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Infinite Ergodic Walks in Finite Connected Undirected Graphs^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Micro-Canonical Ensemble of Equiprobable Walks in Finite Connected Undirected Graphs

## 3. Entropic Pressure and Force in Micro-Canonical Ensemble of Walks

## 4. The Canonical Ensemble of Walks in Finite Connected Undirected Graphs

## 5. The Canonical Ensemble of Intrinsic Random Walks in Finite Connected Undirected Graphs

## 6. Navigation through Graphs over Canonical Ensembles of Walks

## 7. Navigability of Graphs and Graph Nodes over Canonical Ensembles of Walks

## 8. A Grand-Canonical Ensemble of Ergodic Walks in Finite Connected Undirected Graphs

## 9. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Entropic pressure and force in the city spatial graph of Lubbock, Texas (of 10,421 nodes).

**Left**: The nodes of the city graph are colored according to values of entropic pressure (9).

**Right**: The nodes of the city graph are colored according to values of the Fiedler eigenvector belonging to the second largest eigenvalue of the entropic force matrix ${\mathcal{F}}_{ij}$ (10) (or the smallest eigenvalue of the associated Laplacian matrix). The Fiedler eigenvector indicates the direction of fastest decrease of the entropic force over the city spatial graph of Lubbock.

**Figure 3.**Densities of nodes in the membrane graph with respect to the isotropic and anisotropic intrinsic random walks.

**Left**: Density of nodes wrt to the isotropic random walk ${W}_{ij}^{\left(1\right)}$ is proportionate to their degree centrality.

**Right**: The anisotropic random walk ${W}_{ij}^{\left(\infty \right)}$ is confined in the central nodes of the membrane graph.

**Figure 4.**Spectral gaps is maximum (mixing time is minimum) over the canonical ensemble of intrinsic random walks for the anisotropic random walk ${W}_{ij}^{\left(\infty \right)}$.

**Figure 5.**Navigability to the nodes in the membrane graph by the isotropic ${W}_{ij}^{\left(1\right)}$ (left) and anisotropic ${W}_{ij}^{(\infty )}$ (right) intrinsic random walks.

**Figure 6.**The grand canonical probabilities in the membrane graph (

**left**) and in the spatial graph of the city of Lubbock, Texas (

**right**). The highlighted nodes exhibit the long paths growth rates inferior to the topological entropy of the graph, in the thermodynamic limit $n\to \infty $, and therefore have higher relative fugacity in the course of prospective graph structural changes.

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Volchenkov, D. Infinite Ergodic Walks in Finite Connected Undirected Graphs. *Entropy* **2021**, *23*, 205.
https://doi.org/10.3390/e23020205

**AMA Style**

Volchenkov D. Infinite Ergodic Walks in Finite Connected Undirected Graphs. *Entropy*. 2021; 23(2):205.
https://doi.org/10.3390/e23020205

**Chicago/Turabian Style**

Volchenkov, Dimitri. 2021. "Infinite Ergodic Walks in Finite Connected Undirected Graphs" *Entropy* 23, no. 2: 205.
https://doi.org/10.3390/e23020205