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  • 1Department of Mathematics, <a href="https://ror.org/0220mzb33">King's College London</a>, Strand, London WC2R 2LS, United Kingdom.

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We solved level-set percolation for multivariate Gaussians on complex networks using a cavity method. This approach determines local percolation probabilities, revealing correlations with node variances and network structure.

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Area of Science:

  • Statistical physics
  • Network science
  • Complex systems

Background:

  • Level-set percolation is crucial for understanding phase transitions in complex systems.
  • Multivariate Gaussians on networks are modeled using weighted graph Laplacians.
  • Cavity and message passing methods offer powerful tools for analyzing network properties.

Purpose of the Study:

  • To develop a solution for level-set percolation of multivariate Gaussians on complex networks.
  • To determine locally varying percolation probabilities using a cavity approach.
  • To analyze the critical percolation threshold and its dependence on network structure and Gaussian properties.

Main Methods:

  • A cavity or message passing approach was employed to solve the level-set percolation problem.
  • The critical percolation threshold (h_c) was determined by the largest eigenvalue of a weighted nonbacktracking matrix.
  • Analysis was performed on Erdős-Rényi networks, power-law distributed networks, and random regular graphs.

Main Results:

  • Self-consistent determination of locally varying percolation probabilities was achieved.
  • Strong correlations were found between single-node variances of multivariate Gaussians and local percolation probabilities.
  • The critical percolation threshold's behavior was analyzed for different network types and edge weight schemes.

Conclusions:

  • The cavity method provides an effective solution for level-set percolation on complex networks.
  • Network topology and Gaussian properties significantly influence percolation behavior.
  • The study offers insights into the interplay between statistical mechanics and network science.