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Multi-granularity Adaptive Hypergraph Representation Learning via Granular-ball

Sen Zhao, Yifan Guan, Jinyuan Ni, Gaojie Xu, Zhang Xu, Xiaoyu Lian, Yi Liu, Yi Wang, Wei Wang

arXiv:2609.05574v1cs.LG

TL;DR

Hypergraph methods model high-order relationships, but prior hyperedge-generation rules often overlook graph-topology diversity and hyperedge granularity. MGHRL uses adaptive granular-ball splitting and a multi-column network with reversible connections. The authors report that it outperforms state-of-the-art techniques across benchmark datasets.

  • Problem

    Many hyperedge-generation methods rely on predefined rules and overlook graph-topology diversity and the multi-granularity of hyperedges.

    Source: §1 · p. 2

  • Method

    MGHRL adaptively splits granular balls to generate multi-granularity hypergraphs and uses separate network columns with reversible connections to learn across granularities.

    Source: §1 · p. 2, §3.2 · p. 5

  • Results

    MGHRL consistently outperforms state-of-the-art techniques across various benchmark datasets.

    Source: §5 · pp. 9–10

  • Takeaways & Limitations

    The authors conclude that MGHRL improves modeling of complex graph relationships and performs well on node-classification tasks.

    Source: §5 · p. 9

    MGHRL does not capture temporal continuity or structural dynamics, limiting its applicability to video-level dynamic human action recognition.

    Source: §C.1 · p. 14

Abstract

from arXiv · show

Hypergraph representation learning aims to capture high-order information in graphs by constructing hyperedges that simultaneously connect multiple nodes. These hyperedges adapt to the graph's topological features, facilitating the extraction of high-order relationships at multiple granularities. Most prior work relies on predefined definitions to generate hyperedges, overlooking the diversity in graph topological structures and the multi-granularity characteristics of hyperedges. As a result, this limits their ability to effectively and adaptively discover high-order relationships and efficiently process complex structural information. To address this limitation, we propose a novel framework called \underline{M}ulti-\underline{G}ranularity \underline{H}ypergraph \underline{R}epresentation \underline{L}earning (MGHRL). MGHRL introduces an Adaptive Granular Hypergraph Generation strategy, which generates hyperedges at multiple levels of granularity through the adaptive splitting of granular-ball, effectively capturing high-order relationships based on the graph's topological structure. Additionally, we propose a Multi-Granularity Hypergraph Network with multiple sub-networks, capturing features from hyperedges at different granularities and integrating them via hierarchical reversible connections. Experimental results show that MGHRL significantly outperforms baseline models on benchmark datasets.

1 Introduction

Hypergraphs model high-order relationships, but prior hyperedge-generation rules often overlook diverse graph topologies and the multiple granularities of cohesive node groups. MGHRL addresses these gaps with adaptive granular-ball splitting and a multi-column network, and is reported to improve hypergraph representation learning on eight real-world datasets.

1 Introduction

  • Hypergraphs connect multiple nodes in one hyperedge to represent high-order interactions beyond pairwise graph relationships.The introduction identifies social networks and bioinformatics as examples of systems with high-order interactions.Source: §1 · p. 1
  • Prior hyperedge-generation approaches often use predefined rules that overlook diverse graph topologies and hyperedge granularity.The paper also notes that hypergraph modeling often neglects the distinct learning capabilities of hyperedges at different granularities.Source: §1 · pp. 1–2
  • MGHRL adaptively splits granular balls to construct hyperedges at multiple granularities, refining from coarse to fine based on node cohesion and graph topology.This contrasts with fixed-granularity methods that overlook hierarchical structure.Source: §1 · p. 2, Figure 1 · p. 2
  • Its Multi-Granularity Hypergraph Network learns from different granularities in separate columns and progressively refines information through multi-level reversible connections.The framework also emphasizes the role of hyperedge multi-granularity in representation learning.Source: §1 · p. 2
  • MGHRL effectively improves hypergraph representation learning on eight real-world datasets.The introduction reports extensive experiments across eight datasets.Source: §1 · p. 2

2 Related Work

Related work moves from graph neural networks that aggregate neighboring-node features toward hypergraphs, whose hyperedges represent high-order relationships. Existing hypergraph approaches vary in construction and propagation, but often retain predefined criteria for generating hyperedges.

2 Related Work

  • GNN approaches include localized GCN aggregation [1], attention-weighted neighbor aggregation in GAT, neighborhood sampling in GraphSAGE [2], and structure-distinguishing aggregation in GIN.The paper notes that traditional graphs remain limited to pairwise relationships [1] [2].Source: §2 · p. 3
  • Hypergraph methods model high-order relationships, but hyperedge data can be difficult to obtain or publicly access.Examples include HGNN's nearest-node hyperedges [12], HGNN+'s k-Hop hyperedges [13], and HyperGCN's clique expansion.Source: §2 · p. 3
  • The reviewed methods span unified graph–hypergraph frameworks [23], hypergraph diffusion [14], and architecture search, while often relying on predefined hyperedge criteria.The paper identifies graph-structure diversity and hyperedge multi-granularity as commonly overlooked.Source: §2 · p. 3

3 Approach

MGHRL builds a coarse-to-fine granular-ball hypergraph through adaptive splitting, then uses a multi-column network with reversible connections to integrate features across granularities.

