Abstract
We propose highorder hypergraph walks as a framework to generalize graphbased network science techniques to hypergraphs. Edge incidence in hypergraphs is quantitative, yielding hypergraph walks with both length and width. Graph methods which then generalize to hypergraphs include connected component analyses, graph distancebased metrics such as closeness centrality, and motifbased measures such as clustering coefficients. We apply highorder analogs of these methods to real world hypernetworks, and show they reveal nuanced and interpretable structure that cannot be detected by graphbased methods. Lastly, we apply three generative models to the data and find that basic hypergraph properties, such as density and degree distributions, do not necessarily control these new structural measurements. Our work demonstrates how analyses of hypergraphstructured data are richer when utilizing tools tailored to capture hypergraphnative phenomena, and suggests one possible avenue towards that end.
Introduction
In the study of complex systems, graph theory is often perceived as the mathematical scaffold underlying network science [1]. Systems studied in biology, sociology, telecommunications, and physical infrastructure often afford a representation as a set of entities (“vertices”) with binary relationships (“edges”), and hence may be analyzed utilizing graph theoretic methods. Graph models benefit from simplicity and a degree of universality. But as abstract mathematical objects, graphs are limited to representing pairwise relationships between entities. However, realworld phenomena in these systems can be rich in multiway relationships involving interactions among more than two entities, dependencies between more than two variables, or properties of collections of more than two objects.
Hypergraphs are generalizations of graphs in which edges may connect any number of vertices, thereby representing kway relationships. As such, hypergraphs are the natural representation of a broad range of systems, including those with the kinds of multiway relationships mentioned above. Indeed, hypergraphstructured data (i.e. hypernetworks) are ubiquitous, occurring whenever information presents naturally as setvalued, tabular, or bipartite data. Additionally, as finite set systems, hypergraphs have identities related to a number of other mathematical structures important in data science, including finite topologies, simplicial complexes, and Sperner systems. This enables use of a wider range of mathematical methods, such as those from computational topology, to identify features specific to the highdimensional complexity in hypernetworks, but not available using graphs. Although an expanding body of research attests to the increased utility of hypergraphbased analyses, many network science methods have been historically developed explicitly (and often, exclusively) for graphbased analyses. Moreover, it is common that data arising from hypernetworks are reduced to graphs.
Before proceeding, let us consider an example. Figure 1 illustrates two authorpaper datasets, which may be naturally structured as a hypergraph by representing authors as vertices, and the set of authors appearing on each paper as hyperedges.^{Footnote 1} The hypergraph derived from the rightmost network exhibits higherorder relationships by virtue of having papers with 3 authors. Comparing these examples highlights structural information retained and lost between graph and hypergraph representations. For instance, both networks are similar in that each pair of authors A, B, C has coauthored a paper (in fact, exactly two papers) together. This is captured by the coauthorship graph (center), which is therefore identical for these two networks. However, there are also clear differences not captured by the graph representation. For instance, each author appears on 4 versus 2 papers, and each paper features 2 versus 3 authors. Beyond these basic counts, these networks also exhibit more subtle differences: for any pair of authors in the leftmost network, the set of papers they’ve coauthored is different from the set of joint papers between any other pair of authors, whereas in the rightmost, every pair of authors has coauthored exactly the same set of papers. As this toy example suggests, while graphs do capture some properties of hypernetworks, they are insufficient as hypergraph substitutes.
In spite of this incongruity between graph and hypergraph analyses, effectively extending graph theoretical tools to hypergraphs has sometimes lagged or proven elusive. A critical aspect of this is axiomatization: as a generalization there are many, sometimes mutually inconsistent, sets of possible definitions of hypergraph concepts which can yield the same results consistent with graph theory when instantiated to the graph case. In some cases, developing any coherent hypergraph analog poses significant theoretical obstacles. For example, extending the spectral theory of graph adjacency matrices to hypergraphs poses an immediate challenge in that hyperedges may contain more than two vertices, thereby rendering the usual (twodimensional) adjacency matrix insufficient for encoding adjacency relations. In other cases, graph theoretical concepts may be trivially extended to hypergraphs, but in doing so ignore structural nuance native to hypergraphs which are unobservable in graphs. For instance, while edge incidence and vertex adjacency can occur in at most one vertex or edge for graphs, these notions are setvalued and hence quantitative for hypergraphs. Consequently, while subsequent graph walk based notions, such as connectedness, are immediately applicable to hypergraphs, they ignore highorder structure in failing to account for the varying “widths” associated with hypergraph walks.
Due to these challenges, scientists seeking tools to study hypergraphstructured data are frequently left to contend with disparate approaches towards hypergraph research. One approach for grappling with hypergraph complexity is to limit attention to hypergraphs with only uniformly sized edges containing the same number of vertices. Much of the hypergraph research in the mathematics literature, such as in hypergraph coloring [2, 3], the aforementioned spectral theory of hypergraphs [4, 5], hypergraph transversals [6], and extremal problems [7], focus on this kuniform case only. While imposing this assumption facilitates more mathematically sophisticated and structurally faithful analysis of the hypergraphs in question, realworld hypergraph data is unfortunately very rarely kuniform. Consequently, such tools are problematic in lacking applicability to real hypernetwork data. Another approach towards hypergraph research is to limit attention to transformations of (potentially nonuniform) hypergraphs to graphs. Sometimes called the hypergraph line graph, 2section, clique expansion, or onemode projection, such transformations clearly enable the application of graphtheoretic tools to the data. Yet, unsurprisingly, such hypergraphtograph reductions are inevitably lossy [8, 9]. Hence, although affording simplicity, such approaches are of limited utility in uncovering hypergraph structure.
To enable analyses of hypernetwork data that better reflect their complexity but remain tractable and applicable, we believe striking a balance between this faithfulnesssimplicity tradeoff is essential. With this goal at heart, we extend a number of graph analytic tools popular in network science to hypergraphs under the framework of highorder hypergraph walks. We characterize a hypergraph walk as an “swalk”, where the order s controls the minimum walk “width” in terms of edge overlap size. Highorder swalks (\(s>1\)) are possible on hypergraphs whereas for graphs, all walks are 1walks. The hypergraph walkbased methods we develop include connected component analyses, graphdistance based metrics such as closenesscentrality, and motifbased measures such as clustering coefficients. As each of these methods is based fundamentally on the graphtheoretic notion of a walk, we extend them to hypergraphs by using hypergraph walks. Ultimately, our goal is not only to formulate these generalizations in a cogent manner, but to probe whether these tools reveal prevalent and meaningful structure in real hypernetwork data. To the latter end, we compute these measures based on hypergraph walks on three real datasets from different domains and discuss the results.
Our work is organized as follows: in Sect. 2, we provide background definitions and review preliminary topics relevant to hypernetwork theory. In Sect. 3, we define the swalk notion underpinning our subsequent work, discuss related prior research, and reiterate our contributions. In Sect. 4, we introduce swalk based analytical measures, apply them to the aforementioned datasets, and analyze the results. In Sect. 5, we consider three generative hypergraph models, and experimentally test the extent to which the structural properties observed in Sect. 4 can be replicated by synthetic models. Finally, in Sect. 6 we conclude and outline several directions for future research.
Preliminaries
Hypergraphs are generalizations of graphs in which edges may link any number of vertices together. Just as “network” is often used to refer to processes or systems yielding data streams which are graphstructured, we will use the term “hypernetwork” to refer to those yielding hypergraphstructured data. More formally, we define a hypergraph as follows:
Definition 1
A hypergraph\(H=(V,E)\) is a set \(V=\{v_{1},\ldots,v_{n}\}\) of elements called vertices, and an indexed family of sets \(E=(e_{1},\ldots,e_{m})\) called hyperedges in which \(e_{i} \subseteq V\) for \(i=1,\ldots, m\).
When the hypergraph is clear from context, we call its hyperedges simply “edges”. The degree of a vertex is the number of hyperedges to which it belongs, \(d(v)= \vert \{e: v \in e\} \vert \), and the size of a hyperedge is its cardinality, \(\vert e \vert \). A hypergraph in which all hyperedges have size k is called kuniform, and a 2uniform hypergraph is simply a graph.^{Footnote 2} Definitions of hypergraphs given in the literature may differ slightly from author to author. For instance, Bretto’s hypergraph definition [10] is identical to ours, apart from prohibiting empty edges (\(e_{i}\) such that \(e_{i}=\varnothing \)). Berge [11] similarly prohibits empty edges, as well as isolated vertices (\(v_{i}\) such that \(v_{i} \notin \bigcup_{i=1}^{m} e_{i}\)). In contrast, Katona [12] allows empty edges and isolated vertices, but defines \(E=\{e_{1},\ldots,e_{m}\}\) as a set and explicitly prohibits pairs of duplicated edges \(e_{i}=e_{j}\) for \(i\neq j\). In defining E as an (indexed) family of sets, we allow for duplicated edges but require edges be distinguishable by index. Returning to the leftmost authorpaper network in Fig. 1, in the corresponding hypergraph with authors as vertices, the hyperedges corresponding to papers 1 and 4 are examples of duplicate edges: they are equivalent as sets yet distinguishable by paper title. The generality of Definition 1 in permitting isolated vertices, as well as empty, duplicated, and singleton edges is intended to facilitate the application of hypergraphs to real data, which commonly possess such features.
Definition 2
The incidence matrixS of a hypergraph \(H=(V,E)\), is a \(\vert V \vert \times \vert E \vert \) matrix defined by
Under Definition 1, any rectangular Boolean matrix uniquely defines a labeled hypergraph;^{Footnote 3} conversely any labeled hypergraph uniquely defines an incidence matrix. Consequently, there is a bijection between hypergraphs and bicolored graphs. Recall a bicolored graph is a triple \((V,E,f)\) where V is a set of vertices, E is a set of pairs of vertices, and \(f: V \to \{0,1\}\) satisfies \(f(v_{i}) \neq f(v_{j})\) for all \(v_{i},v_{j} \in V\) where \(\{v_{i},v_{j}\} \in E\). Indexing rows and columns by vertices such that \(f(v_{i})=0\) and \(f(v_{j})=1\), respectively, incidence matrices may be uniquely associated with bicolored graphs by defining \(S(i,j)=1\) for \(\{v_{i},v_{j}\}\in E\). Note bicolored graphs specify a fixed bicoloring f and differ from bipartite graphs, which are graphs admitting some bicoloring. Accordingly, a bipartite graph with k connected components has \(2^{k}\) possible bicolorings, each of which may correspond to a distinct hypergraph. In applications, however, the data often comes with a bicoloring (e.g. in an authorpaper network) and hence the terms “bipartite” and “bicolored” graphs have been used synonymously. For the purposes of this work, gearing our exposition towards hypergraphs rather than bicolored graphs is more natural because our approach is settheoretic.^{Footnote 4} Beyond these bijective correspondences, mathematical research on hypergraph categories and their isomorphisms requires careful consideration [13–15].
