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Local clustering in scale-free networks with hidden variables.

Remco van der Hofstad1, A J E M Janssen1, Johan S H van Leeuwaarden1

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Summary

We studied triangle formation in random graphs with power-law distributed degrees. Clustering decreases with degree, and the average clustering coefficient scales with network size, vanishing only in extremely large networks.

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Area of Science:

  • Graph theory
  • Network science
  • Statistical physics

Background:

  • Investigating network structure is crucial for understanding complex systems.
  • Random graphs with power-law degree distributions exhibit unique properties, including infinite variance.

Purpose of the Study:

  • To analyze triangle presence in correlated random graphs with hidden variables.
  • To understand how degree distribution and network size affect clustering.

Main Methods:

  • Analysis of random graphs with hidden variables following a power law (exponent τ∈(2,3)).
  • Characterization of natural cutoff (h_c) and structural cutoff (h_s).
  • Derivation of the scaling of the average clustering coefficient (C) with network size (N).

Main Results:

  • Local clustering decreases with increasing hidden variable (degree).
  • The average clustering coefficient C scales as N^(2-τ)lnN for scale-free networks under specific conditions.
  • Clustering vanishes only for extremely large networks (N≈10^9) when τ is close to 2.

Conclusions:

  • The study provides insights into the clustering behavior of scale-free networks with infinite-variance degrees.
  • Negative degree correlations (disassortative mixing) significantly influence network clustering.
  • The findings have implications for understanding the structure and properties of real-world complex networks.