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Scalable tensor methods for nonuniform hypergraphs

Sinan G. Aksoy, Ilya Amburg, Stephen J. Young

TL;DR

This work addresses scalable analysis of nonuniform hypergraphs by avoiding explicit construction of the order-$r$ adjacency tensor through implicit TTSV algorithms. It presents two complementary paths—unordered blowups and generating-function techniques—to compute TTSV1 and TTSV2 with low-degree polynomial dependence on the maximum edge size $r$, enabling practical tensor-based centrality and clustering. The paper demonstrates that tensor-derived centralities (ZEC and HEC) provide information beyond clique-expansion metrics and can distinguish higher-order structure, including Gram-mate hypergraphs, while CP-based tensor embeddings support clustering with geometry differing from matrix-based Laplacian embeddings. These advances offer scalable, hypergraph-native analytics with potential impact on tasks requiring multiway interaction modeling in networks and data analytics.

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.

Scalable tensor methods for nonuniform hypergraphs

TL;DR

This work addresses scalable analysis of nonuniform hypergraphs by avoiding explicit construction of the order- adjacency tensor through implicit TTSV algorithms. It presents two complementary paths—unordered blowups and generating-function techniques—to compute TTSV1 and TTSV2 with low-degree polynomial dependence on the maximum edge size , enabling practical tensor-based centrality and clustering. The paper demonstrates that tensor-derived centralities (ZEC and HEC) provide information beyond clique-expansion metrics and can distinguish higher-order structure, including Gram-mate hypergraphs, while CP-based tensor embeddings support clustering with geometry differing from matrix-based Laplacian embeddings. These advances offer scalable, hypergraph-native analytics with potential impact on tasks requiring multiway interaction modeling in networks and data analytics.

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 to a low-degree polynomial in , where is the number of vertices and is the maximum hyperedge size. Our algorithms are implicit, avoiding formation of the order 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.
Paper Structure (13 sections, 4 theorems, 43 equations, 7 figures, 1 table, 6 algorithms)

This paper contains 13 sections, 4 theorems, 43 equations, 7 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor times same vector for hypergraphs
  4. Unordered blowup approach
  5. Generating function approach
  6. The hypergraph adjacency tensor in practice
  7. Timing experiments
  8. Centrality
  9. Tensor centralities provide complementary information
  10. Persistence in tensor centrality
  11. Tensor centrality distinguishes Gram mates
  12. Clustering
  13. Conclusion and future work

Key Result

Lemma 3.1

\newlabellem:comb0 Let $e=\{v_1,\dots,v_k\}$ be a hyperedge of a rank $r$ and $\beta(e)$ and $\kappa(e)$ as defined in Definition def:blowup. Then

Figures (7)

  • Figure 1: Efficiency of FFT versus subset expansion for generating function evaluation.
  • Figure 2: Runtimes of TTSV1 algorithms.
  • Figure 3: Total (left) and per-edge (middle) runtimes for hyperedge size $\lvert e\rvert$ as a function of rank $r$, and hyperedge size distributions (right) for DAWN and cooking.
  • Figure 4: Kendall $\tau_B$ rank correlation coefficient of ZEC, HEC, and CEC for the top $k$ nodes.
  • Figure 5: Node rank persistence of the top 10 nodes under ZEC, HEC, and CEC for DAWN and cooking. Purple indicates a rank change from $r-1$ and yellow indicates no change.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2: Nonuniform Hypergraph Adjacency Tensor BANERJEE201714
  • Lemma 3.1
  • Proposition 3.2
  • Proof 1
  • Proposition 3.3
  • Proof 2
  • Definition 4.1: $Z$- and $H$-eigenvector centrality, (ZEC and HEC)
  • Theorem 4.2: Perron-Frobenius theorem for the hypergraph adjacency tensor qi2017tensor