As an upshot of the hypergraphbicolored graph correspondence, a number of complex network analytics for bipartite data extend naturally to hypergraphs, and vice versa. However, interpreting this in light of the fact that bicolored graphs are graphs does not mean graph theoretic methods suffice for studying hypergraphs. Whether interpreted as bicolored graphs or hypergraphs, data with this structure often require entirely different network science methods than (general) graphs. An obvious example is triadic measures like the graph clustering coefficient: these cannot be applied to bicolored graphs since (by definition) bicolored graphs have no triangles. Detailed work developing bipartite analogs of modularity [16], community structure inference techniques [17], and other graphbased network science topics [18] further attests that bipartite graphs (and hypergraphs) require a different network science toolset than for graphs.
Another important topic highlighted by the bicolored graphshypergraph correspondence is the duality of hypergraphs. That is, just as it may be arbitrary to label one partition in a bicolored graph “left” and the other partition “right”, which class of objects one designates as “vertices” versus “hyperedges” in a hypernetwork may also be arbitrarily chosen. However, hypergraph properties and methods may be vertexbased or edgebased, and hence differ depending on which choice is made. To avoid limiting one’s analysis towards either a vertexcentric or edgecentric approach, it may be prudent to consider both the hypergraph and its dual hypergraph. Loosely speaking, the dual of a hypergraph is the hypergraph constructed by swapping the roles of vertices and edges. More precisely:
Definition 3
Let \(H=(V,E)\) be a hypergraph with vertex set \(V=\{v_{1},\ldots,v_{n}\}\) and family of edges \(E=(e_{1},\ldots,e_{m})\). The dual hypergraph of H, denoted \(H^{*}=(E^{*},V^{*})\), has vertex set \(E^{*}=\{e_{1}^{*},\ldots,e_{m}^{*}\}\) and family of hyperedges \(V^{*}=(v_{1}^{*},\ldots,v_{n}^{*})\), where \(v_{i}^{*} := \{e_{k}^{*} : v_{i} \in e_{k}\}\).
Put equivalently, the dual of a hypergraph with incidence matrix S is the hypergraph associated with the transposed incidence matrix, \(S^{T}\). Clearly, \((H^{*})^{*}=H\). Furthermore, observe that two vertices belonging to the same set of edges in H correspond to multiedges in \(H^{*}\) and isolated vertices in H correspond to empty edges in \(H^{*}\). Thus, the generality of our Definition 1 in permitting multiedges, empty edges, and isolated vertices ensures the dual of a hypergraph is also a hypergraph. Indeed, as a formal matter, one could go so far as to always consider that hypergraphs present in dual pairs.^{Footnote 5}
Continuing this line of observation, in the complex networks literature, one of the most oftused tools for studying hypergraph data is its line graph. In the line graph of a hypergraph, each vertex represents a hyperedge, and each edge represents an intersection between a pair of hyperedges. More formally:
Definition 4
Let \(H=(V,E)\) be a hypergraph with vertex set \(V=\{v_{1},\ldots,v_{n}\}\) and family of hyperedges \(E=(e_{1},\ldots,e_{m})\). The line graph ofH, denoted \(L(H)\), is the graph on vertex set \(\{e_{1}^{*},\ldots,e_{m}^{*}\}\) and edge set \(\{\{e_{i}^{*}, e_{j}^{*}\}: e_{i} \cap e_{j} \neq\varnothing \text{ for } i \neq j \}\).
In order to additionally capture information about the size of hyperedge edge intersections, line graphs of hypergraphs may be defined with additional edge weights where \(\{e_{i}^{*},e_{j}^{*}\}\) has weight \(\vert e_{i} \cap e_{j} \vert \). By definition of matrix multiplication, it is easy to see the line graph of a hypergraph with incidence matrix S has edgeweighted adjacency matrix \(S^{T}S\) with diagonal entries all converted to zero. Figure 2 gives an example of a hypergraph, its dual, and their respective line graphs. All hypergraph visualizations in this paper were created using HyperNetX (HNX) [19], a recently released^{Footnote 6} Python library echoing NetworkX [20] for exploratory hypergraph data analytics.
Hypergraph line graphs are also referred to by a plethora of other names. Berge [11] refers to \(L(H)\) as both the “line graph” and “representative graph” of H; Naik [21, 22] refers to \(L(H)\) as the “intersection graph” of H. In the complex networks literature on bipartite graphs or “2mode” graphs, the oftmentioned “onemode projections” are equivalent to hypergraph line graphs. For instance, \(L(H)\) and \(L(H^{*})\) are referred to as the “top and bottom” projections in [18], similarly, [23] dubs these the “column and row” projections. Moreover, \(L(H^{*})\) is commonly referred to as the “2section”, “clique graph”, or “clique expansion” of the hypergraph H, since the edge set of \(L(H^{*})\) is generated by taking all 2element subsets of each edge in H, hence vertices within each hyperedge H form a clique in \(L(H^{*})\). Consequently, if G is a graph, \(L(G^{*})\) is identical to G.
Line graphs play an important role in hypernetwork science. Due to the relative dearth of hypergraph analytic tools, line graphs are often analyzed in place of the hypergraphs they were derived from so that classical network science techniques can be applied. However, hypergraph line graphs are fundamentally limited in several ways. First, line graphs are lossy representations of hypergraphs in the sense that distinct hypergraphs can have identical line graphs. We note such structural loss does not occur for graphs, as Whitney’s theorem [24] states, apart from the triangle and 4vertex star graph, any pair of connected graphs with isomorphic line graphs must be isomorphic. In the case of hypergraph line graphs, Kirkland [9] recently illustrated the structural loss in a severe sense by giving an example of two distinct \(19 \times 19\) incidence matrices S and R respectively, such that both
Put equivalently in the language of hypergraphs: although nonisomorphic, the weighted line graphs of the hypergraphs represented by S and R, as well as those of their dual hypergraphs, are both identical. Kirkland also constructed infinite familities of such pairs of hypergraphs and showed they constitute a vanishingly small proportion of hypergraphs. Accordingly, while one isn’t likely to encounter such pairs of hypergraphs in empirical data, Kirkland’s work illustrates structural properties of hypergraphs may be lost even when simultaneously accounting for hypergraph duality and using weighted line graphs. Consequently, depending on the properties under consideration, the extent to which line graphs faithfully represent hypergraphs may be unclear. Nonetheless, researchers have offered preliminary evidence that some meaningful, albeit incomplete, hypergraph structure can be extracted from their line graphs [23].
Lastly, as noted in [18, 25], another important limitation of hypergraph line graphs is computational: sparse hypergraphs can still yield relatively dense line graphs that may be difficult to analyze or store in computer memory. This can be easily seen by observing that kway edge intersections (guaranteed by a vertex of degree k) in the hypergraph yield \(\binom{k }{2}\) edges in its line graph. Particularly if the hypergraph is large and its vertex degree and edge cardinality distributions are heavily skewed (common features in real world network data), its line graphs may be too dense to analyze computationally or even construct at all.
From graph walks to hypergraph walks
One of the most fundamental concepts in graph theory, underpinning a myriad of areas including Hamiltonian and Eulerian graphs, distance and centrality measures, stochastic processes on graphs and PageRank, is that of a walk. For a graph \(G=(V,E)\), a walk of lengthk is a sequence of vertices \(v_{0},v_{1},\ldots,v_{k}\), such that each pair of successive vertices are adjacent. By definition of a (simple) graph, two adjacent vertices belong to exactly one edge, and conversely, two incident edges intersect in exactly one vertex. Consequently, any valid graph walk can be equivalently expressed as either a sequence of adjacent vertices or as a sequence of incident edges, i.e.
In the setting of hypergraphs, this simple observation no longer holds. Hypergraph edge incidence and vertex adjacency is setvalued and quantitative in the sense that two hyperedges can intersect at any number of vertices, and two vertices can belong to any number of shared hyperedges. This motivates two walk concepts for hypergraphs that are dual but distinct: walks on the vertex level (consisting of successively adjacent vertices), and walks on the edge level (consisting of successively intersecting edges). For ease of presentation, and to be consistent with related prior work, we limit our exposition to edgelevel hypergraph walks. Nonetheless, both notions are captured when duality is considered, as a vertexbased walk on a hypergraph H is simply an edgewalk on the dual hypergraph \(H^{*}\). We define a hypergraph walk as an “swalk” on a hypergraph, where s controls for the size of edge intersection, as follows:
Definition 5
For a positive integer s, an swalk of length k between hyperedges f and g is a sequence of hyperedges,
where for \(j=1,\ldots,k\), we have \(s \le \vert e_{i_{j1}} \cap e_{i_{j}} \vert \) and \(i_{j1} \neq i_{j}\).
When interpreted on the dual hypergraph \(H^{*}\), an swalk corresponds to a sequence of adjacent vertices in which each successive pair of vertices belong to at least s shared hyperedges. Since in a graph a pair of vertices can belong to at most 1 edge, the usual graph walk between vertices x, y on a graph G is equivalent to a 1walk between hyperedges \(x^{*}\), \(y^{*}\) on the dual, \(G^{*}\). Consequently, the \(s=1\) case recovers the usual graph walk and swalks for \(s>1\) are only possible on hypergraphs.
As will become apparent in subsequent sections, a number of basic yet important properties of walks in graphs immediately extend to swalks on hypergraphs. For instance, just as any graph walk ending at vertex \(v_{k}\) can be concatenated with any walk starting at vertex \(v_{k}\) to form another walk, any swalk ending at a particular edge can be concatenated to any other swalk starting at the edge. Consequently, the existence of an swalk between hyperedges defines an equivalence relation under which hyperedges can be partitioned into sconnected components, which we explore in Sect. 4.2. Furthermore, this also ensures the length of the shortest swalk between edges, called sdistance (Sect. 4.3), satisfies the triangle inequality and defines a bonafide distance metric on the hypergraph. Finally, in Sect. 4.4 we explore how one may distinguish between different kinds of swalks in a hierarchical way, and how the subsequent notions of straces, smeanders, spaths, and scycles lend themselves to discerning substructures native to hypergraphs, such as striangles. For readers interested in random walks on hypergraphs, the Appendix includes a brief discussion of recent literature and its relationship to swalks.
Prior work
Many researchers have considered different notions of “highorder walks” on hypergraphs, abstract simplicial complexes, and related set systems. Concepts closely related to swalks have for long appeared in the mathematics literature. Bermond, Heydemann, and Sotteau [33] introduced and analyzed kline graphs of uniform hypergraphs, which are derived from hypergraphs by representing each hyperedge as a vertex, and linking two such vertices if their corresponding hyperedges intersect in at least k vertices. In this way, a (graph) walk on their line graphs corresponds to an swalk on a hypergraph. In [34], Lu and Peng define higher order walks on hypergraphs for kuniform hypergraphs as sequences of edges intersecting in exactlys vertices, where the vertices within each edge are ordered. Their work is related to a rich literature on Hamiltonian cycles in kuniform hypergraphs (e.g. [35, 36]) and takes a spectral approach: these generalized walks are used to define a socalled sLaplacian matrix. Wang and Lee [37] define hypergraph paths as edge sequences in which no successive intersection is a subset of any other. Their motivation is to prove enumeration formulas for certain cycle structures in hypergraphs. In a series of three recent papers [38–40], Kang, Cooley, Koch, and others consider a notion of swalk between stuples of vertices. They conduct a rigorous mathematical analysis of the asymptotic swalk properties of binomial random kuniform hypergraphs, considering hitting times, the evolution of highorder scomponents, and highorder “hypertree” structures. Lastly, in [41, 42], authors of the present work briefly considered the swalk based notion of sdistance as applied to Domain Name System (DNS) cyber data, and the Enron email dataset, respectively.
Contributions
The main contributions of the present work are:

