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Relation between the degree and betweenness centrality distribution in complex networks.

H Masoomy1, V Adami2, M N Najafi2

  • 1Department of Physics, Shahid Beheshti University, 1983969411 Tehran, Iran.

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|May 18, 2023
PubMed
Summary
This summary is machine-generated.

Barthelemy's conjecture on network centrality exponents fails for some correlated time series models. Specifically, the Bak-Tang-Weisenfeld model and fractional Brownian motion violate the predicted relationship between degree and betweenness scaling.

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Area of Science:

  • Complex networks analysis
  • Statistical physics
  • Time series analysis

Background:

  • Centrality measures like degree (k) and betweenness (b) are crucial for classifying complex networks.
  • Barthelemy's conjecture posits a maximal b-k exponent (η_max=2) for scale-free networks, implying δ ≥ γ + 1/2, where γ and δ are scaling exponents for degree and betweenness distributions, respectively.
  • This conjecture has been previously violated in certain models.

Purpose of the Study:

  • To systematically investigate Barthelemy's conjecture using visibility graphs of correlated time series.
  • To determine if the conjecture holds for different correlation strengths and specific models.

Main Methods:

  • Constructing visibility graphs for three models: 2D Bak-Tang-Weisenfeld (BTW) sandpile model, 1D fractional Brownian motion (FBM), and 1D Levy walks.
  • Analyzing the scaling exponents (γ and δ) of degree and betweenness centrality distributions.
  • Examining the b-k exponent (η) and its relation to γ and δ.

Main Results:

  • Barthelemy's conjecture fails for the BTW model and FBM when the Hurst exponent H is less than 0.5, with η > 2 and δ < γ + 1/2 observed.
  • The conjecture remains valid for the Levy process.
  • Failure is attributed to large fluctuations in the b-k scaling relation, violating hyperscaling and leading to anomalous behavior in BTW and FBM.
  • A universal generalized degree distribution function was found for these models, matching the Barabasi-Albert network's behavior.

Conclusions:

  • Barthelemy's conjecture is not universally applicable to all complex networks, particularly those derived from correlated time series with strong fluctuations.
  • The study highlights the importance of considering correlation strength and emergent anomalous behaviors when applying theoretical conjectures in network science.
  • The identified universal degree distribution suggests underlying common scaling mechanisms across different complex network models.