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Large deviations of random walks on random graphs.

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This study explores rare fluctuations in random walks on Erdös-Rényi graphs. A modified random walk explains these large deviations and links degree fluctuations to localization transitions.

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Area of Science:

  • Statistical Physics
  • Network Science
  • Probability Theory

Background:

  • Random walks on graphs are fundamental models in network science.
  • Understanding rare events (large deviations) is crucial for complex system dynamics.
  • Erdös-Rényi random graphs provide a foundational model for network structure.

Purpose of the Study:

  • To analyze rare fluctuations of time-integrated observables for unbiased random walks on Erdös-Rényi graphs.
  • To develop a modified, biased random walk model explaining these large deviations.
  • To investigate the relationship between degree fluctuations, localization transitions, and trajectory entropy.

Main Methods:

  • Analysis of time-integrated functionals of random walks.
  • Construction of a modified, biased random walk.
  • Investigation of observables like the sum of degrees and the sum of their logarithm.
  • Establishing links to maximum entropy random walks.

Main Results:

  • A modified random walk model successfully explains large deviations in the long-time limit.
  • Sudden changes in degree fluctuations are shown to be analogous to dynamical phase transitions.
  • These fluctuations are linked to localization transitions on the random graph.
  • Connections are made between trajectory entropy large deviations and maximum entropy random walks.

Conclusions:

  • The modified random walk provides a framework for understanding rare events in random graph dynamics.
  • The study highlights the interplay between local graph properties (degree) and global walk behavior (localization).
  • Insights into trajectory entropy and its relation to maximum entropy principles are established.