From Graphs to Hypergraphs: Hypergraph Projection and its Remediation

TL;DR

This paper investigates the consequences of representing hypergraphs as graphs, analyzes information loss due to projection, and proposes a learning-based method to reconstruct higher-order structures, validated on real datasets.

Contribution

It provides a theoretical analysis of hypergraph projection effects and introduces a novel learning-based approach for hypergraph reconstruction.

Findings

Identifies two common patterns causing information loss in hypergraph projections.

Quantifies the difficulty of recovering lost higher-order structures without additional information.

Demonstrates the effectiveness of the reconstruction method on real-world datasets with applications in protein ranking and link prediction.

Abstract

We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.