Random walks with resetting on hypergraph

TL;DR

This paper develops a spectral theory-based framework for analyzing random walks with resetting on hypergraphs, deriving key parameters, establishing relationships with simple random walks, and demonstrating improved node ranking and search efficiency.

Contribution

It introduces a novel spectral approach for random walks with resetting on hypergraphs, preserving hypergraph structure and enhancing node ranking accuracy.

Findings

Exact expressions for occupation probability, stationary distribution, and mean first passage time.

A general condition for optimal reset probability and its existence.

The new framework outperforms traditional methods in node ranking and improves search efficiency.

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.