Science Foundation of China (NSFC) through Young Scientist Fund Project No. 12405044. F. B. acknowledges support from the Austrian Science Fund (FWF) through Projects No. 10.55776/PAT1052824 and No. 10.55776/PAT1652425.

The data that support the findings of this Letter are openly available [34]. Code is available as part of the library Hypergraphx [42].