Scalable tensor methods for nonuniform hypergraphs

TL;DR

This paper introduces scalable tensor algorithms for nonuniform hypergraphs that efficiently analyze higher-order interactions, enabling new insights in hypergraph centrality and clustering beyond traditional graph methods.

Contribution

The authors develop low-complexity, implicit tensor algorithms for nonuniform hypergraphs, facilitating practical analysis of higher-order structures.

Findings

Algorithms significantly reduce computational complexity.

Tensor-based measures reveal higher-order structures.

Tensor methods provide complementary insights to graph reduction.

Abstract

While multilinear algebra appears natural for studying the multiway interactions modeled by hypergraphs, tensor methods for general hypergraphs have been stymied by theoretical and practical barriers. A recently proposed adjacency tensor is applicable to nonuniform hypergraphs, but is prohibitively costly to form and analyze in practice. We develop tensor times same vector (TTSV) algorithms for this tensor which improve complexity from $O(n^r)$ to a low-degree polynomial in $r$, where $n$ is the number of vertices and $r$ is the maximum hyperedge size. Our algorithms are implicit, avoiding formation of the order $r$ adjacency tensor. We demonstrate the flexibility and utility of our approach in practice by developing tensor-based hypergraph centrality and clustering algorithms. We also show these tensor measures offer complementary information to analogous graph-reduction approaches on data, and are also able to detect higher-order structure that many existing matrix-based approaches provably cannot.