developing hypergraph generalizations of graph network science measures using the swalk framework,

applying these new measures to real hypernetworks, analyzing and comparing the results, discussing structural insights they reveal, and

experimentally testing the degree to which existing generative hypergraph null models are able to replicate the properties seen in real data captured by these measures.
To make clear how these new hypergraph metrics generalize their graph counterparts, when appropriate we include a definition subheading called “Graph case & equivalence”. This subheading addresses two distinct questions: first, it describes how the definition(s) in question reduce to the graph case when the hypergraph is a graph. As we will see, most of the proposed hypergraph measures reduce to a graph analog by taking \(s=1\) and examining the dual of the graph, \(G^{*}\) (taking the dual converts our edgebased exposition to match the vertexbased notions common in network science).
Second, this heading describes whether the hypergraph measure is equivalent to a graph measure on the hypergraph’s sline graphs (Definition 7), which are generalizations of the aforementioned line graph. In the case of sconnected components and sdistance measures (Sects. 4.2–4.3), these have natural equivalences on, and thus may be obtained via, sline graphs. However, the spath, scycle, and sclustering coefficients (Sect. 4.4) rely on subset information not encoded in, and hence not discernible from, the sline graphs. Furthermore, properties of the hypergraph generative models we consider, such as hypergraph degree distributions and metamorphosis coefficients, also cannot be determined from the sline graphs. A takeaway from this is that our study of swalks and hypernetwork science includes, but extends beyond, the study of sline graphs.
Lastly, we briefly compare our approach with that of the related research surveyed in “Prior work” above: while the present work is similarly based around the concept of highorder hypergraph walks, we utilize them for different ends. In particular, we use this framework to develop network science concepts that are aimed at messy, real hypergraph data. In contrast to all the work mentioned above, apart from that of Wang and Lee [37], we do not assume kuniformity, as real hypernetworks are frequently nonuniform. Furthermore, our methods apply to disconnected hypergraphs and permit duplicate hyperedges—both of which are also common in real data. Additionally, we make design choices to ensure our methods are more computationally tractable in light of the combinatorial explosion inherent in hypergraphs. For instance, as opposed to Lu and Peng who define swalks between arbitrary orderedstuples of vertices,^{Footnote 7} defining our notion of swalk between pairs of unordered hyperedges (or when working with the dual, pairs of single vertices) permits the development of methods more tractable in application to real data.
Hypergraph walk framework
In this section, we explore how analytic tools from network science extend to hypergraphs in the hypergraph walk framework. Within each subsection, we focus on one topic (e.g. sdistance, a hypergraph geodesic), and introduce relevant methods in the “Methods” section. In the “Application to Data” section, we apply these methods to real hypernetworks and analyze the results. Our goal is not to explain why the observed structure exists using domainspecific analyses. Rather, we identify abstract structural properties revealed by these measures, and highlight how these properties differ within each dataset (as we vary the walk order s), as well as across datasets. This illustrates particular properties these measures capture, as well as new insights revealed by considering swalk based metrics. While we take a broader, methodsbased viewpoint here, we believe such an approach may be more useful in guiding future applicationspecific studies of these methods across multiple domains. To that end, we consider three datasets from three domains: corporate governance, biology, and text analysis.
Test data sets
For each dataset, we define the associated hypergraph, review prior graph or hypergraph analyses of these (or closely related) datasets, and discuss basic properties which are summarized in Fig. 3. In this figure and throughout, we use the same notation as in Definition 4 to refer to the dual hypergraph associated with a data set (e.g. LesMis^{*} refers to the dual hypergraph of LesMis, in which the roles of the vertices and edges are swapped relative to how they are defined below). Figure 3 plots the edge cardinality distribution and pairwise edge intersection cardinality distribution. For instance, the point \((x,y)=(3,100)\) on the “edges” distribution means there are 100 edges which consist of exactly 3 vertices in that hypergraph; the same point on the “pairwise edge intersection” distribution means there are 100 distinct, unordered pairs of hyperedges whose intersection contains exactly 3 vertices. We remind the reader the edge cardinality distribution of the dual hypergraph \(H^{*}\) is the same as the vertex degree distribution of H.
The table in Fig. 3 highlights basic statistics for each hypergraph. The number of “multiedges” is the number of edges that duplicate (in the sense of set equality) another edge, i.e. \(\vert \{i: e_{i}=e_{j} \text{ for } j< i\} \vert \). The maximum edge size and edge intersection sizes, reported below the multiedge counts, are particularly pertinent because they determine the range of interest for our measures: the former determines the largest value of s for which swalk based measures are defined while the latter determines the maximum value of s for which swalk based measures are nontrivial. Finally, “density” measures the number of vertexhyperedge memberships relative to the number of possible vertexhyperedge memberships. Put equivalently, this is the number of nonzero entries in the incidence matrix S divided by the product \(\vert V \vert \cdot \vert E \vert \). By definition, density is always the same for the hypergraph and its dual, whereas the other reported values are edgebased and may differ.
CompBoard

Data set. A companyboard network. Vertices represent people and hyperedges represent company boards. A vertex belongs to a hyperedge if that person sits on the company board. The data consists of 4573 companies and 32,189 people. Companies are identified by ticker symbols, excluding any location or exchange code suffixes (e.g. Vodafone group is represented solely by VOD, not VOD.L or VOD.O) taken from the NYSE, AMEX, and NASDAQ stock exchange listings^{Footnote 8} on 10/1/2018. The data was collected from publicly available^{Footnote 9} board director information listed on Reuters. Board director names were cross referenced against age data to better distinguish different people with the same full name.

Prior work. Companyboard network studies are historically rooted in corporate elite theory, focusing on companies which share a common board member called interlocking directorates. Many such studies focus on line graph representations of the network, linking companies whose boards interlock. For instance, Conyon and Muldoon [43] studied the smallworld properties of companyboard networks from the US, UK, and Germany, focusing on the clustering coefficient and average path length of the line graphs. In [44], Newman compares the clustering coefficient of a companyboard network line graph to that of a random model.^{Footnote 10} Levine and Roy [47] appear to be among the first to analyze bipartite representations of companyboard networks directly, rather than solely line graphs. They considered topics such as the average path length, connected component sizes, and proposed a “rubberband model”^{Footnote 11} to cluster the bipartite network. Later, Robins and Alexander devised a bipartite global clustering coefficient, based on the ratio of bipartite 4cycles to 3paths, to measure “the extent to which directors remeet one another on two or more boards” [48]. In Sect. 4.4, we propose a new notion of hypergraph clustering coefficients and explain how it compares to that of Robins and Alexander, as well as graph clustering coefficients measured on the line graph. More generally, since an “interlocking directorate” is represented by a hyperedge intersection, our methods can be interpreted in this context as not only based on the existence of interlocks (i.e. a pure line graph analysis) but also their size and relative set relationships.

Basic properties. The edge size distribution shows the sizes of company boards are tightly concentrated around 7–10 members and drop off sharply at either end: only about 3% of companies have fewer than 4 members, and 3% have more than 14. In contrast, the edge size distribution of the dual hypergraph is monotonically decreasing, showing more than 99% of board members belong to between 1–3 company boards. The pairwise edge intersection distribution for the hypergraph and its dual similarly exhibit a sharp decrease, and the range of these distributions imply different companies share up to a maximum of 12 board members, while different members serve on up to 5 common company boards. Among the three datasets, CompBoard is the sparsest: it contains about 0.03% of possible vertexhyperedge memberships, as opposed to 0.33% and 2.68% for Diseasome and LesMis, respectively.
Diseasome

Data set. A human genedisease network from [49]. Vertices represent genes and hyperedges represent genetic disorders. A vertex belongs to a hyperedge if mutations in that gene are implicated by that disease. The data consists of 903 genes and 516 diseases.

Prior work. Goh et al. [49] collected the list of genes, disorders, and their associations from the Online Mendelian Inheritance in Man (OMIM) [50] compendium in 2005. Their study considered the line graphs of the hypergraph and its dual, which they dubbed the Human Disease Network and Disease Gene Network. They show the size of the largest connected component in these networks differed with those generated by random models. In Sect. 4.2 we study a generalized notion of highorder connected components and compare these against those of random hypergraph models in Sect. 5.1. For a broader discussion of the potential applications of hypergraphs and hypergraph statistics in biology and genomics, see [51].

