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Local time of random walks on graphs.

Václav Zatloukal1

  • 1Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1, Czech Republic.

Physical Review. E
|November 16, 2021
PubMed
Summary

We developed a general method to calculate the average occupation time for random walks on graphs. This approach uses the resolvent of the transition matrix for accurate analysis.

Area of Science:

  • Graph theory
  • Probability theory
  • Stochastic processes

Background:

  • Random walks are fundamental models in various fields.
  • Understanding the local time (occupation time) is crucial for analyzing random walk behavior.
  • Existing methods for calculating local time averages can be computationally intensive or limited in scope.

Purpose of the Study:

  • To present a general method for calculating sample-path averages of local time functionals for discrete-time random walks on generic graphs.
  • To provide a computationally tractable approach for analyzing random walk behavior.

Main Methods:

  • The study focuses on discrete-time random walks on generic graphs.
  • A general method is presented for calculating sample-path averages of local time functionals.

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  • The method utilizes the resolvent of the transition matrix.

    • Main Results:

    • A general formula is derived for computing sample-path averages of local time functionals.
    • The method is applicable to any discrete-time random walk on a generic graph.
    • The resolvent of the transition matrix is identified as a key component for calculation.

    Conclusions:

    • The developed method offers a powerful tool for analyzing the local time of random walks.
    • This approach simplifies the calculation of occupation time averages, enhancing the study of graph-based stochastic processes.
    • The findings have implications for network analysis, statistical physics, and computer science.