Level-set percolation of Gaussian random fields on complex networks

Area of Science:

• Complex Networks
• Statistical Physics
• Graph Theory

Background:

• Level-set percolation is crucial for understanding phase transitions in complex systems.
• Multivariate Gaussian distributions on graphs are used in various fields, including machine learning and statistical physics.
• Weighted graph Laplacians are essential for analyzing graph structure and dynamics.

Purpose of the Study:

• To provide an explicit solution for level-set percolation of multivariate Gaussians on complex networks.
• To develop a self-consistent method for determining locally varying percolation probabilities.
• To analyze the heterogeneous microstructure of percolation problems.

Main Methods:

• Utilizing a cavity or message passing approach.
• Analyzing the heterogeneous microstructure of the percolation problem.
• Self-consistent determination of locally varying percolation probabilities.

Main Results:

• An explicit solution for level-set percolation was derived.
• The method allows for the evaluation of percolation probabilities in both locally treelike graphs and the thermodynamic limit.
• The analysis accounts for the heterogeneous microstructure of complex networks.

Conclusions:

• The developed method offers a robust framework for studying percolation phenomena in complex networks with multivariate Gaussian distributions.
• The cavity approach effectively captures the local variations in percolation probabilities.
• The findings are applicable to random graphs within the configuration model class.