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Line Hypergraph Convolution Network: Applying Graph Convolution for Hypergraphs

Sambaran Bandyopadhyay, Kishalay Das, M. Narasimha Murty

arXiv:2002.03392v1cs.SIcs.LG

TL;DR

Simple-graph convolution handles pairwise edges, but hypergraphs represent higher-order relationships that occur in real-world interactions. The paper applies graph convolution to hypergraphs through a weighted, attributed line graph, and reports performance on par with or better than state-of-the-art hypergraph neural networks.

  • Problem

    Conventional graph convolution is suited to pairwise edges, leaving a need to model higher-order relationships represented by hypergraphs.

    Source: §1 · p. 1

  • Method

    LHCN maps a hypergraph to a weighted, attributed line graph, applies graph convolution there, and maps learned labels back to hypergraph nodes.

    Source: §1 · p. 2, §4.2 · p. 6

  • Results

    LHCN performs on par with, and often improves on, state-of-the-art hypergraph neural networks; it beats HyperGCN by roughly 8.4% on Cora, while HyperGCN leads on Pubmed.

    Source: §1 · p. 2, §5.3 · p. 9

  • Takeaways & Limitations

    Hypergraph construction remains a practical issue, and learning the transformation rule from data is an open direction.

    Source: §6 · p. 11

    Information propagation assumes nodes connected by a hyperedge tend to have similar embeddings and labels, aligning with the hypergraph Laplacian and homophily.

    Source: §4.3 · p. 7

Abstract

from arXiv · show

Network representation learning and node classification in graphs got significant attention due to the invent of different types graph neural networks. Graph convolution network (GCN) is a popular semi-supervised technique which aggregates attributes within the neighborhood of each node. Conventional GCNs can be applied to simple graphs where each edge connects only two nodes. But many modern days applications need to model high order relationships in a graph. Hypergraphs are effective data types to handle such complex relationships. In this paper, we propose a novel technique to apply graph convolution on hypergraphs with variable hyperedge sizes. We use the classical concept of line graph of a hypergraph for the first time in the hypergraph learning literature. Then we propose to use graph convolution on the line graph of a hypergraph. Experimental analysis on multiple real world network datasets shows the merit of our approach compared to state-of-the-arts.

1 Introduction

Conventional graph convolution targets pairwise relationships, while real-world interactions can be higher-order and difficult to represent with simple graphs. The paper introduces LHCN, which applies graph convolution through a hypergraph line graph, and reports performance on par with or better than existing hypergraph neural networks.

1 Introduction

  • Graph neural networks have improved semi-supervised graph tasks such as node classification, and GCN aggregates transformed node attributes over neighborhoods.GCN learns its transformation matrix by minimizing cross-entropy on labeled nodes.Source: §1 · p. 1
  • Simple-graph convolution is suited to pairwise relations, but real-world interactions may involve higher-order relationships.A co-authorship example illustrates that a group of authors may be related without every pair being directly connected.Source: §1 · pp. 1–2
  • Hypergraphs represent higher-order relations by allowing an edge to connect more than two vertices, but computation on them is more complicated and expensive.The paper motivates learning algorithms that can use hypergraphs to represent real-world interactions.Source: §1 · p. 2
  • Existing approaches transform hypergraphs through clique or star expansion, while tensor-based methods typically assume uniform hyperedge sizes.Recent hypergraph neural networks use clique expansion or a hypergraph Laplacian to apply convolution.Source: §1 · p. 2
  • The proposed LHCN maps a hypergraph to a weighted, attributed line graph, applies graph convolution there, and maps labels back to hypergraph nodes.The method is presented as applicable to non-uniform hypergraphs.Source: §1 · p. 2
  • LHCN performs on par with, and often improves on, state-of-the-art hypergraph neural networks.The paper reports experiments on popular node-classification datasets.Source: §1 · p. 2

2 Related Work

Prior hypergraph analysis commonly converts hypergraphs to simple graphs or uses tensors, while neural approaches apply graph convolution through expansions or hypergraph Laplacians. LHCN contributes a different graph-convolution approach for hypergraphs.

2 Related Work

  • Graph neural network work includes semi-supervised node classification, GraphSAGE aggregation for inductive classification, GAT attention, and DGI's mutual-information objective.The cited literature also includes surveys of network representation learning and graph neural networks [31] [12].Source: §2 · p. 3
  • Hypergraph analysis often transforms the structure by clique or star expansion, while tensor methods typically assume uniform hyperedges.Other tensor-based work addresses clustering, and hypergraph literature also covers link prediction, submodular hypergraphs, and active learning.Source: §2 · p. 3
  • Existing hypergraph neural networks include HGNN's hypergraph-Laplacian convolution, dynamic hypergraph construction, and HyperGCN's maximum-distance regularizer,,.HyperGCN uses a hypergraph Laplacian to create a weighted simple graph before applying GCN,.Source: §2 · pp. 3–4
  • LHCN belongs to hypergraph neural networks and proposes a novel way to apply graph convolution on hypergraphs.The paper contrasts this contribution with prior approaches based on clique expansion or hypergraph Laplacians.Source: §2 · pp. 3–4

3 Problem Statement and Notations Used

The paper formulates node classification on an attributed undirected hypergraph, using both its structure and node features. The goal is to predict labels for unlabelled nodes from a subset of labeled nodes.

