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Random walks with resetting on hypergraph

Fei Ma, Xincheng Hu, Haobin Shi, Wei Pan, Ping Wang

Abstract

Hypergraph has been selected as a powerful candidate for characterizing higher-order networks and has received increasing attention in recent years. In this article, we study random walks with resetting on hypergraph by utilizing spectral theory. Specifically, we derive exact expressions for some fundamental yet key parameters, including occupation probability, stationary distribution, and mean first passage time, all of which are expressed in terms of the eigenvalues and eigenvectors of the transition matrix. Furthermore, we provide a general condition for determining the optimal reset probability and a sufficient condition for its existence. In addition, we build up a close relationship between random walks with resetting on hypergraph and simple random walks. Concretely, the eigenvalues and eigenvectors of the former can be precisely represented by those of the latter. More importantly, when considering random walks, we abandon the traditional approach of converting hypergraph into a graph and propose a research framework that preserves the intrinsic structure of hypergraph itself, which is based on assigning proper weights to neighboring nodes. Through extensive experiments, we show that the new framework produces distinct and more reliable results than the traditional approach in node ranking. Finally, we explore the impact of the resetting mechanism on cover time, providing a potential solution for optimizing search efficiency.

Random walks with resetting on hypergraph

Abstract

Hypergraph has been selected as a powerful candidate for characterizing higher-order networks and has received increasing attention in recent years. In this article, we study random walks with resetting on hypergraph by utilizing spectral theory. Specifically, we derive exact expressions for some fundamental yet key parameters, including occupation probability, stationary distribution, and mean first passage time, all of which are expressed in terms of the eigenvalues and eigenvectors of the transition matrix. Furthermore, we provide a general condition for determining the optimal reset probability and a sufficient condition for its existence. In addition, we build up a close relationship between random walks with resetting on hypergraph and simple random walks. Concretely, the eigenvalues and eigenvectors of the former can be precisely represented by those of the latter. More importantly, when considering random walks, we abandon the traditional approach of converting hypergraph into a graph and propose a research framework that preserves the intrinsic structure of hypergraph itself, which is based on assigning proper weights to neighboring nodes. Through extensive experiments, we show that the new framework produces distinct and more reliable results than the traditional approach in node ranking. Finally, we explore the impact of the resetting mechanism on cover time, providing a potential solution for optimizing search efficiency.
Paper Structure (17 sections, 98 equations, 5 figures, 1 table)

This paper contains 17 sections, 98 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Hypergraph and its matrix representations
  4. Random walks on hypergraph
  5. Some parameters related to simple random walk on hypergraph
  6. Occupation probability
  7. Stationary distribution
  8. Mean first passage time
  9. Random walks with resetting on hypergraph
  10. Eigenvalues and eigenvectors
  11. Some fundamental parameters
  12. Optimal resetting for random walks on hypergraph
  13. Applications
  14. Node ranking
  15. Optimizing cover time
  16. ...and 2 more sections

Figures (5)

  • Figure 1: An example hypergraph indicating a co-authorship network where each node represents an author, and each hyperedge connects all the authors who contributed to the same paper.
  • Figure 2: The scatter plot of the normalized ranking obtained from random walks on the hypergraph.
  • Figure 3: The scatter plot of the normalized ranking obtained from random walks on the corresponding clique graph.
  • Figure 4: Comparison of the normalized ranking $P_i^{\infty} / \max_j P_j^{\infty}$ on the hypergraph and $Q_i^{\infty} / \max_j Q_j^{\infty}$ on the corresponding clique graph.
  • Figure 5: Cover time under different reset rates. The reset rate ranges from $0$ to $0.001$, with a step size of $0.00002$. The red dashed line indicates case $\gamma=0$.