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This study introduces a generalized random graph model with conditional link probabilities, revealing its connection to statistical mechanics. The findings offer new frameworks for analyzing complex networks and addressing data limitations in various applications.

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Area of Science:

  • Complex Systems and Network Science
  • Statistical Mechanics
  • Graph Theory

Background:

  • Random graphs are essential for modeling complex networks.
  • Exponential random graphs generate networks with specific statistical moments.
  • Existing models lack mechanisms for link probability conditioning based on other links.

Purpose of the Study:

  • To generalize exponential random graphs by introducing conditional link probabilities.
  • To develop a statistical mechanical formalism for these generalized graphs.
  • To explore the application of renormalization group transformations and disorder effects.

Main Methods:

  • Developed a generalized random graph model with interaction terms.
  • Derived closed-form renormalization group transformations for pairwise interactions.
  • Introduced disorder to study renormalization group flow and its equivalence to drift diffusion.

Main Results:

  • Established a closed-form renormalization group transformation for maximum coordination number two.
  • Demonstrated that higher coordination numbers lack exact transformations, mirroring lattice systems.
  • Showcased the formal equivalence between disorder-induced renormalization group flow and anisotropic drift diffusion.

Conclusions:

  • Certain pairwise conditioning effects on random graphs are irrelevant at long wavelengths.
  • The model provides a systematic framework for inference and reconstruction with data limitations.
  • Applications include modeling social networks, opinion dynamics, and neural networks.