Basic properties. The edge size distributions of Diseasome and its dual show the most genes implicated by a disease is 41 while the most diseases implicated by a gene is 11. The pairwise edge intersection size distribution show 94% of pairs of diseases implicating common genes share exactly 1 gene; conversely, examining this distribution for the dual hypergraph reveals 98% of gene pairs associated with a common disease share exacltly 1 disease. Among the three datasets, Diseasome and its dual features the narrowest range of pairwise edge intersection sizes, with maximum edge sizes of 5 and 4, respectively.
LesMis

Data set. A characterscene network from [52]. Vertices represent characters and hyperedges represent scenes from Victor Hugo’s novel, Les Misérables. There are 80 characters and 402 scenes.

Prior work. This dataset was collected by Donald Knuth [52] and can be structured according to different granularities in which hyperedges represent the scenes, chapters, books, or volumes of the novel. The line graph of the LesMis hypergraph, often dubbed the Les Mis coappearence network, has appeared frequently in network science literature for the purpose of demonstrating clustering or modularity methods [53, 54], or centrality and ranking methods [55]. With regard to the latter, we apply our proposed hypergraph centrality measure to rank LesMis characters in Sect. 4.3.

Basic properties. In LesMis and its dual, the largest hyperedge features 9 characters and 137 scenes, respectively, with the latter hyperedge (unsurprisingly) corresponding to the protagonist, Jean Valjean. Compared against other datasets, the edge intersection size distributions are particularly distinct. LesMis^{*} features the largest range of edge intersection sizes across all datasets. Both LesMis and its dual are also notable for featuring the most edge intersections relative to the number of possible edge intersections: respectively, 22% and 8% of all pairs of edges in LesMis and its dual intersect, whereas this ratio is an order of magnitude smaller for Diseasome and its dual and two orders of magnitude smaller for CompBoard and its dual.
Connected components
Methods Under Definition 1, the graph notions of connectedness and connected components extend naturally to the swalk framework.
Definition 6
For a hypergraph \(H=(V,E)\), a subset of hyperedges \(C \subseteq E\) is called sconnected if there exists an swalk between all \(f,g \in C\) and is further called an sconnected component if there is no sconnected \(J \subseteq E\) such that \(C \subsetneq J\).
Since for any \(e \in E\), there can be no swalk from e to any other hyperedges if \(\vert e \vert < s\), the order of an sconnected component is bounded above by \(\vert E_{s} \vert \), where \(E_{s}=\{e \in E: \vert e \vert \geq s\}\). More precisely, for any positive integer s and hypergraph H, the edges in \(E_{s}\) can always be partitioned into sconnected components. We call a hypergraph sconnected if \(E_{s}\) is sconnected.
While an sconnected component of a hypergraph H is an equivalence class of edges, a vertexbased notion of sconnected components is obtained by simply applying the above definition to the dual hypergraph \(H^{*}\). In comparing these edge and vertexbased notions, note the number of 1connected components for H and \(H^{*}\) are always the same: it is straightforward to see, in either case, the number of such components is the same as the number of nontrivial connected components (i.e. excluding isolated vertices) in the bipartite graph representation of the hypergraph. In this sense, edge and vertexbased connectedness are equivalent for \(s=1\) and whenever H is a graph. However, for \(s\geq 2\), the number of sconnected components for H and \(H^{*}\) may differ. Hence, sconnectedness in hypergraphs is richer and more varied for highorders, yielding dual but distinct vertex and edgebased notions.
An effective way of visualizing and studying basic properties of sconnected components is via its sline graph. As previously mentioned, sline graphs were studied for kuniform hypergraphs by Bermond, Heydemann, and Sotteau [33] as early as 1977. A definition for the general case may be stated as follows:
Definition 7
Let \(H=(V,E)\) be a hypergraph with vertex set \(V=\{v_{1},\ldots,v_{n}\}\) and edge set \(E \supseteq E_{s}\) where \(E_{s}=\{e \in E: \vert e \vert \geq s\}=\{e_{1},\ldots,e_{k}\}\) for an integer \(s \ge 1\). The sline graph ofH, denoted \(L_{s}(H)\), is the graph on vertex set \(\{e_{1}^{*},\ldots,e_{k}^{*}\}\) and edge set \(\{\{e_{i}^{*}, e_{j}^{*}\}: \vert e_{i} \cap e_{j} \vert \geq s \text{ for } i \neq j \}\).
In other words, each vertex in the sline graph represents a hyperedge with at least s vertices in the hypergraph, and two vertices are linked in the sline graph if their corresponding hyperedges intersect in at least s vertices in the hypergraph. In this way, the 1line graph is simply the line graph from Definition 4 and the connected components of the sline graph represent the sconnected components of the hypergraph. Hence, we have:
Graph case & equivalence
If H is a graph, H is connected iff \(H^{*}\) is 1connected. A hypergraph H is sconnected iff \(L_{s}(H)\) is connected.
Table 1 presents examples of two hypergraphs and their associated sline graphs. Observe that both hypergraphs have identical 1line graphs. Nonetheless, comparing their sline graphs for \(s=2,3,4\) suggests differences otherwise lost when solely considering the (usual) line graph, which sline graphs generalize.
Although more general, sline graphs are still subject to a limitation underlying (the usual) hypergraph line graphs: they do not uniquely identify a hypergraph, up to isomorphism. For instance, while we previously observed the two authorpaper networks in Fig. 1 to be different, the corresponding hypergraphs formed by letting hyperedges denote authors have the same sline graphs for \(s=1,2\) and either trivial or empty sline graphs for \(s>2\). More generally, Kirkland’s aforementioned work [9] shows even when duality is considered, sline graphs may not uniquely identify a hypergraph. Nevertheless, sline graphs can be utilized to determine a number of informative swalk properties, including sdistance, which we explore in the next section. It is worth repeating, however, the study of highorder hypergraph swalks is not limited to sline graphs. As we will see in Sect. 4.4, sline graphs cannot distinguish between finer classes of swalks, such as smeanders and spaths, and consequently cannot be used to compute sclustering coefficients, for example. Returning again to the examples in Fig. 1, we will see that while these hypergraphs have identical nontrivial sline graphs, they are distinguished by their spaths.
Application to data
Table 2 presents the scomponents of LesMis, Diseasome, and CompBoard \(s=1,\ldots,5\) by visualizing their sline graphs. A pagesized version of this table is included in the Appendix.
Qualitative differences are readily apparent from the visualization. For LesMis, the majority of hyperedges are contained within a giant component for \(s=1,\ldots,5\). This means one can link most characters with each other via a pathway of characters cooccurring in at least one to five scenes together. For \(s=1\) in Diseasome, we similarly observe a giant component; however, for \(s\geq 2\), this giant component fragments into small, roughly equally sized components. Here, as s increases from one to five, many of the sharedgene pathways linking diseases for \(s=1\) break down. By \(s=5\), the scomponents consist almost entirely of isolated hyperedges (apart from a single pair of closely related diseases,“Diabetes Mellitus” and “Mature onset diabetes of the young (MODY)”) implying diseases associated with 5 or more genes do not share 5 or more of those genes with other disorders. The most dramatic fragmentation occurs for CompBoard. For \(s=1\), 74% of the companies are contained within the giant component (pictured in the lowerleft hand corner), while for \(s=2\), this drops to 0.5%. This affirms shared boardmember pathways linking companies almost always rely on single shared board members.
To quantify these changes in sconnected component sizes more rigorously, we compute several entropybased measures on the sconnected component size probability distribution, \(\boldsymbol{p}_{s}=\langle p^{s}_{1},\ldots,p^{s}_{k} \rangle \), defined by taking \(p^{s}_{j}\) to be the fraction of hyperedges in \(E_{s}\) that are in the j’th scomponent. For a discrete probability distribution p, its Shannon entropy is given by \(H(\boldsymbol{p})=\sum_{i=1}^{k} p_{i} \log _{2}(p_{i})\). However, direct comparisons of Shannon entropy on our data may be problematic, as the number of hyperedges in \(E_{s}\) and number of scomponents varies not only between datasets, but also as s varies within each dataset, thereby complicating crosscomparisons of the (unitless) Shannon entropy. To facilitate more meaningful entropy comparisons, we consider the normalized entropy
called psmoothness by the authors [56]. This normalization derives from the fact that H achieves its maximum value of \(\log _{2}(k)\) for the uniform distribution, \(\langle \frac{1}{k},\ldots,\frac{1}{k} \rangle \), and a minimum value of 0 for a fully skewed distribution, e.g. \(\langle 0,\ldots, 0, 1\rangle \). For the special case in which \(k=1\), one takes the limiting definition as \(k \to 1\), and defines \(\widetilde{H}(\boldsymbol{p}):= 1\).
We consider the psmoothness of the scomponent size distribution \(\boldsymbol{p}_{s}\), which we denote \(\widetilde{H}_{s} = \widetilde{H}(\boldsymbol{p}_{s})\). If all scomponents are equally sized, then \(\widetilde{H}_{s}\) is 1, whereas if the disparity between component sizes is maximal (e.g. \(\vert E_{s} \vert 1\) hyperedges in one scomponent, and 1 hyperedge in the other), then \(\widetilde{H}_{s}\) approaches 0. In this sense, psmoothness reflects how smooth or uniform the scomponent sizes are, but may not reflect how dispersedly the hyperedges are distributed among sconnected components. For that purpose, we consider an additional measure, again from [56], aptly called dispersion. The dispersion of the scomponent size distribution compares the number of scomponents to the number of possiblescomponents on a logarithmic scale, i.e.
where \(C_{s}\) denotes the set of sconnected components and \(E_{s}\) denotes the set of shyperedges.
Figure 4 plots the psmoothness and dispersion for \(s=1,\ldots, s_{\max }\), where \(s_{\max }= \max_{f,g \in E} \vert f \cap g \vert \). For \(s>s_{\max }\), the scomponents are either all isolated hyperedges, or nonexistent. In all datasets, both dispersion and psmoothness tend to increase in s, although, as evident from LesMis, this increase is not always monotonic. LesMis^{*} exhibits lower values of psmoothness for each of \(s=1,\ldots,5\) relative to those for corresponding values of s in the other datasets, consistent with the highly skewed distribution reflecting the large component we observed in the visualization. CompBoard exhibits a large separation between psmoothness and dispersion for \(s=1\). In this case, while the component size distribution is still skewed—and hence has low psmoothness—the remaining scomponents consist of many isolated hyperedges, reflected in the high dispersion value. Lastly, for Diseasome, psmoothness is maximal while dispersion is minimal for \(s=1\), and for \(s \geq 2\) both psmoothness and dispersion closely coincide at values near 1. This reflects the fragmentation of a single giant component into many scomponents (hence the high dispersion) that are equally sized (hence the high psmoothness).
Distance and centrality
Methods
Under Definition 1, it is straightforward to show the length of the shortest swalk serves as a distance metric function over a set of hyperedges. More precisely
Proposition 1
Let\(H=(V,E)\)be a hypergraph and\(E_{s}=\{e\in E: \vert e \vert \geq s \}\). Define thesdistance function\(d_{s}: E_{s} \times E_{s} \to \mathbb{Z}_{\geq 0}\)by
Then\((E_{s},d_{s})\)is a metric space.
We omit the proof, as the triangle inequality can be proved constructively, and the other metric space axioms follow immediately from Definition 3.
Graph case & equivalence
If H is a graph, then the graph distance between vertices x and y in H is equivalent to the 1distance between hyperedges \(x^{*}\) and \(y^{*}\) in \(H^{*}\). For a hypergraph H, the sdistance between x and y is equivalent to the graph distance between \(x^{*}\) and \(y^{*}\) in \(L_{s}(H)\). Consequently, the forthcoming sdistance based measures in Definitions 8–9 are equivalent to their graph counterparts on \(L_{s}(H)\) and, whenever H is a graph, reduce to their graph counterparts on \(H^{*}\) for \(s=1\).
With sdistance serving as hypergraph geodesic distance, hypergraph sanalogs of local and global distancebased graph invariants easily extend.
Definition 8
Let \(H=(V,E)\) be a hypergraph.

