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A two-steps tensor eigenvector centrality for nodes and hyperedges in hypergraphs

Qing Xu, Chunmeng Liu, Changjiang Bu, Jihong Shen

TL;DR

A new tensor-based centrality measure for general hypergraphs that captures a higher-order mutual reinforcement mechanism: a node's importance is determined by the importance of its incident hyperedges and the other nodes within these hyperedges; symmetrically, a hyperedge's importance is determined by the importance of its constituent nodes and the other hyperedges containing these nodes.

Abstract

Hypergraphs have been a powerful tool to represent higher-order interactions, where hyperedges can connect an arbitrary number of nodes. Quantifying the relative importance of nodes and hyperedges in hypergraphs is a fundamental problem in network analysis. In this paper, we propose a new tensor-based centrality measure for general hypergraphs. We use a third-order tensor to represent the relationship between nodes and hyperedges. The tensor's positive Perron vector is defined as the centrality vector of the hypergraph. The existence and uniqueness of this centrality vector are guaranteed by the Perron-Frobenius theorem for tensors. This new centrality measure captures a higher-order mutual reinforcement mechanism: a node's importance is determined by the importance of its incident hyperedges and the other nodes within these hyperedges; symmetrically, a hyperedge's importance is determined by the importance of its constituent nodes and the other hyperedges containing these nodes. We further provide a combinatorial interpretation by proving that the centrality vector represents the limit geometric capacity of two-steps expansion trees. We illustrate the centrality measure on real-world hypergraph datasets.

A two-steps tensor eigenvector centrality for nodes and hyperedges in hypergraphs

TL;DR

A new tensor-based centrality measure for general hypergraphs that captures a higher-order mutual reinforcement mechanism: a node's importance is determined by the importance of its incident hyperedges and the other nodes within these hyperedges; symmetrically, a hyperedge's importance is determined by the importance of its constituent nodes and the other hyperedges containing these nodes.

Abstract

Hypergraphs have been a powerful tool to represent higher-order interactions, where hyperedges can connect an arbitrary number of nodes. Quantifying the relative importance of nodes and hyperedges in hypergraphs is a fundamental problem in network analysis. In this paper, we propose a new tensor-based centrality measure for general hypergraphs. We use a third-order tensor to represent the relationship between nodes and hyperedges. The tensor's positive Perron vector is defined as the centrality vector of the hypergraph. The existence and uniqueness of this centrality vector are guaranteed by the Perron-Frobenius theorem for tensors. This new centrality measure captures a higher-order mutual reinforcement mechanism: a node's importance is determined by the importance of its incident hyperedges and the other nodes within these hyperedges; symmetrically, a hyperedge's importance is determined by the importance of its constituent nodes and the other hyperedges containing these nodes. We further provide a combinatorial interpretation by proving that the centrality vector represents the limit geometric capacity of two-steps expansion trees. We illustrate the centrality measure on real-world hypergraph datasets.
Paper Structure (8 sections, 4 theorems, 17 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 8 sections, 4 theorems, 17 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The two-steps eigenvector centrality of general hypergraphs
  4. The definition of hypergraph two-steps eigenvector centrality (HTEC)
  5. Combinatorial interpretation: geometric capacity of two-steps expansion trees
  6. Numerical illustration and algorithm
  7. Experiments on real-world hypergraphs
  8. Conclusion

Key Result

Lemma 1

Perron Th If $\mathcal{A}$ is a nonnegative weakly irreducible tensor, then its spectral radius $\rho(\mathcal{A})$ is the unique positive eigenvalue of $\mathcal{A}$, and there exists a unique positive eigenvector $\mathbf{x}$ (up to a positive scaling factor) corresponding to $\rho(\mathcal{A})$.

Figures (5)

  • Figure 1: A schematic diagram of converting a hypergraph to a bipartite graph.
  • Figure 2: Illustration and comparison of expansion trees rooted at node $v_3$ with depth $t=2$. The left panel shows the simple graph. The middle panel depicts the linear expansion tree $\mathcal{L}_2(v_3)$. The right panel depicts the two-steps expansion tree $\mathcal{T}_2(v_3)$.
  • Figure 3: A non-uniform sunflower hypergraph with hyperedge size from 2 to 7.
  • Figure 4: Scatter plots of node and edge centralities by HTEC, Linear, Max and Log-exp model centralities.
  • Figure 5: Similarity of top-k ranked nodes and hyperedges by HTEC, Linear, Max and Log-exp model centralities.

Theorems & Definitions (11)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • Lemma 3
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 1 more