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Width of percolation transition in complex networks.

Tomer Kalisky1, Reuven Cohen

  • 1Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel. kaliskt@mail.biu.ac.il

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 12, 2006
PubMed
Summary

The percolation transition in complex networks has a measurable width, not a sharp threshold. This width depends on network size and structure, impacting how systems behave near critical points.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Complex Systems

Background:

  • The percolation transition in finite systems is characterized by a region of non-zero width (Deltap(c)), not a sharp threshold.
  • Understanding this transition is crucial for analyzing the robustness and behavior of complex networks.

Purpose of the Study:

  • To investigate the width of the percolation transition region in complex networks.
  • To determine how network topology, specifically average cluster length and degree distribution, influences this transition width.
  • To analyze the survivability of percolation clusters near the critical probability.

Main Methods:

  • Analytical derivations and numerical simulations were employed to study percolation phenomena.
  • The study analyzed Erdos-Renyi graphs and scale-free networks with varying degree distributions.

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  • Survivability (S(p,l)) was calculated as a function of probability (p) and cluster size (l).

    • Main Results:

    • The percolation transition width (Deltap(c)) scales with the average cluster length (l) as Deltap(c) approximately p(c)/l.
    • For Erdos-Renyi graphs, the exponent nu(opt) related to cluster length is 1/3.
    • For scale-free networks (3
    • Cluster survivability near criticality follows S(p,l) approximately exp[(p-p(c))l/p(c)].

    Conclusions:

    • The finite-size scaling of the percolation transition width is determined by network-specific exponents.
    • Within a probability range defined by |p-p(c)|
    • This research provides a quantitative understanding of percolation transitions in diverse complex networks.