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Subgraphs in random networks.

S Itzkovitz1, R Milo, N Kashtan

  • 1Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2003
PubMed
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We developed new equations to predict subgraph counts in complex networks. These findings reveal how subgraph frequencies change in scale-free networks, aiding in network motif detection.

Area of Science:

  • Network Science
  • Statistical Physics
  • Complex Systems

Background:

  • Understanding subgraph distribution is crucial for modeling complex systems.
  • Classic Erdos networks have Poissonian degree distributions, with predictable subgraph scaling.
  • Many real-world networks exhibit non-Poissonian degree distributions, requiring new models.

Purpose of the Study:

  • To derive approximate equations for average subgraph counts in random sparse directed networks with arbitrary degree sequences.
  • To establish scaling rules for subgraph appearances in directed scale-free networks.
  • To analyze the impact of the degree distribution's power-law exponent on subgraph frequencies.

Main Methods:

  • Developed approximate equations for subgraph counts in random sparse directed networks.

Related Experiment Videos

  • Analyzed scaling rules for directed scale-free networks with power-law degree distributions (P(k) ~ k^-gamma).
  • Investigated transitions between different scaling regimes based on the exponent gamma.

    • Main Results:

    • Identified three distinct scaling regimes for subgraph appearances ( ~ N^alpha) based on the degree exponent gamma.
    • Derived specific scaling exponents (alpha) for each regime: alpha=n-g+s-1 (gamma<2), alpha=n-g+s+1-gamma (2gamma(c)).
    • Found that certain subgraphs appear significantly more frequently in scale-free networks compared to Erdos networks.

    Conclusions:

    • The derived equations accurately predict subgraph distributions in random sparse directed and scale-free networks.
    • The findings highlight critical transitions in subgraph scaling behavior controlled by network degree distribution properties.
    • Results have significant implications for identifying network motifs and understanding the structure of complex real-world networks.