A Set of Master Variables for the Two-Star Random Graph.

Area of Science:

• Statistical physics
• Network science
• Graph theory

Background:

• Exponential random graph models are crucial for network analysis.
• The two-star model is the simplest with edge interactions.
• Analyzing large networks (N → ∞) requires robust theoretical frameworks.

Purpose of the Study:

• To develop a novel method for analyzing large exponential random graphs.
• To introduce auxiliary 'master variables' for controlling the thermodynamic limit.
• To provide explicit control over 1/N corrections in graph theory.

Main Methods:

• Introduction of auxiliary 'master variables' to manage the thermodynamic limit (N → ∞).
• Application of these variables to analyze both dense and sparse graph regimes.
• Derivation of corrections to the mean-field solution without functional integrals.

Main Results:

• The method recovers the Park-Newman mean-field solution for dense graphs with explicit 1/N correction control.
• The first subleading correction to the Park-Newman result is computed, quantifying nonextensive free energy contributions.
• A compact derivation of the Annibale-Courtney solution for sparse graphs is achieved, bypassing complex functional integral methods.

Conclusions:

• The proposed 'master variable' approach offers a powerful and simplified framework for large N random graph analysis.
• This method provides explicit control over corrections, enhancing the accuracy of theoretical predictions.
• It offers a more accessible alternative to existing techniques for studying sparse and dense random graph models.