Renormalization of interacting random graph models

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.