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Hypernetwork Science via High-Order Hypergraph Walks
Sinan G. Aksoy, Cliff Joslyn, Carlos Ortiz Marrero, Brenda Praggastis, Emilie Purvine
TL;DR
Graph methods cannot fully represent real-world multi-way relationships, motivating a hypergraph-native framework based on high-order walks. Applied to real hypernetworks and generative models, it reveals interpretable structure lost by graph methods while showing that basic properties do not necessarily determine these measurements.
Problem
Graph representations capture pairwise relationships but omit structural information in real-world multi-way interactions.
Method
The paper defines quantitative high-order s-walks and extends connected-component, distance-based, motif, and clustering analyses from graphs to hypergraphs.
Results
High-order measures reveal varied, interpretable structural properties in real hypernetworks that are otherwise lost in graph-based analyses, while three models provide no consistent match across s.
Takeaways & Limitations
Hypergraph s-walks provide a framework for analyzing hypergraph-native structure beyond usual graph walks.
Takeaways & Limitations
s-line graphs do not uniquely identify a hypergraph up to isomorphism.
Abstract
from arXiv · showhide
We propose high-order hypergraph walks as a framework to generalize graph-based 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 distance-based metrics such as closeness centrality, and motif-based measures such as clustering coefficients. We apply high-order analogs of these methods to real world hypernetworks, and show they reveal nuanced and interpretable structure that cannot be detected by graph-based 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 hypergraph-structured data are richer when utilizing tools tailored to capture hypergraph-native phenomena, and suggests one possible avenue towards that end.
1 Introduction
Graphs provide simple, broadly applicable foundations for network science, but they can lose higher-order information that hypergraphs represent naturally. This work develops high-order hypergraph-walk methods to extend common graph analyses while balancing structural faithfulness, simplicity, and tractability.
- Motivation: Graphs model binary relationships simply and universally, but their representations can lose structural information present in complex systems.The introduction motivates this limitation across biological, sociological, telecommunications, and physical-infrastructure systems.
- Motivation: Hypergraphs represent k-way relationships and naturally capture set-valued, tabular, or bipartite data across many systems.The paper presents hypernetwork data as ubiquitous because hypergraphs generalize graph edges to connect any number of vertices.
- Challenges: Graph-to-hypergraph generalization remains difficult because competing definitions can agree in the graph case while producing inconsistent hypergraph concepts.The introduction identifies axiomatization as a critical challenge in extending graph-theoretic tools.
- Contribution: The paper addresses this challenge with high-order hypergraph walks that extend graph analytic tools while balancing faithfulness, simplicity, and tractability.An s-walk uses order s to control the minimum walk width through edge-overlap size.
- Contribution: The resulting framework extends connected components, closeness centrality, and clustering coefficients to hypergraphs using walk-based definitions.The methods generalize graph-distance and motif-based analyses through hypergraph walks, including s > 1 walks unavailable in ordinary graphs.
2 Preliminaries
This section defines hypergraphs, their incidence-based representations, duality, and line graphs, while emphasizing that line graphs can lose structural information and become computationally impractical. It motivates hypergraph-native methods despite the correspondence between hypergraphs and bicolored graphs.
- Hypergraph definitions: A hypergraph consists of vertices and an indexed family of hyperedges, which may contain any number of vertices; vertex degree counts incident hyperedges, and edge size is cardinality.Uniformity is defined by common hyperedge size: k-uniform hypergraphs have edges of size k, while 2-uniform hypergraphs are graphs.
- Representations: Incidence matrices provide a Boolean representation of hypergraphs, with entries indicating vertex–hyperedge membership and a bijection between labeled hypergraphs and rectangular Boolean matrices.The same correspondence also gives a bijection between hypergraphs and bicolored graphs.
- Duality: Hypergraph duality transposes the incidence matrix, exchanging vertices and hyperedges while preserving the hypergraph framework under the definitions allowing duplicated, empty, and isolated elements.The dual operation is involutive: (H*)* = H.
- Line graphs: Line graphs replace hyperedges with vertices and encode pairwise hyperedge intersections, but distinct non-isomorphic hypergraphs can have identical weighted line graphs, losing structural information.This loss can persist for the dual hypergraphs, although such indistinguishable pairs form a vanishingly small proportion of hypergraphs.
- Line graphs: Sparse hypergraphs can produce dense line graphs because a vertex of degree k yields k-way edge intersections, making large, skewed real-world networks difficult or impossible to store and analyze.The problem is especially acute when vertex-degree and hyperedge-cardinality distributions are heavily skewed.
3 From Graph Walks to Hypergraph Walks
The section introduces s-walks as a hypergraph-native generalization of graph walks, capturing quantitative edge overlap at vertex and edge levels. It establishes the framework for extending network-science measures, analyzing real hypernetworks, and testing generative null models.
- Hypergraph walks: Hypergraph incidence is set-valued and quantitative, motivating dual but distinct walks on vertices and hyperedges.Two hyperedges may intersect in multiple vertices, while two vertices may share multiple hyperedges.
- Hypergraph walks: An s-walk requires successive hyperedges to overlap in at least s vertices, with s = 1 recovering ordinary graph walks.For s > 1, such walks are possible only on hypergraphs.
- Hypergraph walks: Concatenation of s-walks implies that mutual s-walk reachability among hyperedges defines an equivalence relation.This extends a basic compositional property of graph walks to hypergraphs.
- Contributions: The work develops hypergraph generalizations of graph network-science measures using the s-walk framework.The framework is applied to real hypernetworks to compare results and identify structural insights.
- Contributions: Generative hypergraph null models are experimentally tested for their ability to replicate measure-captured properties of real data.The evaluation focuses on properties seen in real data through the proposed measures.
