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Hyper-distance oracles in hypergraphs

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Abstract

We study point-to-point distance estimation in hypergraphs, where the query is parameterized by a positive integer s, which defines the required level of overlap for two hyperedges to be considered adjacent. To answer s-distance queries, we first explore an oracle based on the line graph of the given hypergraph and discuss its limitations: The line graph is typically orders of magnitude larger than the original hypergraph. We then introduce HypED, a landmark-based oracle with a predefined size, built directly on the hypergraph, thus avoiding the materialization of the line graph. Our framework allows to approximately answer vertex-to-vertex, vertex-to-hyperedge, and hyperedge-to-hyperedge s-distance queries for any value of s. A key observation at the basis of our framework is that as s increases, the hypergraph becomes more fragmented. We show how this can be exploited to improve the placement of landmarks, by identifying the s-connected components of the hypergraph. For this latter task, we devise an efficient algorithm based on the union-find technique and a dynamic inverted index. We experimentally evaluate HypED on several real-world hypergraphs and prove its versatility in answering s-distance queries for different values of s. Our framework allows answering such queries in fractions of a millisecond while allowing fine-grained control of the trade-off between index size and approximation error at creation time. Finally, we prove the usefulness of the s-distance oracle in two applications, namely hypergraph-based recommendation and the approximation of the s-closeness centrality of vertices and hyperedges in the context of protein-protein interactions.

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Notes

  1. The s-distance is often defined as the length of the shortest s-path. We subtract 1 to align with graph theory conventions where the distance between edges is the number of vertices in a shortest path between them, making adjacent edges connected by a path of length 2 but at distance 1.

  2. An alternative definition of vertex-to-vertex s-distance that is a metric could be considered. Let the dual hypergraph be the one obtained by swapping the roles of vertices and hyperedges: Hyperedges become vertices, and each vertex in the original hypergraph becomes a hyperedge that connects all the vertices in the dual that correspond to the hyperedges of the original hypergraph by which it was contained. We can compute the hyperedge-to-hyperedge s-distance in this dual hypergraph and obtain a metric vertex-to-vertex s-distance in the original hypergraph.

    However, note that the vertex-to-vertex s-distance as in Definition 3 and the hyperedge-to-hyperedge s-distance in the dual hypergraph yield distinct results. For instance, s-connected vertices in the hypergraph may be at infinite distance in the dual. In fact, an s-path in the hypergraph is a sequence of hyperedges such that consecutive hyperedges share at least s common vertices, whereas an s-path in the dual is a sequence of vertices such that consecutive vertices belong to at least s common hyperedges.

    Both definitions are valid and could be adopted depending on the applications at hand. Our framework can also handle this alternative definition of vertex-to-vertex s-distance, by simply applying it to the dual hypergraph. Of course, this leads to a separate oracle that could not be used to answer hyperedge-to-hyperedge s-distance queries in the original hypergraph. Therefore, henceforth we focus on the vertex-to-vertex s-distance as per Definition 3. This allows to answer all three types of queries with a single oracle.

  3. http://www.sociopatterns.org/datasets.

  4. http://konect.cc/networks.

  5. https://snap.stanford.edu/biodata.

  6. In contrast to hyperedges, for \(s > 1\), a vertex v may belong to different s-connected components, as it may be in hyperedges that overlap only in v.

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  1. CENTAI Institute, Turin, Italy

    Giulia Preti, Gianmarco De Francisci Morales & Francesco Bonchi

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  1. Giulia Preti
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Preti, G., De Francisci Morales, G. & Bonchi, F. Hyper-distance oracles in hypergraphs. The VLDB Journal (2024). https://doi.org/10.1007/s00778-024-00851-2

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