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General expression for the component size distribution in infinite configuration networks.

Ivan Kryven1

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This study introduces a simple equation to determine connected component size distributions in random networks based on degree distributions. The findings reveal how network properties influence component sizes, crucial for understanding network structures.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Computer Science

Background:

  • Random networks are fundamental models in various scientific disciplines.
  • Understanding the size distribution of connected components is key to characterizing network structure and function.
  • Existing methods for calculating component size distributions can be computationally intensive.

Purpose of the Study:

  • To derive a simple, computationally efficient equation for the size distribution of connected components in infinite configuration networks.
  • To analytically determine the relationship between degree distribution parameters and component size distribution asymptotes.
  • To investigate the impact of heavy-tailed degree distributions on component size distributions.

Main Methods:

  • Derivation of a novel analytical equation relating degree distribution to component size distribution.
  • Numerical computation using the derived equation for validation and analysis.
  • Asymptotic analysis of the component size distribution based on network parameters.

Main Results:

  • A simple equation is presented for calculating component size distribution from arbitrary degree distributions.
  • The asymptotic behavior of component size distribution is shown to depend on the first three moments, scale, and exponent of the degree distribution.
  • Heavy-tailed degree distributions can lead to multiple asymptotic modes in component size distributions, which may also exhibit heavy tails.

Conclusions:

  • The derived equation offers a fast and stable method for computing component size distributions in random networks.
  • Network component size distributions are highly sensitive to the underlying degree distribution, particularly its tail behavior.
  • This work provides valuable insights into the structural properties of complex networks and their relationship to degree distributions.