3 Problem Statement and Notations Used

  • The input is an undirected hypergraph with nodes and hyperedges, where each hyperedge connects a subset of nodes and the incidence matrix records membership.An entry is 1 when the corresponding hyperedge contains the node and 0 otherwise.Source: §3 · p. 4
  • Each node has a d-dimensional feature vector, forming the feature matrix X.Source: §3 · p. 4
  • Given labels for a subset of nodes, the objective is to learn a function that predicts labels for unlabelled nodes using both hypergraph structure and node attributes.Source: §3 · p. 4

4 Solution Approach: LHCN

LHCN converts an attributed hypergraph into a weighted line graph for graph convolution, then maps learned labels and representations back to hypergraph nodes.

4.1 Line Graph and Extension to Hypergraphs

  • Each hyperedge becomes a weighted line-graph node, linked to intersecting hyperedges; its attributes average member-node features, and its training label is the majority label among labeled members.Source: §4.1 · pp. 5–6

4.2 Convolution Network on the Line Graph

  • The method applies a two-layer normalized GCN with self-loops to line-graph features, learns parameters by cross-entropy and ADAM, and predicts labels for all line-graph nodes.Source: §4.2 · p. 6

4.3 Information Propagation to the Hypergraph

  • Learned hyperedge labels map to hypernodes by majority vote, while node representations average representations of incident hyperedges.Source: §4.3 · p. 7
  • For real-world sparse graphs with constant s, total runtime is O(m + n).Source: §4.3 · p. 7

5 Experimental Evaluation

Across three citation-network datasets, LHCN improves on the closest baseline on Cora, while HyperGCN performs best on Pubmed. The evaluation also examines learned representations and training behavior as training-set size varies.

5 Experimental Evaluation

  • The experiments focus on semi-supervised node classification and do not evaluate unsupervised tasks such as hypernode clustering or hyperedge prediction.The authors note this matches the objective of graph convolution and prior hypergraph-neural-network literature.Source: §5 · p. 7

5.1 Datasets and Baseline Algorithms Used

  • The evaluation uses Cora, Citeseer, and Pubmed citation hypergraphs, where citations define hyperedges and paper attributes are bag-of-words vectors.Cora and Citeseer use binary attributes; Pubmed uses tf-idf attributes.Source: §5.1 · p. 8

5.2 Experimental Setup

  • LHCN is evaluated on 80–20% random train-test splits using a two-layer GCN on the generated line graph.Experiments use three datasets and report approximately 16, 21, and 867 seconds for Cora, Citeseer, and Pubmed, respectively.Source: §5.2 · pp. 8–9

5.3 Results on Node Classification

  • LHCN outperforms the closest baseline HyperGCN by roughly 8.4% on Cora, while HyperGCN performs best on Pubmed and many methods, including LHCN, reach around 70% accuracy.Results average ten runs with an 80–20% random train-test split; Table 2 reports accuracy and standard deviation.Source: §5.3 · p. 9, Table 2 · p. 9

5.4 Hypernode Visualization

  • LHCN’s t-SNE visualizations show hypernode representations with different node classes separated to a good extent across the datasets.HyperGCN representations are omitted because their visual quality was not good.Source: §5.4 · pp. 9–10, Figure 4 · p. 10

5.5 Further Analysis on LHCN

  • Training loss decreases quickly at first and converges, while node-classification performance improves as the training share rises from 50% to 95%.The loss trend is attributed to ADAM and learning-rate reductions every 100 epochs.Source: §5.5 · p. 10, Figure 5 · p. 10

6 Conclusion and Future Work

The conclusion presents graph convolution on a weighted, attributed line-graph transformation as a promising hypergraph approach. It identifies meaningful hypergraph construction as a practical issue and suggests learning the transformation rule from data as future work.

6 Conclusion and Future Work

  • LHCN applies graph convolution to a hypergraph through a weighted, attributed line-graph transformation.The conclusion describes experimental results as promising and improving on state of the art in hypergraph graph neural networks.Source: §6 · p. 11
  • Constructing a hypergraph that lets algorithms extract meaningful information remains a practical issue.Source: §6 · p. 11
  • The authors identify learning the rule for transforming a hypergraph to a line graph from data as future work.Source: §6 · p. 11

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