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Fast Approximation of Centrality and Distances in Hyperbolic Graphs

  • V. Chepoi
  • F. F. Dragan
  • M. Habib
  • Y. Vaxès
  • H. Alrasheed
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

We show that the eccentricities (and thus the centrality indices) of all vertices of a \(\delta \)-hyperbolic graph \(G=(V,E)\) can be computed in linear time with an additive one-sided error of at most \(c\delta \), i.e., after a linear time preprocessing, for every vertex v of G one can compute in O(1) time an estimate \(\hat{e}(v)\) of its eccentricity \(ecc_G(v)\) such that \(ecc_G(v)\le \hat{e}(v)\le ecc_G(v)+ c\delta \) for a small constant c. We prove that every \(\delta \)-hyperbolic graph G has a shortest path tree, constructible in linear time, such that for every vertex v of G, \(ecc_G(v)\le ecc_T(v)\le ecc_G(v)+ c\delta \). We also show that the distance matrix of G with an additive one-sided error of at most \(c'\delta \) can be computed in \(O(|V|^2\log ^2|V|)\) time, where \(c'< c\) is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity.

Notes

Acknowledgements

The research of V.C., M.H., and Y.V. was supported by ANR project DISTANCIA (ANR-17-CE40-0015).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • V. Chepoi
    • 1
  • F. F. Dragan
    • 2
  • M. Habib
    • 3
  • Y. Vaxès
    • 1
  • H. Alrasheed
    • 4
  1. 1.Laboratoire d’Informatique et Systèmes, Aix-Marseille Univ, CNRS, and Univ. de Toulon Faculté des Sciences de LuminyMarseilleFrance
  2. 2.Algorithmic Research Laboratory, Department of Computer ScienceKent State UniversityKentUSA
  3. 3.Institut de Recherche en Informatique Fondamentale, University Paris Diderot - Paris7ParisFrance
  4. 4.Information Technology DepartmentKing Saud UniversityRiyadhSaudi Arabia

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