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Clique percolation in random graphs.

Ming Li1, Youjin Deng1,2, Bing-Hong Wang1

  • 1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China.

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Summary
This summary is machine-generated.

Clique percolation theory reveals distinct phase transitions in Erdős-Rényi graphs. The fraction of cliques shows continuous transition, while vertex fraction exhibits a step-like transition for l>1.

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Area of Science:

  • Graph theory
  • Statistical physics
  • Network science

Background:

  • Clique percolation is a generalization of classical percolation theory.
  • It studies the connectivity of cliques within a graph structure.
  • Understanding phase transitions in complex networks is crucial.

Purpose of the Study:

  • To develop a theoretical framework for clique percolation in Erdős-Rényi graphs.
  • To determine critical points and order parameters for clique percolation.
  • To analyze the nature of phase transitions in terms of clique and vertex fractions.

Main Methods:

  • Theoretical analysis of clique percolation.
  • Derivation of exact solutions for critical points.
  • Calculation of order parameters for different transition types.
  • Comparison with existing simulation results.

Main Results:

  • The fraction of cliques in the giant clique cluster undergoes a continuous phase transition.
  • The fraction of vertices in the giant clique cluster shows a discontinuous, step-function-like transition for l>1.
  • For l=1, the vertex fraction exhibits a continuous phase transition.
  • At the critical point, the order parameter for l>1 is a constant dependent on k and l, not 0 or 1.

Conclusions:

  • Theoretical findings align with simulation data, validating the approach.
  • Provides a theoretical basis for understanding clique percolation phenomena.
  • Clarifies the distinct behaviors of clique and vertex fractions during phase transitions.