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Scaling properties of d-dimensional complex networks.

Samuraí Brito1, Thiago C Nunes2, Luciano R da Silva2,3

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This study analyzes d-dimensional geographically located networks, revealing three distinct interaction regimes based on the scaling variable α_{A}/d. These findings offer insights into network structure and dynamics.

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Area of Science:

  • Network science
  • Statistical physics
  • Complex systems

Background:

  • The study of networks is interdisciplinary with broad scientific applications.
  • Few studies have focused on the properties of geographically embedded d-dimensional networks.
  • Understanding network scaling and interactions is crucial for various scientific fields.

Purpose of the Study:

  • To investigate the scaling properties of d-dimensional geographically located networks.
  • To analyze the impact of preferential attachment involving Euclidean distances on network evolution.
  • To identify and characterize different interaction regimes within these networks.

Main Methods:

  • Numerical analysis of network properties including connectivity, shortest path, degree distribution entropy, and clustering coefficient.
  • Simulation of d-dimensional networks (d=1,2,3,4) with preferential attachment incorporating Euclidean distances (r_{ij}^{-α_{A}}).
  • Examination of network behavior across a range of α_{A} values and dimensions.

Main Results:

  • Observed universal scaling behavior of network properties as a function of the scaled variable α_{A}/d.
  • Identified three distinct interaction regimes: very-long-range (non-Boltzmannian), moderately-long-range (non-Boltzmannian), and short-range (Boltzmannian-like).
  • Characterized the transitions between regimes, noting specific critical values for α_{A}/d, such as 1 and 1/2 for different network properties.

Conclusions:

  • The study confirms the existence of three distinct interaction regimes in d-dimensional geographically located networks.
  • The scaling properties are governed by the ratio α_{A}/d, unifying observations across different dimensions.
  • Findings provide a framework for understanding the interplay between geography, preferential attachment, and network structure.