(i)
The seccentricity of a hyperedge f is \(\displaystyle\max_{g \in E_{s}} d_{s}(f,g)\).

The sdiameter is the maximum seccentricity over all edges in \(E_{s}\), while the sradius is the minimum.


(ii)
The averagesdistance of H is \(\displaystyle\binom{ \vert E_{s} \vert }{2}^{1} \displaystyle\sum_{f,g \in E_{s}} d_{s}(f,g)\).

(iii)
The scloseness centrality of a hyperedge f is \(\displaystyle\frac{ \vert E_{s} \vert 1}{\displaystyle\sum_{g \in E_{s}} d_{s}(f,g)}\).
Important caveats arise when applying Definition 8 to real data. As we’ve observed, H may contain more than one scomponent for some values of s, in which case the sdistance between some pairs of edges is infinite. Consequently, the seccentricity of every edge (and hence sdiameter and sradius) and mean sdistance are all infinite; similarly, the scloseness centrality of every edge is trivially 0. Similar to how these issues are sometimes addressed for graphs, one alternative is to compute these measures on only the largest scomponent. Depending on the analyst’s aims, this approach might be satisfactory, particularly if the majority of hyperedges in \(E_{s}\) are contained within the largest scomponent, as was seemingly the case in LesMis^{*}.
However, restricting to the largest component may be unsatisfactory in cases where the largest scomponent does not constitute the overwhelming majority of edges in \(E_{s}\), as in CompBoard for \(s\geq 2\). In such cases, one may wish to compute seccentricity on a percomponent basis, taking the extrema over all scomponents as the sdiameter and sradius. One may similarly compute mean sdistance or scloseness percomponent, however, it is unclear how to properly synthesize these values in order to obtain (in the former case) a single global numerical measure or (in the latter case) a ranking over all hyperedges in the entire network. Instead of a percomponent approach, an elegant alternative for averaging graph distances in disconnected graphs, advocated by Newman [57], is to use the harmonic mean instead of the arithmetic. This approach was adopted by Latora and Marchiori [58] to define network efficiency as the reciprocal of the harmonic mean path length, proposed as a quantitative measure of smallworldness. Latora and Marchiori termed this measure “efficiency” in reference to how efficiently information might be exchanged over the network. Later, a similar approach was adopted by Rochat [59] to define the harmonic closeness centrality index of vertices in a disconnected graph. Extending these notions to the hypergraph context, a more practical definition of the aforementioned sdistance based notions is given by:
Definition 9
Let \(H=(V,E)\) be a hypergraph and let \(C_{s}\) denote the set of its sconnected components.

(i)
The seccentricity of a hyperedge \(f \in C\) where \(C \in C_{s}\) is \(\displaystyle\max_{g \in C} d_{s}(f,g)\).

The sdiameter is the maximum seccentricity over all edges in \(E_{s}\), while the sradius is the minimum.


(ii)
The averagesefficiency of H is \(\displaystyle\binom{ \vert E_{s} \vert }{2}^{1} \displaystyle\sum_{ \substack{f,g \in E_{s} \\ f\neq g}} \frac{1}{d_{s}(f,g)}\).

(iii)
The harmonicscloseness centrality index of a hyperedge f is \(\displaystyle\frac{1}{ \vert E_{s} \vert 1}\displaystyle\sum_{ \substack{g \in E_{s} \\ f \neq g}} \frac{1}{d_{s}(f,g)}\).
We take the limiting value of 0 for the summand in (ii) and (iii) when f and g are in different scomponents. Both (ii) and (iii) are numerical quantities between 0 and 1, with larger values indicative of closer sdistances between hyperedges, either globally (in the former case) or locally (in the latter case). If \(\vert E_{s} \vert =1\), they are undefined. In practice, one may ignore such isolated hyperedges when computing centrality or assign them a value of 0 by convention, analogous to how [60, p. 221] sets the closeness centrality value of an isolated vertex in a graph to 0.
Application to data
We compute three of the aforementioned sdistancebased measures: the average sefficiency index, the sdiameter (i.e. the maximum seccentricity over all scomponents) and the harmonic scloseness centrality index.
Figure 5 plots the maximum sdiameter (over all scomponents) and average sefficiency for the hypergraph and its dual. Larger values of average sefficiency imply smallersdistances among the hyperedges in question. For some of the data (e.g. LesMis, Diseasome, Diseasome^{*}) average sefficiency and sdiameter tends to decrease as s increases. In these networks, the shortest swalks linking hyperedges tend to become longer (or infinite) as s is increased. However, for CompBoard^{*}, average sefficiency increases in s for each \(s\geq 2\). This suggests that, among company board members who sit on multiple boards, those who sit on more boards tend to (on average) be closer to one another in sdistance. LesMis^{*} exhibits a similar phenomena regarding average sefficiency, where for characters appearing in at least \(s\geq 22\) scenes, the more scenes they appear in, the closer they are to each other in sdistance.
Turning to sdiameter, it is possible for sdiameter (taken as the maximum over all scomponents) to increase or decrease in s. In the former scenario, as s is increased, shorter swalks linking hyperedges may disappear, and those edges may only be linked via longer swalks, thereby increasing sdiameter. LesMis exhibits this most prominently, where sdiameter increases from 5 to 9 as s increases from 1 to 2. On the other hand, if increasing s eliminates allswalks between pairs of hyperedges, then these hyperedges are separated into different scomponents in which hyperedges may be closer to each other. In such cases, the sdiameter may decrease, as in Diseasome. Consistent with our intuition from the Diseasome visualization in Fig. 2, this sdiameter drop reflects the fragmentation of the network into small components; accordingly average sefficiency also drops because of the infinite sdistances between edges in different scomponents.
Lastly, Fig. 6 (top row) lists the top 15 hyperedges in CompBoard, Diseasome, and LesMis^{*} for \(s=1,2,3\), as ranked according to their harmonic scloseness centrality. Boxes enclosing hyperedges indicate a tie in scloseness centrality. Comparing the ordinal rankings across the datasets, for some data the the top 15 ranked hyperedges for \(s=1\) remain within the top ranked for \(s=2\) (e.g. for LesMis^{*}, 10 remain in the top 15) whereas in other data, the top ranked hyperedges may change completely (e.g. in CompBoard, none of the top 15 companies with highest 1closeness centrality remain in the top 15 for 2closeness centrality).
A drop in a hyperedge’s rank from \(s=1\) to 2 may indicate short pathways linking that hyperedge to others rely on sparse hyperedge intersections. For example, in Diseasome we observe that while “Colon cancer” and “Breast cancer” remain in the top 3 ranked hyperedges for \(s=1,2,3\), “Diabetes mellitus” drops from having the second largest 1centrality, to having the 34th largest 2centrality. “Diabetes mellitus” shares genes with 24 other diseases and hence this hyperedge intersects with 24 other hyperedges. However, of these 24 diseases, “Diabetes mellitus” shares at least 2 genes with only two diseases: “Obesity” and “Mature Onset Diabetes of the Young (MODY)”. Thus, any 2walk between “Diabetes mellitus” and another disease can only go through one of these diseases, which (in this case) results in larger average 2distance between diabetes and other diseases, relative to the average 2distance between other pairs of diseases. In contrast, “Breast cancer” shares at least 2 genes with 9 other diseases, and (on average) can be linked to other diseases via a shorter 2walk than for “Diabetes mellitus”.
To more rigorously explore these changes in ordinal rankings by scloseness, we compute Kendall’s \(\tau _{B}\) rank correlation coefficient between the top k ranked hyperedges for one value of s and the rankings of those same hyperedges under another value of s. We compute this coefficient for each of \(k=10,\ldots, \vert E \vert \) and for each ordered pair of svalues from \(\{1,2,3\}\). Hyperedges with equal scloseness centrality are considered tied in rank, and we assign the minimum scloseness centrality score of 0 to any hyperedge with fewer than s vertices. Kendall’s \(\tau _{B}\) ranges from −1 (if the ordinal rankings are perfectly inverted) to 1 (if the ordinal rankings are identical), and is explicitly formulated to handle ties in rank [61]. Figure 6 plots results for CompBoard, Diseasome, and LesMis^{*}. CompBoard exhibits an absence of correlation for the 1closeness rankings when compared against 2 or 3, and a stronger correlation for the 3closeness rankings compared against the 2closeness rankings. When all hyperedges in the network are considered (i.e. for \(k= \vert E \vert \), given by the rightmost points in each plot), the 1closeness rankings of LesMis^{*} exhibits the strongest correlations between the 2 and 3closeness rankings.
Paths, cycles, and clustering coefficients
Methods
So far, our methods have centered solely around the base definition of swalk. However, just as graph walks may be distinguished into finer classes such as trails, paths, circuits and cycles, swalks may also be distinguished from each other and organized hierarchically. As we’ll show, doing so allows one to define highorder substructures native to hypergraphs, such as striangles, that cannot be determined from their sline graphs.
Definition 10
For a hypergraph \(H=(V,E)\), let the sequence of hyperedges \({\omega }=(e_{i_{0}},e_{i_{1}},\ldots, e_{i_{k}})\) be an swalk of length k. For ease of notation let \(I_{j}=e_{i_{j1}} \cap e_{i_{j}}\) be the j’th intersection. The swalk ω may be further defined as:

(i)
An strace if \({i_{x}} \neq {i_{y}}\) for all \(x\neq y\) (all hyperedges are pairwise distinct by label).

