High-order Line Graphs of Non-uniform Hypergraphs: Algorithms, Applications, and Experimental Analysis

TL;DR

This paper introduces efficient parallel algorithms for high-order line graph expansions of hypergraphs, significantly reducing computational costs and enabling large-scale spectral analysis on shared-memory machines.

Contribution

The authors develop the first framework for hypergraph spectral analysis using high-order line graphs, with algorithms that are faster and more memory-efficient than existing methods.

Findings

Algorithms are orders of magnitude faster than existing sparse matrix multiplication methods.

Achieve 5-31x speedup over previous heuristic algorithms for s-line graph computation.

Enable analysis of large hypergraph datasets on a single shared-memory machine.

Abstract

Hypergraphs offer flexible and robust data representations for many applications, but methods that work directly on hypergraphs are not readily available and tend to be prohibitively expensive. Much of the current analysis of hypergraphs relies on first performing a graph expansion -- either based on the nodes (clique expansion), or on the edges (line graph) -- and then running standard graph analytics on the resulting representative graph. However, this approach suffers from massive space complexity and high computational cost with increasing hypergraph size. Here, we present efficient, parallel algorithms to accelerate and reduce the memory footprint of higher-order graph expansions of hypergraphs. Our results focus on the edge-based $s$-line graph expansion, but the methods we develop work for higher-order clique expansions as well. To the best of our knowledge, ours is the first framework to enable hypergraph spectral analysis of a large dataset on a single shared-memory machine. Our methods enable the analysis of datasets from many domains that previous graph-expansion-based models are unable to provide. The proposed $s$-line graph computation algorithms are orders of magnitude faster than state-of-the-art sparse general matrix-matrix multiplication methods, and obtain approximately $5-31{\times}$ speedup over a prior state-of-the-art heuristic-based algorithm for $s$-line graph computation.