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Maximal Entropy Random Walk

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Prof. Dr. Peter Grassberger

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Jülich Supercomputing Center, Jülich Research Center, D-52425 Jülich, Germany
Interests: statistical physics; complex networks; bioinformatics; percolation

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Dear Colleagues,

Ordinary random walks on graphs are defined such that entropy is locally maximized, i.e. the choice of the next move is uniform at each time step. If the graph is finite and regular (i.e., all nodes have the same degree), such walks have also globally maximal entropy: In the limit $T\to\indty$, all walks of length $T$ have the same probability $p \sim e^{-hT}, where $h$ is the entropy of the graph. This is no longer so for non-regular graphs. But one can always define modified next-move probabilities such that the resulting walks have uniform and globally maximal entropy. The existence of such "maximal entropy random walks" has been known since the seminal works of Bowen, Ruelle, Parry and others, but many of their fascinating properties have been discovered only recently. In particular, they found applications in community detection, link prediction, image analysis, and quasispecies evolution -- and they bear intriguing similarities with quantum mechanics, such as e.g. localization in inhomogeneous systems.

Prof. Dr. Peter Grassberger
Guest Editor

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Keywords

networks
random walks
maximum entropy
community detection
link prediction
image analysis
neutral quasispecies evolution
quantum mechanics
localization

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16 pages, 976 KiB

Open AccessArticle

Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects

by Alexey V. Gulyaev and Mikhail V. Tamm

Entropy 2023, 25(9), 1318; https://doi.org/10.3390/e25091318 - 09 Sep 2023

Cited by 1 | Viewed by 1089

Abstract

It is known that maximal entropy random walks and partition functions that count long paths on graphs tend to become localized near nodes with a high degree. Here, we revisit the simplest toy model of such a localization: a regular tree of degree p with one special node (“root”) that has a degree different from all the others. We present an in-depth study of the path-counting problem precisely at the localization transition. We study paths that start from the root in both infinite trees and finite, locally tree-like regular random graphs (RRGs). For the infinite tree, we prove that the probability distribution function of the endpoints of the path is a step function. The position of the step moves away from the root at a constant velocity

v = (p - 2) / p

. We find the width and asymptotic shape of the distribution in the vicinity of the shock. For a finite RRG, we show that a critical slowdown takes place, and the trajectory length needed to reach the equilibrium distribution is on the order of

\sqrt{N}

instead of

{log}_{p - 1} N

away from the transition. We calculate the exact values of the equilibrium distribution and relaxation length, as well as the shapes of slowly relaxing modes. Full article

(This article belongs to the Special Issue Maximal Entropy Random Walk)

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16 pages, 2447 KiB

Open AccessArticle

Identifying Vital Nodes in Hypergraphs Based on Von Neumann Entropy

by Feng Hu, Kuo Tian and Zi-Ke Zhang

Entropy 2023, 25(9), 1263; https://doi.org/10.3390/e25091263 - 25 Aug 2023

Cited by 1 | Viewed by 1140

Abstract

Hypergraphs have become an accurate and natural expression of high-order coupling relationships in complex systems. However, applying high-order information from networks to vital node identification tasks still poses significant challenges. This paper proposes a von Neumann entropy-based hypergraph vital node identification method (HVC) that integrates high-order information as well as its optimized version (semi-SAVC). HVC is based on the high-order line graph structure of hypergraphs and measures changes in network complexity using von Neumann entropy. It integrates

s

-line graph information to quantify node importance in the hypergraph by mapping hyperedges to nodes. In contrast, semi-SAVC uses a quadratic approximation of von Neumann entropy to measure network complexity and considers only half of the maximum order of the hypergraph’s

s

-line graph to balance accuracy and efficiency. Compared to the baseline methods of hyperdegree centrality, closeness centrality, vector centrality, and sub-hypergraph centrality, the new methods demonstrated superior identification of vital nodes that promote the maximum influence and maintain network connectivity in empirical hypergraph data, considering the influence and robustness factors. The correlation and monotonicity of the identification results were quantitatively analyzed and comprehensive experimental results demonstrate the superiority of the new methods. At the same time, a key non-trivial phenomenon was discovered: influence does not increase linearly as the

s

-line graph orders increase. We call this the saturation effect of high-order line graph information in hypergraph node identification. When the order reaches its saturation value, the addition of high-order information often acts as noise and affects propagation. Full article

(This article belongs to the Special Issue Maximal Entropy Random Walk)

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