3 Approach

  • The approach combines adaptive granular-hypergraph generation with a multi-granularity hypergraph network to model complex hypergraph structures.The generation strategy adaptively refines initialized coarse granular balls through binary splitting, while the network integrates information across granularities.Source: §3 · p. 3

3.1 Adaptive Granular Hypergraph Generation

  • AGHG starts with coarse granular balls and repeatedly divides them according to structural and feature similarity, producing a topological covering from local patterns to global structure.The decomposition is framed as seeking topological coherence and semantic meaning, with ball quality representing structural or topological quality.Source: §3.1 · p. 4

3.1.1 Coarse-grained granular-ball initialization

  • Initialization selects high-degree nodes as coarse-ball centers and assigns remaining nodes to their closest center by breadth-first search.The passage describes this initialization as balancing granular-ball quality with algorithm efficiency and scalability.Source: §3.1.1 · pp. 4–5

3.1.2 Fine-grained binary splitting

  • Granular balls are split when their child balls' combined quality exceeds the parent's, and the resulting balls define multi-granularity hyperedges.The method evaluates ball connectivity as an indicator of cohesion and richer high-order relationships.Source: §3.1.2 · p. 5

3.2 Multi-Granularity Hypergraph Network

  • MGHN uses multiple columns with reversible connections to integrate information across granularities, while each column learns from a distinct graph replica with decoupled parameters.Its columns progressively disentangle lower-level features from higher-level semantics and fuse them through reversible connections.Source: §3.2 · pp. 5–6
  • A node embedding is updated by aggregating a weighted sum of hyperedge embeddings, while the non-linear residual fusion module integrates feature representations from current and previous columns.The reversible scaling module applies channel-level scaling with a learnable parameter initially set to 1, constrained to a minimum absolute value of 1 × 10^-3 to support stable gradient flow.Source: §3.2 · pp. 6–7

4 Experiment

MGHRL achieves the highest node-classification accuracy across seven benchmark datasets, while ablations and sensitivity tests examine its components and parameter choices. A case study illustrates multi-granularity hyperedges in citation networks.

4.1 Datasets

  • The evaluation covers three citation networks and four social-media networks, with training, validation, and test splits of 6:2:2.Citation-network tasks predict paper topics; social-media tasks predict developer fields or page categories.Source: §4.1 · p. 7

4.2 Baselines

  • MGHRL is compared with graph neural networks and hypergraph neural networks using node-classification accuracy.The baselines include GCN, GAT, GraphSAGE, and GDGIN, alongside HyperGCN, HGNN, HGNN+, and other HNNs.Source: §4.2 · p. 7

4.3 Experimental Setting

  • Experiments use four columns, hidden dimension 16, and eight attention heads, with Adam, cross-entropy loss, and learning rate 0.0001.The reported setup also uses ReLU and regularization parameter 5 × 10−4.Source: §4.3 · p. 7

4.4 Comparison Experiments

  • MGHRL achieves the highest accuracy across seven benchmark datasets, with social-network improvements up to 4.17%.The comparison uses node-classification accuracy; the table reports test accuracy and marks statistically significant improvements at p < 0.01.Source: §4.4 · p. 7, Table 2 · p. 8

4.5 Ablation Studies

  • Removing AGHG in favor of k-Hop causes performance drops across datasets, while replacing MGHN with HGNN causes a larger decline, particularly on Citeseer.The ablation variants are MGHRL - w/o AGHG and MGHRL - w/o MGHN.Source: §4.5 · p. 8, Table 3 · p. 8

4.6 Hyper-parameter Sensitivity Analysis

  • Performance peaks at hidden dimension d = 16, improves with more columns before declining at c = 5, and consistently improves as attention heads increase.The decline at c = 5 is attributed likely to over-smoothing of node features.Source: §4.6 · pp. 8–9

4.7 Case Study

  • In Cora, PubMed, and CiteSeer, AGHG hyperedges reveal local and global structures, including clusters without direct citations and cross-domain patterns.Figure 4 visualizes representative node subsets; distinct colors denote node categories and circles mark AGHG-constructed hyperedges.Source: §4.7 · p. 9, Figure 4 · p. 9

5 Conclusion

MGHRL combines adaptive granular-ball splitting with multi-column network structures to model hyperedges across granularities. The conclusion reports stronger modeling of complex graph relationships and performance above state-of-the-art techniques across benchmark datasets.

