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Area of Science:

  • Network Science
  • Statistical Physics
  • Graph Theory

Background:

  • Random walks are fundamental to analyzing network dynamics.
  • Understanding first return time distributions is crucial for network processes.
  • Existing methods struggle with accurate approximations on large-scale networks.

Purpose of the Study:

  • To propose a novel approximation for the first return time distribution of random walks on undirected networks.
  • To provide an accurate analytical tool for network analysis.
  • To investigate the influence of network structure on random walk behavior.

Main Methods:

  • Combining a message-passing solution for short-term behavior.
  • Employing a mean-field approximation for long-term behavior.
  • Testing the approximation on diverse large graph classes.

Main Results:

  • Excellent agreement between the proposed approximation and true distributions.
  • Demonstrated accuracy across various large graph structures.
  • Quantified the relative importance of local versus global network structure.

Conclusions:

  • The approximation effectively captures first return time distributions.
  • Network local structure is highly influential.
  • Global network structure's impact is primarily through the total number of edges.