- Graph case & equivalence: s-connected components and s-distance measures have natural s-line-graph equivalences, whereas s-path, s-cycle, and s-clustering coefficients use unavailable subset information.Most measures reduce to graph analogs by taking s = 1 on the dual graph, but not all are recoverable from s-line graphs.
4 Hypergraph Walk Framework
The framework defines hypergraph-native connectivity, line-graph, component-distribution, and distance measures parameterized by walk width s. Applied to real networks, these measures reveal structure that ordinary line graphs and graph-based analyses can miss, while also requiring care when networks fragment into multiple s-components.
- s-Connected Components: s-connected components partition hyperedges of size at least s, with component order bounded above by |E_s|.Here E_s = {e ∈ E : |e| ≥ s}; a hypergraph is s-connected when E_s is s-connected.
- s-Connected Components: Applying the component definition to the dual hypergraph yields vertex-based s-connectivity, while H and H* always have the same number of 1-connected components.The edge-based and vertex-based notions differ beyond this shared 1-connectivity count.
- s-Line Graphs: For s = 2, 3, 4, s-line graphs distinguish hypergraphs that have identical 1-line graphs, revealing differences lost by the usual line graph.However, s-line graphs still do not uniquely identify a hypergraph up to isomorphism; the author-paper example has identical s-line graphs for s = 1, 2.
- Distance Measures: When multiple s-components exist, inter-component s-distances are infinite, making s-diameter, s-radius, mean s-distance, and s-closeness centrality respectively problematic or trivial.Restricting analysis to the largest component can be unsatisfactory when it does not contain most edges, as in CompBoard for s ≥ 2.
- Distance Measures: LesMis’s s-diameter increases from 5 to 9 as s increases from 1 to 2, because shorter s-walks disappear and remaining connections require longer walks.Conversely, eliminating all s-walks between pairs can decrease s-diameter, while fragmentation can also reduce average s-efficiency through infinite inter-component distances.
5 Comparison with Generative Hypergraph Null Models
The study compares three hypergraph null models with progressively stronger controls over density, degree and edge-size distributions, and community structure. None consistently reproduces the original datasets’ high-order s-walk properties, indicating these properties are not determined solely by the controlled inputs.
- Null-model design: ER, CL, and BTER provide tiered control over density, degree and edge-size distributions, and metamorphosis coefficients, respectively.ER controls density; CL additionally controls degree and edge-size distributions; BTER controls all three property groups.
- Null-model design: Metamorphosis coefficients quantify community structure through bipartite 4-cycles (butterflies) and 3-paths (caterpillars), which BTER is designed to match alongside degree distributions.BTER matches average metamorphosis coefficients for vertices and hyperedges conditioned on degree and cardinality.
- Results: ER produces more dissimilar s-component size distributions than CL and BTER, as shown by higher Kolmogorov–Smirnov distance across increasing s.KS distance ranges from 0 to 1, with smaller values indicating greater similarity to the original hypergraph.
- Results: For average s-efficiency, ER generally underperforms CL and BTER as s increases, while BTER notably outperforms both on CompBoard for s ≥2.CL and BTER perform comparably for LesMis∗ and Diseasome; average s-efficiency values of 0 are omitted from the log-scale plots.
- Interpretation: None of the three models consistently matches high-order s-walk properties across s, suggesting these measures are not explained solely by preserved inputs such as degree distributions.The authors caution that the experiment does not provide generalized guidance about which model best preserves particular s-walk properties.
6 Conclusion
The paper proposes hypergraph s-walks as a framework for extending network-science analyses to hypergraphs while capturing hypergraph-native structure in real data. It identifies generalization to richer hypergraph types and scalable computation as key open directions.
- Conclusion: Hypergraph s-walks extend popular graph-analytic tools to hypergraphs and reveal varied, interpretable, significant structural properties in real data.These properties can otherwise be lost when hypergraphs are analyzed with graph-based methods.
- Open directions: Future work should generalize s-walk measures to weighted, directed, ordered, and temporal hypergraphs.The paper identifies these extensions as theoretically and practically interesting open questions.
- Open directions: Efficient algorithms remain needed because some current s-walk methods are not scalable to massive hypergraphs.Computing s-centrality through the s-line graph can become infeasible for large hypergraphs with skewed degree distributions as s-line-graph density increases quadratically in the maximum degree.
A Hypergraph random walks and s-walks
Hypergraph random walks can be defined as vertex-state Markov chains that select an incident hyperedge and then a vertex within it. The s-walk framework provides s-weighted and s-stratified alternatives, while walk-derived Laplacians support clustering methods.
- Random walks: A standard hypergraph random walk selects an incident hyperedge and then a vertex within that edge, using random choices or prescribed weights.This process defines a vertex-state discrete-time Markov chain.
- Random walks: Random-walk transition matrices can be symmetrized into hypergraph Laplacians used by spectral clustering and non-negative matrix factorization.Proposed Laplacians include Zhou’s normalized hypergraph cut formulation, along with alternatives by Rodriguez and Bolla.
- s-walks: The s-walk framework defines random walks either as s-weighted walks or as an s-stratified set of walks.These are presented as two distinct ways of using the framework to define random walks.
- s-walks: In an s-weighted walk, hyperedge intersection cardinalities determine transition weights, equivalent to a weighted graph walk with adjacency matrix S^T S.Here, S denotes the hypergraph incidence matrix.
B The s-connected components of the data
This section presents results for the s-connected components of three datasets: LesMis∗, Diseasome, and CompBoard.
- The section covers the LesMis∗, Diseasome, and CompBoard datasets.