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Percolation in random graphs with higher-order clustering.

Peter Mann1,2,3, V Anne Smith2, John B O Mitchell1

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Summary
This summary is machine-generated.

This study extends percolation theory for complex networks. We analyze clustered networks with cycles and cliques, finding that larger cycles mimic treelike structures while cliques deviate, influenced by degree assortativity.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Graph Theory

Background:

  • Percolation theory and generating functions are standard for analyzing complex networks.
  • Current models often assume locally treelike network structures, excluding short-range loops.
  • Clustered networks with cycles and cliques present a challenge to existing theoretical frameworks.

Purpose of the Study:

  • To adapt the generating function formulation of percolation theory for clustered networks.
  • To investigate the impact of simple cycles and cliques on network critical properties.
  • To derive analytical approximations for the giant component size in such networks.

Main Methods:

  • Utilizing the generating function formulation of percolation theory.
  • Generalizing the Molloy-Reed criterion for clustered network analysis.
  • Deriving approximate analytical solutions for the giant component size in Poisson and power-law clustered networks.

Main Results:

  • Networks with larger simple cycles exhibit behavior closer to treelike assumptions.
  • Networks with larger cliques deviate significantly from treelike behavior.
  • Degree assortativity strongly influences the deviation caused by cliques.

Conclusions:

  • The generating function approach can be extended to analyze clustered networks beyond treelike assumptions.
  • The presence of cycles and cliques alters network critical properties and giant component size.
  • Understanding these deviations is crucial for accurate complex network analysis, especially concerning degree assortativity.