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Complex networks emerging from fluctuating random graphs: analytic formula for the hidden variable distribution.

Sumiyoshi Abe1, Stefan Thurner

  • 1Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan. suabe@sf6.so-net.ne.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 26, 2005
PubMed
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Complex networks are built from fluctuating random graphs, analogous to superstatistics connecting statistical mechanics through temperature fluctuations. A quantum method reveals a hidden variable distribution formula, generating network degree distributions.

Area of Science:

  • Statistical mechanics
  • Complex networks
  • Network science

Background:

  • Superstatistics links Boltzmann-Gibbs statistical mechanics to generalizations via temperature fluctuations.
  • Complex networks can be modeled using fluctuating Erdös-Rényi random graphs.

Purpose of the Study:

  • To construct complex networks from fluctuating random graphs.
  • To derive an exact analytic formula for the hidden variable distribution.
  • To generate a generic degree distribution using the Poisson transformation.

Main Methods:

  • Analogy to superstatistics.
  • Construction of complex networks from fluctuating Erdös-Rényi random graphs.
  • Application of a quantum-mechanical method to derive the hidden variable distribution formula.

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Main Results:

  • An exact analytic formula for the hidden variable distribution was derived.
  • The formula describes the nature of fluctuations in the network.
  • A generic degree distribution is generated through the Poisson transformation.
  • For static scale-free networks, the hidden variable distribution decays as a power law and diverges at the origin.

Conclusions:

  • The study provides a novel method for constructing complex networks with fluctuating properties.
  • The derived hidden variable distribution offers insights into network structure and dynamics.
  • This approach unifies concepts from statistical mechanics and network science.