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Hyperbolic random geometric graphs: Structural and spectral properties.

Kevin Peralta-Martinez1, J A Méndez-Bermúdez2, José M Sigarreta3

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This study analyzes hyperbolic random geometric graphs (HRGs) using random matrix theory (RMT). HRGs share key average properties with Euclidean random graphs, particularly in vertex count and spectral characteristics.

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Area of Science:

  • Complex Networks
  • Random Matrix Theory
  • Geometric Graph Theory

Background:

  • Hyperbolic random geometric graphs (HRGs) offer a complex model for network analysis.
  • Understanding their structural and spectral properties is crucial for network science.
  • Existing research often focuses on Euclidean counterparts, necessitating exploration of hyperbolic spaces.

Purpose of the Study:

  • To conduct a comprehensive numerical investigation of HRG structural and spectral properties.
  • To apply random matrix theory (RMT) methods to analyze these graphs.
  • To compare the properties of HRGs with Euclidean random geometric graphs.

Main Methods:

  • Numerical simulations of HRGs G(n,ρ,α,ζ) within a Poincaré disk.
  • Analysis of average structural properties: non-isolated vertices, topological indices, clustering coefficients.
  • Application of RMT measures: eigenvalue spacing ratios (rR, rC), inverse participation ratio, Shannon entropy of eigenvectors.

Main Results:

  • The average number of non-isolated vertices 〈Vx(G)〉 is a function of the average degree 〈k〉, following 〈Vx(G)〉≈n[1-exp(-γ〈k〉)].
  • Normalized spectral measures 〈rR(G)〉, 〈rC(G)〉, and 〈S(G)〉 exhibit scaling behavior with a parameter ξ∝〈k〉n^δ.
  • HRGs and Euclidean random graphs share fundamental average structural and spectral properties.

Conclusions:

  • HRGs exhibit predictable structural and spectral behaviors that can be analyzed using RMT.
  • The findings highlight similarities between hyperbolic and Euclidean random graph models.
  • This research provides insights into the statistical mechanics of complex networks in hyperbolic spaces.