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Percolation on sparse networks.

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We present a new message passing method to model network resilience and disease spread. This approach accurately calculates key percolation metrics, offering insights into network behavior.

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Area of Science:

  • Network science
  • Statistical physics
  • Epidemiology

Background:

  • Percolation theory models network resilience (e.g., Internet) and disease spread.
  • Existing models may struggle with complex network structures like loops.

Purpose of the Study:

  • To reformulate percolation as a message passing process.
  • To develop accurate calculation methods for percolation metrics.
  • To determine the percolation threshold using network properties.

Main Methods:

  • Message passing reformulation of percolation.
  • Derivation of exact equations for cluster sizes.
  • Analysis of fixed points of the message passing process.
  • Utilizing the nonbacktracking matrix eigenvalue.

Main Results:

  • Accurate calculation of percolating cluster size and average cluster size.
  • High accuracy even on networks with numerous short loops.
  • Identification of the percolation threshold via the leading eigenvalue of the nonbacktracking matrix for networks with few loops.

Conclusions:

  • The message passing approach provides an effective framework for percolation studies.
  • This method offers accurate predictions for network resilience and spread dynamics.
  • The nonbacktracking matrix eigenvalue is a key determinant of the percolation threshold in certain network types.