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Infinite Ergodic Walks in Finite Connected Undirected Graphs.

Dimitri Volchenkov1

  • 1Department of Mathematics and Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA.

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Summary
This summary is machine-generated.

We introduce entropic force and pressure to quantify graph defects and their impact on long walks. This helps analyze network structure, mobility, and navigability in finite graphs.

Keywords:
entropic force and pressuregraph node’s fugacitygraph node’s navigabilitygraph’s navigationstatistical ensembles of walks

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Area of Science:

  • Statistical Mechanics
  • Network Science
  • Graph Theory

Background:

  • Ensembles of walks (micro-canonical, canonical, grand canonical) are studied in finite graphs.
  • Infinite walk length highlights sensitivity to structural irregularities and defects.

Purpose of the Study:

  • To describe structural imbalance, anisotropy, and navigability in finite graphs using properties of infinitely long walks.
  • To introduce novel concepts of entropic force and pressure to quantify the impact of graph defects.

Main Methods:

  • Analysis of walk ensembles in the thermodynamic limit (infinite walk length).
  • Development of entropic force and pressure metrics to characterize graph defects.

Main Results:

  • Infinitely long walks reveal graph structural properties like imbalance and anisotropy.
  • New entropic force and pressure concepts quantify defect effects on mobility and navigation.
  • Node fugacity is introduced for analyzing network dynamics during expansion or shrinking.

Conclusions:

  • The study provides new tools to analyze complex network structures and dynamics.
  • Entropic force and pressure offer insights into graph navigability and defect impact.
  • Findings are applicable to understanding and predicting network behavior.