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Maximum-Entropy Random Walks on Hypergraphs

Anqi Dong, Anzhi Sheng, Xin Mao, Can Chen

Abstract

Random walks are fundamental tools for analyzing complex networked systems, including social networks, biological systems, and communication infrastructures. While classical random walks focus on pairwise interactions, many real-world systems exhibit higher-order interactions naturally modeled by hypergraphs. Existing random walk models on hypergraphs often focus on undirected structures or do not incorporate entropy-based inference, limiting their ability to capture directional flows, uncertainty, or information diffusion in complex systems. In this article, we develop a maximum-entropy random walk framework on directed hypergraphs with two interaction mechanisms: broadcasting where a pivot node activates multiple receiver nodes and merging where multiple pivot nodes jointly influence a receiver node. We infer a transition kernel via a Kullback--Leibler divergence projection onto constraints enforcing stochasticity and stationarity. The resulting optimality conditions yield a multiplicative scaling form, implemented using Sinkhorn--Schrödinger-type iterations with tensor contractions. We further analyze ergodicity, including projected linear kernels for broadcasting and tensor spectral criteria for polynomial dynamics in merging. The effectiveness of our framework is demonstrated with both synthetic and real-world examples.

Maximum-Entropy Random Walks on Hypergraphs

Abstract

Random walks are fundamental tools for analyzing complex networked systems, including social networks, biological systems, and communication infrastructures. While classical random walks focus on pairwise interactions, many real-world systems exhibit higher-order interactions naturally modeled by hypergraphs. Existing random walk models on hypergraphs often focus on undirected structures or do not incorporate entropy-based inference, limiting their ability to capture directional flows, uncertainty, or information diffusion in complex systems. In this article, we develop a maximum-entropy random walk framework on directed hypergraphs with two interaction mechanisms: broadcasting where a pivot node activates multiple receiver nodes and merging where multiple pivot nodes jointly influence a receiver node. We infer a transition kernel via a Kullback--Leibler divergence projection onto constraints enforcing stochasticity and stationarity. The resulting optimality conditions yield a multiplicative scaling form, implemented using Sinkhorn--Schrödinger-type iterations with tensor contractions. We further analyze ergodicity, including projected linear kernels for broadcasting and tensor spectral criteria for polynomial dynamics in merging. The effectiveness of our framework is demonstrated with both synthetic and real-world examples.
Paper Structure (21 sections, 9 theorems, 63 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 21 sections, 9 theorems, 63 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensors
  4. Directed Hypergraphs
  5. MERWs on Graphs
  6. MERWs for Higher-Order Broadcasting
  7. Broadcasting MERWs on Uniform Hypergraphs
  8. Extension to Non-Uniform Hypergraphs
  9. Ergodicity
  10. Sinkhorn--Schrödinger Scaling
  11. MERWs for Higher-Order Merging
  12. Merging MERWs on Uniform Hypergraphs
  13. Extension to Non-Uniform Hypergraphs
  14. Ergodicity
  15. Sinkhorn--Schrödinger Scaling
  16. ...and 6 more sections

Key Result

Proposition 1

Assume that the constraint set $\mathcal{B}(\textbf{p})$ is nonempty. Problem prob:1 admits a unique optimal broadcasting transition tensor $\mathscr{B}^\star \in \mathcal{B}(\textbf{p})$. Moreover, there exist strictly positive vectors $\textbf{u}, \textbf{v} \in \mathbb R^n_{++}$ such that $\maths where $\odot$ and $\circ$ denote entry-wise multiplication and vector outer product, respectively.

Figures (4)

  • Figure 1: Broadcasting MERWs on directed hypergraphs. Each colored region represents a directed hyperedge, with arrows indicating activation from a pivot (tail) node to a set of receiver (head) nodes (e.g., $v_1\rightarrow \{v_2,v_3\}$).
  • Figure 2: Merging MERWs on directed hypergraphs. Each colored region represents a directed hyperedge, with arrows indicating activation from a set of pivot (tail) nodes to a receiver (head) node (e.g., $\{v_2,v_3\}\rightarrow v_1$).
  • Figure 3: Mixing curves of the projected node-level dynamics for non-uniform broadcasting MERWs under different weights $\lambda_2$ and $\lambda_3$.
  • Figure 4: Mixing curves of the node-level dynamics for non-uniform merging MERWs under different weights $\lambda_2$ and $\lambda_3$.

Theorems & Definitions (22)

  • Definition 1: Ergodicity
  • Definition 2: Broadcasting
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Proposition 3
  • proof
  • ...and 12 more