(ii)
An smeander if ω is an strace in which \(I_{x} \neq I_{y}\) for all \(x\neq y\) (all intersections are pairwise distinct).

(iii)
An spath if ω is an smeander in which \(I_{x} \setminus I_{y} \neq \varnothing \) for all \(x\neq y\) (no intersection is included in another).
Graph case & equivalence
If H is a graph, a 1trace on \(H^{*}\) is equivalent to a walk on H in which vertices are distinct but edges may be repeated. Furthermore, if H is a graph, smeanders and spaths on \(H^{*}\) are both equivalent to a graph path on H. However, if H is a hypergraph, a path in \(L_{s}(H)\) does not necessarily correspond to an spath in H. Consequently, the forthcoming spath based triadic notions in Definition 11 cannot be obtained from \(L_{s}(H)\) but reduce to their usual graph counterparts on \(H^{*}\) for \(s=1\) whenever H is a graph.
We note Wang and Lee [37] also define hypergraph paths using the same subset condition stated in Definition 10 above. The notions of swalk, strace, smeander, and spath form a nested hierarchy: every strace, smeander, or spath is an swalk; every smeander and spath is an strace; and every spath is an smeander. However, in each case, the reverse may not be true (e.g. an smeander may not be an spath). With regard to sdistance (Sect. 4.3), it is straightforward to show constructively that if there exists an swalk (resp. strace, smeander) of length k between two hyperedges, there exists an strace (resp. smeander, spath) of length at most k. This implies the length of the shortest swalk between two hyperedges is equivalent to the length of the shortest spath; consequently, sdistance as given by the length of the shortest swalk is equivalent to that given by the length of the shortest spath.
While not having ramifications for the notion of sdistance, the finer classes of swalks above provide a means, within the swalk framework, to define highorder substructures or motifs that cannot be determined from the sline graph. To define an example of these substructures, we require the notion of a closed walk. Analogous to its usage in graph theory, we call an swalk closed if \({i_{0}}={i_{k}}\), and call a closed spath an scycle. As a point of clarification, closed straces, meanders, or paths are still considered valid straces, meanders or paths (that is, only the terminal edges are exempt from the strace requirement that all edges be distinct by label). Using scycles, we define hypergraph sanalogs of triadic measures commonly applied to graph data. Whereas graph triadic notions like the local clustering coefficient [62] are defined for vertices, the sanalogs below are defined for hyperedges, keeping consistent with the rest of our presentation. We remind the reader vertexbased notions are obtained by simply applying the below definition to the dual hypergraph, \(H^{*}\).
Definition 11
For a hypergraph H, an striangle is a closed spath of length 3 and an swedge is an spath of length 2. For an swedge \(e_{0}\), f, \(e_{2}\), we say f is the center of the swedge.

(i)
The slocal clustering coefficient of a hyperedge \(f \in E_{s}\) is given by
$$ s\text{LCC}(f)= \textstyle\begin{cases} \frac{\text{number of }s\text{triangles containing }f}{\text{number of }s\text{wedges centered at }f} & \text{if }f\text{ is the center of an }s\text{wedge}, \\ 0 & \text{otherwise}. \end{cases} $$ 
(ii)
The sglobal clustering coefficient of a hypergraph H is given by
$$ s\text{GCC}(H)= \frac{3 \cdot \text{total number of }s\text{triangles}}{\text{total number of }s\text{wedges}}. $$
In the same way as for the LCC of graphs, one may obtain a global measure for the sLCC of a hypergraph by taking the mean slocal clustering coefficient over all edges in \(E_{s}\).
Figure 7 illustrates examples of three different hypergraphs induced by a closed swalk \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{1}\) of length 3; namely, from left to right: a closed strace that is not an smeander, a closed smeander that is not an spath, and a closed spath of length 3 (i.e. an striangle). Observe the 1line graphs of all three of these hypergraphs consists of a single triangle, while only the rightmost pictured hypergraph in Fig. 7 is a 1triangle (as well as a 2triangle). Another example may be found by reconsidering the authorpaper networks in Fig. 1: for the hypergraphs constructed by letting hyperedges denote authors, the 2walk A, B, C, A is a 2triangle for the leftmost network, while the same walk is a 2trace for the rightmost. These examples illustrate striangles cannot be determined from line graphs (a fact that is unsurprising, since sline graphs do not encode the subset relationships stipulated for spaths).
Since other definitions of hypergraph clustering coefficients have appeared in the complex networks literature, it is worth clarifying how these notions compare to ours. Estrada [63] proposes a global hypergraph clustering coefficient as a ratio of (non swalk based) hypergraph triangles to hypergraph wedges. More precisely, Estrada defines a hypertriangle as an alternating vertexhyperedge sequence with three distinct vertices and three distinct hyperedges such that for each subsequence \(v_{i}\), \(e_{k}\), \(v_{j}\), we have that \(v_{i},v_{j} \in e_{k}\) (put equivalently, these are 6cycles in the bipartite representation of the hypergraph). Thus, returning to the rightmost hypergraph pictured in Fig. 7, the alternating sequence given by interlacing the pair of vertex and hyperedge triples \((v_{1},v_{3},v_{6})\) and \((e_{1},e_{2},e_{3})\) constitutes a triangle, as does the same pair with \(v_{1}\) replaced with \(v_{2}\). It is easy to see the existence of an striangle implies the existence of at least one such hypertriangle as defined by Estrada; however, the converse is not necessarily true (e.g. while the hypergraph pictured in the center of Fig. 7 contains many such hypertriangles, neither this hypergraph nor its dual contain any striangles). In this sense, Estrada’s notion of clustering and ours are fundamentally different.
Other proposed notions of hypergraph clustering differ to ours in being based on averaging various pairwise set theoretic measures between pairs of hyperedges or vertices. For instance, Latapy, Magnien and Vecchio [18] propose a pairwise clustering coefficient between hyperedges \(e_{i}\), \(e_{j}\) as \(\frac{ \vert e_{i} \cap e_{j} \vert }{ \vert e_{i} \cup e_{j} \vert }\), which is the Jaccard similarity coefficient between the sets of vertices constituting the two hyperedges (or, when applied to the dual, the Jaccard similarity between the sets of hyperedges to which two vertices belong). They then define a local and global notion of hypergraph clustering by averaging this quantity. Zhou and Nakhleh [64] propose local and global hypergraph clustering coefficients based on the pairwise excess overlap between hyperedges. As described and studied further by the authors in [8], excess overlap measures the proportion of the vertices in exactly one of the edges that are neighbors of vertices in only the other edge. Lastly, notions of bipartite graph clustering proposed in the literature, (applicable to hypergraphs via the bicolored graphhypergraph correspondence mentioned in Sect. 2) are frequently based on bipartite 4cycles [48, 65]. In the language of hypergraphs, a bipartite 4cycle is a subhypergraph on two hyperedges and two vertices. Hence (in addition to again not being based in highorder swalks) these bipartite 4cycle based notions of clustering differ from our striangle based notions in involving only pairs (rather than triples) of hyperedges.
Application to data
Figure 8 plots the mean sLCC and sGCC (left block) as well as the proportion of triangles and wedges in sline graph that correspond to striangles and swedges in the hypergraph (right block) for each of our datasets. Recall every triangle and wedge in the sline graph represents a closed swalk of length 3 and strace of length 2 which, in turn, may or may not be an striangle or swedge, respectively. For all three datasets, a higher proportion of wedges in the sline graph correspond to swedges compared with the proportion of triangles in \(L_{s}(H)\) that correspond to striangles.
On the other hand, the datasets exhibit different behavior regarding the absolute size of these proportions, as well as how these proportions vary as s varies. For LesMis^{*}, a relatively larger proportion of triangles in \(L_{s}\) correspond to striangles than for CompBoard^{*}. Furthermore, the proportions of striangles to swedges are much greater, both on average locally (given by the mean sLCC) as well as globally (given by the sGCC) than for CompBoard^{*}. In contrast, CompBoard^{*} exhibits an extremely small proportion of triangles in \(L_{s}(H)\) corresponding to striangles. This means whenever there is a triad of board members where each pair belong to common company boards, it is almost always the case that for at least one pair of board members, the set of companies in common are either identical (i.e. forming strace that is not an smeander) or subsets of each other (i.e. an smeander that is not an spath). In general, striangles in both CompBoard and its dual are scarce, reflected by extremely low sLCC and sGCC coefficients.
In the context of a companyboard network an swedge is a generalization of a different represenatives’ interlock,^{Footnote 12} a topic prominent in the corporate governance literature [47, 48, 66]. Furthermore, pairs of hyperedges having an sdistance of 1 and 2 represent socalled “direct interlocks”^{Footnote 13} and “third company interlocks”,^{Footnote 14} respectively, between competing companies. This parallel illustrates how spath and sdistance based notions may provide a generalized framework for describing and measuring phenomenon already important to particular domains. For CompBoard, since the aforementioned interlocks between competing companies are regulated by Section 8 of the Clayton Act [66], it is unsurprising our results show swedges (and hence striangles) are relatively rare.
Comparison with generative hypergraph null models
Graph generation serves farranging purposes across scientific disciplines. Generative graph models are used for benchmarking, algorithm testing, and creating surrogate graphs to protect the anonymity of restricted data. Here, we apply hypergraph generative models as null models to experimentally test the significance of the highorder properties explored in Sect. 4. By “null model”, we mean a generative model that controls for certain basic features of the data. Such models may be utilized to test whether observed measurements in the data are necessarily consistent with controlled features. For example, in the Erdős–Rényi graph model, the user specifies the desired number of vertices n and edgeprobability p; hence, by controlling n and p one can generate ensembles of random graphs with the same expected edge density. A subsequent comparison of measurements on given graph data against those on random graphs with the same edge density tests whether the measured features can be explained as sole consequences of edge density. To the extent to which the properties of the real and synthetic graphs diverge, this provides evidence the properties under question cannot be explained as sole consequences of the structural properties preserved by the model.
In comparison to their graph counterparts, generative hypergraph models are relatively few. While work on random uniform hypergraphs dates back to at least the 1970s, researchers have recently begun developing a wider variety of hypergraph models, both for uniform hypergraphs [38–40, 67] and nonuniform hypergraphs [8, 68–71]. We consider three generative hypergraph models from [65], which can be thought of as hypergraph interpretations of the graph models Erdős–Rényi (ER) [72], Chung–Lu (CL) [73], and Block TwoLevel Erdős–Rényi (BTER) [74, 75]. These models were originally presented as “bipartite models” in [65], with similar acknowledgment of the bicolored graphhypergraph correspondence discussed in Sect. 2. While these models were inspired from their graph counterparts and named as such, there may be multiple ways of conceiving these models in the hypergraph/bicolored graph setting, as is often the case with graphtohypergraph extensions. In fact, others have proposed nonuniform hypergraph analogs of Erdős–Rényi and Chung–Lu (see [8] and [71], respectively) differing to those considered here with regard to the inputs required, the model itself, and the definition of hypergraph assumed.
We’ve chosen these particular models for several reasons. First, they can generate nonuniform hypergraphs in accordance with the full generality of Definition 1. Notably, all three of these models permit duplicated edges, which occur frequently in hypergraphstructured data and are highly prevalent on our particular data (see Fig. 3). One might expect duplicate edges to also be common in authorpaper networks, occurring whenever the same set of authors write multiple papers together (e.g. papers 1, 4 for authors A, B in the leftmost network of Fig. 1). These joint papers suggest stronger relationships amongst the authors in question; disregarding duplicate edges ignores this and skews measurements, such as scentrality, meant to capture such properties. In contrast to the models we consider, the aforementioned nonuniform hypergraph ER and CL models proposed by [8] and [71] do not permit duplicated hyperedges, but treat hyperedges themselves as multisets.
Secondly, taken as a suite, these models provide tiered control over three fundamental properties: (1) vertexhyperedge density, (2) vertex degree and edge cardinality distributions, and (3) metamorphosis coefficients, a measure of community structure from [65] which we return to shortly. Specifically, ER controls for vertexhyperedge density, CL controls for density as well as degree distributions, and BTER controls for all three of the aforementioned properties. Taken in sequence, ER, CL and BTER can each be conceived formally as a generalization of the previous model. All three models afford scalable implementations and [65, 76] report results on hypergraphs generated using these models with hundreds of millions vertexhyperedge memberships; open source implementations are available^{Footnote 15} as part of The Chapel HyperGraph Library (CHGL, [76]), a prototype HPC library [77] for largescale hypergraph generation and analysis written in the emerging programming language of Chapel.
Three generative hypergraph models
We define the generative models we consider below, and then briefly compare their properties.

