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Markov processes on weighted modular graphs lose ergodicity in the thermodynamic limit, preventing distant nodes from being reached. A critical threshold for a parameter σ exists in finite systems, impacting node accessibility.

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Area of Science:

  • Statistical physics
  • Network theory
  • Complex systems

Background:

  • Markov processes and first-passage problems are fundamental in modeling dynamic systems.
  • The Dyson hierarchical model provides a framework for studying systems with hierarchical interactions.
  • Understanding ergodicity is crucial for analyzing the long-term behavior of stochastic processes.

Purpose of the Study:

  • To investigate Markov processes and first-passage times on weighted, modular graphs.
  • To generalize the Dyson hierarchical model by introducing distance-dependent, modulated coupling strengths.
  • To analyze the conditions under which ergodicity is lost and node accessibility is affected.

Main Methods:

  • Analysis of Markov processes on generalized Dyson hierarchical graphs.
  • Study of first-passage time problems in weighted, modular networks.
  • Investigation of the thermodynamic limit and finite-size effects.

Main Results:

  • Ergodicity is lost in the thermodynamic limit for these networks, rendering distant nodes unreachable.
  • A threshold value for the modulation parameter σ is identified in finite systems; large σ values hinder reachability.
  • For generic hierarchical graphs with p-plets, ergodicity is broken in the thermodynamic limit, but no σ threshold emerges due to slow diameter growth.

Conclusions:

  • Weighted, modular graphs exhibit non-ergodic behavior under certain conditions, limiting exploration of the network.
  • The parameter σ plays a critical role in determining node accessibility in finite-sized systems.
  • Network topology, specifically the growth rate of the diameter, influences the presence of critical thresholds.