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Multifractality of complex networks.

Shuhei Furuya1, Kousuke Yakubo

  • 1Department of Mathematical Informatics, The University of Tokyo, Tokyo 113-8656, Japan. sfuruya@stat.t.u-tokyo.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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PubMed
Summary
This summary is machine-generated.

Scale-free networks exhibit complex multifractal structures, not a single fractal dimension. This finding reveals that network properties arise from significant variations in local node density.

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Area of Science:

  • Complex systems
  • Network science
  • Fractal geometry

Background:

  • Scale-free networks are crucial in various domains.
  • Traditional fractal analysis assumes a unique fractal dimension.
  • This assumption may oversimplify complex network structures.

Purpose of the Study:

  • To investigate the fractal properties of scale-free networks.
  • To determine if a unique fractal dimension adequately characterizes these networks.
  • To explore the concept of multifractality in network structures.

Main Methods:

  • Analytical derivations.
  • Numerical simulations.
  • Analysis of mass exponents (τ(q)) for deterministic, stochastic, and real-world networks.
  • Mean-field approximation for a general expression of τ(q).

Main Results:

  • Scale-free networks possess multifractal structures, not a single fractal dimension.
  • The mass exponents τ(q) are nonlinear functions of q, confirming multifractal scaling.
  • A general expression for τ(q) was derived for certain fractal scale-free networks.
  • Multifractality stems from significant fluctuations in local node density.

Conclusions:

  • The fractal nature of scale-free networks is better described by multifractality.
  • Understanding multifractal scaling is key to characterizing network structures.
  • Fluctuations in local node density are fundamental to the observed multifractal properties.