1
Erdős–Rényi, \(\operatorname{ER}(n,m,p)\). The user specifies three scalar parameters: the desired number of vertices n, desired number of hyperedges m, and vertexhyperedge membership probability, \(p \in [0,1]\). For each of the nm vertexhyperedge pairs, the probability of membership is the same,
$$ \Pr (v \in e)=p. $$ 
2
Chung–Lu, \(\operatorname{CL}({{\vec{d_{v}}}}, {{\vec{d_{e}}}})\). The user specifies a desired vertex degree sequence \(\vec{d_{v}}=(d_{v_{1}},\ldots,d_{v_{n}})\) and desired hyperedge size sequence \(\vec{d_{e}}=(d_{e_{1}},\ldots,d_{e_{m}})\), which (in order to be realizable by a hypergraph) satisfy \(c=\sum_{i=1}^{n} d_{v_{i}}=\sum_{i=1}^{m} d_{e_{i}}\). The probability a vertex belongs to a hyperedge is proportional to the product of the desired vertex degree and edge size, i.e.
$$ \Pr (v_{i} \in e_{j}) =\frac{d_{v_{i}}\cdot d_{e_{j}}}{c}. $$To ensure this probability is always less than 1, one may further require the input sequences satisfy \(\max_{i,j} d_{v_{i}}d_{e_{j}} < c\).

3
Block TwoLevel Erdős–Rényi, \(\operatorname{BTER}(\vec{d_{v}}, \vec{d_{e}}, \vec{m_{v}}, \vec{m_{e}})\). In addition to the desired vertex degree and edge size sequences mentioned in Chung–Lu, the user also specifies desired vertex and edge metamorphosis coefficients, \(\vec{m_{v}}\) and \(\vec{m_{e}}\), which, as clarified further below, are measures of community structure based on the prevalence of small, dense substructures in the hypergraph. The BTER model is designed to output a hypergraph that matches the input degree distribution and metamorphosis coefficients. The BTER model proceeds in two phases: in the first, metamorphosis coefficients are approximately matched by grouping vertices and hyperedges into small, disjoint sets called affinity blocks and applying the Erdős–Rényi model on each block. In the second, the degree distributions are matched by running the Chung–Lu model on the excess desired degrees, thereby linking the blocks. As formal details of the BTER model are complicated, the reader is referred to [65] for a full specification.
For ER the expected number of vertexhyperedge memberships is \(pnm\), and hence this simple model can be used to generate random hypergraphs with a specified vertexhyperedge membership density. We reported this density for our datasets in Fig. 3. For the CL model, each vertex v achieves its userspecified desired degree \(d_{v}\) in expectation since
An identical argument also shows each hyperedge e achieves its desired size \(d_{e}\) in expectation. In this way, CL not only matches the desired vertexhyperedge membership density in expectation like ER, but additionally matches the vertex degree and edge size distributions in expectation. We reported these degree distributions for our datasets in Fig. 3.
The CL model is a generalization of the ER model in the sense that the ER can be obtained from CL by taking the degree and edge size sequences to be constant, i.e.
Lastly, the BTER model (which, as explained in [65], utilizes the CL model as a subroutine) can be understood as a generalization of CL. The BTER model is designed to match not only vertex and edge size distributions, but also perdegree metamorphosis coefficients. A complete definition of metamorphosis coefficients is involved; interested readers are referred to [65] for full details. Nonetheless, to elucidate how metamorphosis coefficients are interpreted in the hypergraph setting, we provide a highlevel description.
Metamorphosis coefficients are measures of network community structure based on counts of bipartite 4cycles, also called butterflies, and bipartite 3paths, also called caterpillars. In the language of hypergraphs, a butterfly is a subhypergraph consisting of two vertices and two edges intersecting in those two vertices; a caterpillar is an edge with two vertices intersecting with another edge in one of those vertices. The authors in [65] define metamorphosis coefficients for vertices within each of the two partitions of a bipartite graph, based on the ratios of butterfly to caterpillar counts those vertices participate in. Stated equivalently, this defines metamorphosis for the vertices and hyperedges of a hypergraph. If a hyperedge e has a large metamorphosis coefficient, this means a large proportion of the edges that e intersects with intersect in (at least) 2 vertices; dually, if vertex v has large metamorphosis, then a large proportion of vertices v shares an edge with share (at least) 2 edges. For example, in Fig. 1 each author in the leftmost network repeats a coauthorship with someone on 1 out of 3 of their other papers, and thus has metamorphosis \(\frac{1}{3}\); in the rightmost, each author repeats a coauthorship on all their other papers, and thus has metamorphosis 1. The BTER model is designed to match degree distributions, as well as the average metamorphosis coefficients for vertices and hyperedges of a given degree and cardinality, respectively.
Taken as a suite, these three models serve well as nullmodels since each provides successively more control over hypergraph structure than the previous, providing the flexibility to choose different tiers of structural nuance for the generated hypergraphs. In the next section, we run each model multiple times on each dataset, and study how well each model replicated swalk properties. By (for example) “running CL on LesMis”, we mean extracting the model inputs (in this case, the vertex degrees and hyperedge sizes) from the data, and using the Chung–Lu model to generate a hypergraph under these inputs.
Comparison
Figure 9 compares swalk based properties of LesMis^{*}, Diseasome, and CompBoard against those of synthetic hypergraphs generated by ER, CL, and BTER. For each dataset, we generate 100 instances of each synthetic model and compute the properties in question for each instance. The plot reports the average values observed over the 100 trials, for each s.
In the leftmost column in Fig. 9, we use Kolmogorov–Smirnov (KS) distance to compare the scomponent size probability distributions of the original and synthetic hypergraphs. KS distance is normalized between 0 and 1, with smaller KSvalues indicating greater similarity.^{Footnote 16} Comparing the models as s increases, the ER model exhibits higher KS distance than for CL and BTER, indicating scomponent size distributions that are more dissimilar to the original. All three models seem to exhibit larger KS distances for larger s values, although in some cases (e.g. for CL and BTER on LesMis^{*}) this increase is not monotonic in s. One notable exception is CompBoard, in which all models exhibit much larger KS distance for \(s=1\) than \(s=2\). This can likely be attributed to the large number of isolated hyperedges observed in 1components of CompBoard: in contrast, all three models tend to output hypergraphs in which the majority of hyperedges are contained within a single giant 1component.
Turning to sdistances, we compare the original sdiameter (center right) and average sefficiency (far right) to those of the model’s synthetic hypergraphs. As the average sefficiency plots are in logscale, average values of 0 (which occur whenever no two hyperedges intersect in s vertices) are not plotted. ER tends to have lower sdiameter and average sefficiency as s increases, when compared to CL and BTER. For LesMis^{*} and Diseasome, CL and BTER seem to perform comparably; for CompBoard, however, BTER does noticeably better than both in matching average sefficiency for \(s \geq 2\), although still diverging from the original values considerably for \(s \geq 5\). Lastly, we consider the model’s performance with regard to mean slocal clustering coefficients (center left). For all three datasets, ER produces smaller clustering coefficients than observed in the original data, for all s. For CL and BTER, the mean slocal clustering coefficients sometimes exceed those of the original data for small values of s (e.g. for \(s \leq 2\) on Diseasome), while for some larger values of s (e.g. for \(13 \leq s \leq 19\) on LesMis^{*}), BTER and CL produce smaller local clustering coefficients than those of the original data.
Taking a broader view of these results, none of these three models are able to provide a consistent, close match across values of s. This suggests the swalkbased measures in question cannot be explained as sole consequences of the model inputs (e.g. degree distributions for the Chung–Lu model), that are preserved in expectation in the output hypergraphs. Nonetheless, this experiment should not be extrapolated to provide generalized guidance on which model best preserves certain swalk properties. Depending on the properties of the data in question, it may be the case that ER (the least accurate model on our data) provides a closer match than CL or BTER. In order to provide more a comprehensive approach to such questions, it would be of interest to determine conditions on model inputs under which certain swalk properties of the output hypergraphs can be tightly bounded or controlled. While such work is outside the scope of the present paper, the aforementioned research by Kang, Cooley, and Koch [38–40] illustrates establishing guarantees on even basic highorder walk based properties in random hypergraphs (such as the size of the largest scomponent) requires sophisticated probabilistic analysis.
Conclusion
The prevalence and complexity of hypernetwork data necessitates analytic methods that are both applicable and able to capture hypergraphnative phenomena. We have proposed hypergraph swalks provide a framework under which graph analytic tools popular in network science extend more meaningfully to hypergraphs. In applying these measures to real data, we’ve explored how they may reveal varied, interpretable, and significant structural properties of the data otherwise lost when analyzing hypergraphs under the lens of the usual graph walk. The methods we’ve focused on—connected component analyses, distancebased measures, highorder motifs and clustering coefficients—are meant to illustrate the breadth of tools to which this approach is relevant. However, ours is clearly far from a comprehensive exploration. We conclude by outlining several lines of future work that highlight the limits of our approach.
One immediate open question concerns how the methods we’ve developed may be generalized further. For instance, it would be of both theoretical and practical interest to develop tractable swalk based measures for weighted hypergraphs (with realvalued vertex and/or edge weights), directed hypergraphs (in which each edge’s vertices are either in its “head” or “tail”), ordered hypergraphs (in which each edge’s vertices are totally ordered), or temporal hypergraphs (consisting of sequences of hypergraphs). With regard to the latter topic, our work does not address how hypergraphs, and the structural properties we observed, evolve throughout time. The suite of generative models we considered are effective as structural null models, but do not explicitly posit a process or mechanism through which hypernetworks grow. In contrast, other researchers have put forth and studied hypergraph evolution mechanisms, such as a preferential attachment inspired model for nonuniform hypergraphs [78, 79]. An analysis of these, or the development of entirely new, temporal hypergraph models might shed insight into how highorder structural properties put forth here emerge in network topology.
Lastly, another open direction lies in devising efficient computational methods for the swalk measures put forth here. We did not explore the algorithmic aspects underlying these methods. In some cases, the methods we utilized—while sufficient on our data—were not scalable to massive hypergraph data (e.g. computing scentrality via the sline graph quickly becomes infeasible for large hypergraphs with skewed degree distributions, as the density of sline graphs increase quadratically in the maximum vertex degree). Developing algorithms that leverage the sparsity of the hypergraph (rather than resorting to computation on dense sline graphs) would help facilitate the application of these methods to largerscale data. Furthermore, just as researchers have begun developing efficient schemes for computing atomic bipartite graph motifs such as cycles of length 4 [80, 81], work in a similar vein would prove useful for enabling largescale striangle counting in hypergraphs.
Notes
 1.
One could also have formed a hypergraph by taking papers as vertices and hyperedges as the set of papers each author has written. In this case, by virtue of having authors with 4 papers, the hypergraph derived from the leftmost network exhibits higherorder relationships. The hypergraph obtained by swapping the roles of vertices is called the dual hypergraph. Duality is an essential consideration in hypernetwork science, which we discuss further in the Preliminaries section.
 2.
More precisely, if a 2uniform hypergraph contains duplicated hyperedges, it is a multigraph.
 3.
By “labeled hypergraph” we mean a hypergraph in which each vertex and edge are distinguishable via an assignment of distinct labels—this is not meant to be confused with socalled attributed hypergraphs in which the vertices and edges have associated metadata.
 4.
That is, our focus in this work is on hyperedge incidences and hyperwalks that arise from sequences of incident hyperedges. Hyperedges themselves are defined explicitly for hypergraphs, but only implicitly for bicolored graphs (as the neighborhood of a vertex in the color class designated for hyperedges). For this reason, framing our exposition using the language of arbitrary set systems is natural, whereas adopting the constrained language of bicolored graphs would be cumbersome and confusing.
 5.
Note, however, this is not necessarily true when restricted to the graph case: for a graph G, its dual \(G^{*}\) is 2uniform (and hence still a graph) if and only if G is 2regular, in which case G is a cycle or disjoint union of cycles.
 6.
 7.
Consequently, the sLaplacian matrix they study is \(n^{\underline{s}} \times n^{\underline{s}}\), where n denotes the number of vertices and \(x^{\underline{k}}=\binom{x }{k} k!\) denotes the falling factorial. Even for a modestly sized hypergraph on \(n=10^{4}\) vertices with \(s=20\), this matrix has size \(n^{\underline{s}} \approx 10^{80}\), approximately the number of atoms in the known universe.
 8.
List of companies on these exchanges obtained from https://www.nasdaq.com/screening/companylist.aspx
 9.
 10.
 11.
Described as a physical device consisting of two horizontal bars that support “hooks” representing companies and board member nodes, with rubber bands that “join the appropriate hooks and physically represent the inclusion between persons and boards”
 12.
Defined in [66] as “the linking of two companies by a third company having different representatives on the board of the two companies.”
 13.
A direct interlock occurs when two company boards have 1 or more members in common. [47]
 14.
Defined in [66] as “the linking of two companies by one company having a director on the board of a second, …which has directors in common with a third company, …which in turn has directors on the board of a competitor of the first company.”
 15.
 16.
For example, the green point at \((13,0.2)\) in the LesMis^{*} plot means that, over 100 trials, the average KSdistance between the 13component size distribution of original LesMis^{*} dataset, and that of a CL hypergraph, is 0.2. In cases where the synthetic graph had no hyperedges containing at least s vertices (and hence an empty scomponent size distribution) we define KS distance between the original scomp distribution as 1 (the maximum).
Abbreviations
 ER:

