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Level-set percolation of Gaussian random fields on complex networks.

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We solved level-set percolation for multivariate Gaussians on complex networks by analyzing microstructure and using a cavity approach. This provides a self-consistent method for determining local percolation probabilities.

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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.