5 Conclusion

  • MGHRL uses adaptive granular-ball splitting and multi-column network structures to capture hyperedge characteristics across multiple levels.The conclusion associates this approach with a nuanced understanding of local and global structures.Source: §5 · p. 9
  • MGHRL significantly enhances the modeling of complex relationships in graph-structured data.The conclusion specifically reports superior performance in node classification tasks.Source: §5 · p. 9
  • MGHRL consistently outperforms state-of-the-art techniques across various benchmark datasets.The conclusion characterizes this performance as demonstrating robustness and effectiveness.Source: §5 · p. 10

A.1 Proofs regarding Proposition 2

The proof verifies that the graph-based distance satisfies the standard metric conditions. It then identifies the closed metric balls as basic neighborhoods and states that their family forms a finite topological covering of the node set.

A.1 Proofs regarding Proposition 2

  • The proof checks non-negativity, symmetry, and the triangle inequality for the graph-based distance.It uses shortest-path distance to support non-negativity and symmetry.Source: §A.1 · p. 13
  • The distance is concluded to define a metric and induce a metric topology over the nodes.Source: §A.1 · p. 13
  • Closed metric balls centered at nodes are presented as basic neighborhoods that form a finite topological covering of the node set.Source: §A.1 · p. 13

A.2 Proofs regarding Proposition 3

The proof checks that the Granular-ball covering satisfies coverage, locality, and neighborhood-basis properties in the topology induced by the graph-based metric. It presents these properties as the foundation for localized graph decomposition.

A.2 Proofs regarding Proposition 3

  • The covering is constructed so every node belongs to at least one closed metric ball.Each ball is centered at a node and has a positive radius.Source: §A.2 · p. 13
  • Closed metric balls preserve local structural coherence and support locality-aware computations on the graph.The passage notes that closed balls approximate open neighborhoods in discrete graph spaces.Source: §A.2 · p. 13
  • The finite collection of Granular-balls is asserted to satisfy the neighborhood-basis property and form localized subspaces aligned with the topology.Source: §A.2 · p. 13
  • Together, the three properties are presented as the theoretical foundation for topology-respecting, localized graph decomposition and processing.Source: §A.2 · p. 13

B Time Complexity Analysis of AGHG Algorithm

The analysis describes coarse initialization and fine-grained refinement as the two phases of AGHG. It concludes that the algorithm has subquadratic overall time complexity and is suited to large-scale sparse and medium-density graphs.

B Time Complexity Analysis of AGHG Algorithm

  • The coarse-grained initialization performs |V| iterations, selecting a high-degree node and then running BFS in each iteration.Each BFS explores the graph up to a predefined structural threshold.Source: B Time Complexity Analysis of AGHG… · p. 14
  • Fine-grained refinement recursively splits coarse clusters using binary partitioning, dual-source BFS traversals, and local node reassignments.The analysis says a |V| scaling factor keeps the number of recursive operations sublinear in the total node count.Source: B Time Complexity Analysis of AGHG… · p. 14
  • AGHG has subquadratic overall time complexity and is described as suitable for large-scale sparse and medium-density graphs.The passage also describes a favorable trade-off between computational efficiency and clustering accuracy compared with classical graph clustering methods.Source: B Time Complexity Analysis of AGHG… · p. 14

C Limitations and Future Research

MGHRL advances multi-granular high-order modeling for static hypergraphs, but its limitations include applications involving temporal graphs.

C Limitations and Future Research

C.1 Limitations

MGHRL is designed for static hypergraphs and does not model temporal evolution or time-driven granularity changes. The paper proposes temporal extensions and dynamic-system validation as future directions, and reports no known ethical concerns or negative societal impacts.

C.1 Limitations

  • MGHRL assumes static node relationships and does not explicitly model temporal evolution of node and hyperedge structures.Source: §C.1 · p. 14
  • In human pose estimation, MGHRL cannot capture temporal continuity and structural dynamics, limiting its applicability to video-level dynamic action recognition.Source: §C.1 · p. 14
  • In traffic modeling, its static hyperedge partitioning cannot adapt to evolving high-order regional structures, diminishing effectiveness for dynamic traffic tasks.Source: §C.1 · p. 14
  • MGHRL lacks temporal modeling and cannot represent granularity transitions such as differing spatial scales during peak and off-peak traffic hours.Source: §C.1 · p. 15
  • MGHRL has not been validated on temporal benchmarks including NTU RGB+D, METR-LA, or PEMS-BAY, leaving generalization to real-world temporal tasks unclear.Source: §C.1 · p. 15

C.2 Future Research Directions

  • Future work proposes temporally consistent hyperedges, integration with temporal graph neural networks, and temporal columns or reversible connections.Source: §C.2 · p. 15
  • The paper proposes validating improved MGHRL on dynamic datasets to assess temporal high-order pattern identification and prediction of future interactions.Source: §C.2 · p. 15

D Broader Impacts

  • The work reports no known ethical concerns or negative societal impacts because it involves no human subjects, sensitive data, or ethically sensitive applications.Source: D Broader Impacts · p. 15
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