Erdős–Rényi
 CL:

Chung–Lu
 BTER:

Block TwoLevel Erdős–Rényi
 KS:

Kolmogorov–Smirnov
 LCC:

Local Clustering Coefficient
 GCC:

Global Clustering Coefficient
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Acknowledgements
We would like to thank numerous colleagues for helpful discussions, including Marcin Zalewski, Francesca Grogan, Katy Nowak, Dustin Arendt, Stephen Young, Brett Jefferson, and Louis Jenkins. We also thank referees for thoughtful comments which improved the manuscript.
Availability of data and materials
The Diseasome dataset is available at https://github.com/gephi/gephi/wiki/Datasets. The LesMis dataset is available at https://wwwcsfaculty.stanford.edu/~knuth/sgb.html. The CompBoard dataset may be made available upon request to the corresponding author, subject to institutional approval.
Funding
This work was partially funded under the High Performance Data Analytics (HPDA) program at the Department of Energy’s Pacific Northwest National Laboratory. PNNL Information Release: PNNLSA144766. Pacific Northwest National Laboratory is operated by Battelle Memorial Institute under Contract DEACO676RL01830.
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SGA proposed the hypergraph walk framework, designed and implemented the experiments, and wrote the paper. CJ, COM, BP, and EP refined the measures, analyses, and exposition. All authors read, edited, and approved the final manuscript.
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Appendix
Appendix
Hypergraph random walks and swalks
Although not the focus of the present work, the study of random walks is intimately related to many branches of graph and hypergraph theory, underlying analytic methods such as PageRank, diffusion processes such as chipfiring and loadbalancing in distributed networks, as well as clustering methods. One popular way of defining a random walk on a hypergraph \(H=(V,E)\) is as a discrete time Markov Chain, \(X_{1},X_{2},\ldots \) , with state space V, such that at time t, if \(X_{t}=v_{t}\), then \(X_{t+1}\) is determined by

1
Selecting an edge e containing \(v_{t}\), either uniformly at random or according to given edge weights.

2
Selecting a vertex v from e, either uniformly at random or according to given vertex weights.

3
Setting \(X_{t+1}=v\).
This process defines a probability transition matrix, symmetrizations of which yield certain Laplacian matrices frequently used as inputs to clustering methods, like spectral clustering [26] and nonnegative matrix factorization [27]. For instance, Zhou [28] proposed a hypergraph Laplacian which may be used to cluster a hypergraph’s vertices according to a normalized hypergraph cut criterion. Other hypergraph Laplacian matrices have been proposed by Rodriguez [29] and Bolla [30]. However, Agarwal [31] proved these Laplacians, while defined on the hypergraph, are nonetheless closely related to Laplacians of graphs derived from the hypergraph, such as the aforementioned 2section and bipartite graph. Consequently, neither these Laplacians, nor the clustering methods that utilize them, make full use of the higherorder relationships present in hypergraphs but absent in graphs. Nonetheless, recent work by Chitra and Raphael [32], has identified a potential culprit underlying this shortcoming: these Laplacians are based on random walks featuring socalled edgeindependent vertex weights. They show edgedependent vertex weights (i.e. each vertex has a collection of weights, one for each hyperedge to which it belongs) is a necessary, albeit not sufficient, criterion for defining a random walk on a hypergraph that isn’t equivalent to some random walk on the 2section.
We note there are at least two ways of utilizing the swalk framework to define random walks: (1) as an sweighted random walk, and (2) as an sstratified set of random walks. In the former, the intersection cardinalities between hyperedges serve as the weights for transitioning, which is equivalent to a weighted random walk on the graph with adjacency matrix \(S^{T}S\), where S is the hypergraph incidence matrix. In this case, Laplacian matrices derived from this walk are still subject to Agarwal’s aforementioned criticism. In the latter case, one considers a set of random walks, one for each sline graph (see Definition 7), which may be either weighted or unweighted. However, whether and how this set of random walks might be utilized to define a Laplacian (or whether there is a different way to utilize swalks to define stochastic processes of interest) is an interesting topic we leave to future work.
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Aksoy, S.G., Joslyn, C., Ortiz Marrero, C. et al. Hypernetwork science via highorder hypergraph walks. EPJ Data Sci. 9, 16 (2020). https://doi.org/10.1140/epjds/s13688020002310
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Keywords
 Hypergraph
 Highorder walk
 Generative model