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This study explores scale-free networks and non-Gibbsian statistics, revealing universal q-exponential degree distributions in geographically-located networks. These findings link network growth to q-statistics, with implications for complex systems analysis.

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Area of Science:

  • Complex networks
  • Statistical physics
  • Network science

Background:

  • Scale-free networks exhibit deep connections with non-Gibbsian statistics.
  • Degree distributions in the thermodynamical limit often follow q-exponential forms, optimizing nonadditive entropy (Sq).
  • The q-exponential form recovers Boltzmann-Gibbs entropy as q approaches 1.

Purpose of the Study:

  • Introduce and study d-dimensional geographically-located networks.
  • Investigate the growth of these networks with preferential attachment involving Euclidean distances.
  • Reveal the connection between these network properties and q-statistics.

Main Methods:

  • Numerical verification of degree distributions.
  • Analysis of d-dimensional geographically-located networks (d=1, 2, 3, 4).
  • Study of preferential attachment incorporating Euclidean distances.

Main Results:

  • Q-exponential degree distributions were numerically verified.
  • Universal dependencies on the ratio αA/d were observed for both q and k.
  • The q=1 limit (Boltzmann-Gibbs) is rapidly achieved by increasing αA/d.

Conclusions:

  • Geographically-located networks exhibit universal q-exponential degree distributions.
  • The findings establish a clear link between network structure and q-statistics.
  • Network properties are governed by the ratio αA/d, with implications for complex systems.