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Hyperbolic mapping of complex networks

Hyperbolic mapping of complex networks. Dmitri Krioukov CAIDA/UCSD F. Papadopoulos, M. Kitsak, A. Vahdat, M. Bogu ñá MMDS 2010 Stanford, June 2010. Node density. Node degree. R – disk radius. K – disk curvature. maximized. Degree distribution. Clustering.

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Hyperbolic mapping of complex networks

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  1. Hyperbolic mappingof complex networks Dmitri KrioukovCAIDA/UCSDF. Papadopoulos, M. Kitsak, A. Vahdat, M. BoguñáMMDS 2010Stanford, June 2010

  2. Node density Node degree R – disk radius K – disk curvature maximized Degree distribution Clustering

  3. Connection probabilityas the Fermi-Dirac distribution • connection probability p(x) – Fermi-Dirac distribution • hyperbolic distance x – energy of links/fermions • disk radius R – chemical potential • two times inverse sqrt of curvature 2/ – Boltzmann constant • parameter T – temperature

  4. Chemical potential Ris a solution of • number of links M – number of particles • number of node pairs N(N1)/2 – number of energy states • distance distribution g(x) – degeneracy of state x • connection probability p(x) – Fermi-Dirac distribution

  5. Cold regime 0T1 • Chemical potential R(2/)ln(N/) • Constant  controls the average node degree • Clustering decreases from its maximum at T0 to zero at T1 • Power law exponent  does not depend on T, (2/)1

  6. Phase transition T1 • Chemical potential R diverges as ln(|T1|)

  7. Hot regime T1 • Chemical potential RT(2/)ln(N/) • Clustering is zero • Power law exponent  does depend on T,T(2/)1

  8. Two famous limits at T • Classical random graphs(random graphs with given average degree):T;  fixed • Configuration model(random graphs with given expected degrees):T;  ; T/ fixed

  9. From structure To function

  10. Navigation efficiency(2.1; T0) • Percentage of successful greedy paths99.99% • Percentage of shortest greedy paths100% • Percentage of successful greedy paths after removal of x% of links or nodes • x10%  99% • x30%  95%

  11. Mapping the real Internetusing statistical inference methods • Measure the Internet topology properties • N, k, , c • Map them to model parameters • R, , , T • Place nodes at hyperbolic coordinates (r,) • k ~ e(Rr)/2 • ’s are uniformly distributed on [0,2] • Apply the Metropolis-Hastings algorithm to find ’s maximizing the likelihood that Internet is produced by the model

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