Community and hyperedge inference in multiple hypergraphs

TL;DR

The paper addresses the challenge of revealing latent high-order structure across multiple interrelated hypergraphs. It proposes the Multi-Hypergraph Stochastic Block Model (MHSBM), a generative framework that jointly models within-hypergraph hyperedges and inter-hypergraph edges using latent node memberships $u^l$, within-hypergraph affinities $w^l$, and inter-hypergraph affinities $w^{ll'}$, with Poisson edge formation and an additional hyperedge internal degree $\theta^l$ to capture nonuniform node influence. The model demonstrates strong performance in community detection, hyperedge prediction, and inter-hypergraph edge prediction across real-world, multi-domain, and single-hypergraph settings, leveraging cross-hypergraph information to improve inference. These results underscore the practical value of cross-hypergraph integration for understanding high-order organization in complex systems and point to extensions in adaptive fusion and temporal dynamics.

Abstract

Hypergraphs, capable of representing high-order interactions via hyperedges, have become a powerful tool for modeling real-world biological and social systems. Inherent relationships within these real-world systems, such as the encoding relationship between genes and their protein products, drive the establishment of interconnections between multiple hypergraphs. Here, we demonstrate how to utilize those interconnections between multiple hypergraphs to synthesize integrated information from multiple higher-order systems, thereby enhancing understanding of underlying structures. We propose a model based on the stochastic block model, which integrates information from multiple hypergraphs to reveal latent high-order structures. Real-world hyperedges exhibit preferential attachment, where certain nodes dominate hyperedge formation. To characterize this phenomenon, our model introduces hyperedge internal degree to quantify nodes' contributions to hyperedge formation. This model is capable of mining communities, predicting missing hyperedges of arbitrary sizes within hypergraphs, and inferring inter-hypergraph edges between hypergraphs. We apply our model to high-order datasets to evaluate its performance. Experimental results demonstrate strong performance of our model in community detection, hyperedge prediction, and inter-hypergraph edge prediction tasks. Moreover, we show that our model enables analysis of multiple hypergraphs of different types and supports the analysis of a single hypergraph in the absence of inter-hypergraph edges. Our work provides a practical and flexible tool for analyzing multiple hypergraphs, greatly advancing the understanding of the organization in real-world high-order systems.

Figures (6)

• Figure 1: The advantages of multi-hypergraph: an illustrative example. The left plot shows the partial hypergraph structure of the protein complex HMC and its associated gene clusters within the gene-protein multi-hypergraph. Gene IDs are marked in blue and protein IDs in red within the ground-truth community. All other protein and gene IDs outside this community are marked in black. Gray lines represent gene-protein encoding relationships, while hyperedges are represented by irregular closed curves. The central plot illustrates the protein complexes and gene clusters detected by MHSBM-single (without inter-hypergraph edge information), whereas the right displays results from inter-hypergraph informed detection. This example demonstrates the advantage of multi-hypergraph modeling, where the model improves detection accuracy by leveraging complementary inter-hypergraph edge information.
• Figure 2: Effect of inter-hypergraph edges on the performance of MHSBM. We show the metric variations of MHSBM on the multi-hypergraph across different removal proportions of inter-hypergraph edges. The horizontal axis represents the proportion of inter-hypergraph edges removed between the hypergraph (ranging from 30% to 100%). The vertical axis shows the model's F1-score, CS, and NMI metrics for each hypergraph (H1/H2/H3). This plot shows that the performance of MHSBM has a declining trend, with some fluctuations, as inter‑hypergraph edges decrease.
• Figure 3: Comparison of community detection algorithms on multi-hypergraphs. We show the performance of MHSBM, Hy-MMSBM, and Hypergraph-MT on multi-hypergraphs. For each model, seven distinct random seeds are used for initialization, resulting in seven sets of outcomes from which the mean values are computed.
• Figure 4: Community configuration on the hospital dataset.a Metrics of MHSBM, Hy-MMSBM, and Hypergraph-MT. For MHSBM and Hy-MMSBM models, we set the assortative to false. b Results of MHSBM and Hy-MMSBM under different initializations of $w$. Results above the horizontal axis are the detection results with the assortative parameter set to False, and those below the axis are the results with assortative set to True. c Community affinities. The first row of plots shows the community affinities of the true communities in H0 and H1, the second row displays the community affinities captured by Hy-MMSBM, and the third row shows the community affinities captured by MHSBM.
• Figure 5: Hyperedge prediction on the Gene-Disease dataset. We evaluate the prediction performance by the mean and standard deviation of the AUC under different maximum hyperedge sizes D in the test set. The line plots depict the AUC variation trends of methods, with shaded regions indicating the standard deviation of AUC values. The figure illustrates the stability of the MHSBM for hyperedge prediction at varying hyperedge sizes.