Hyperbolic mapping of complex networks
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Hyperbolic mappingof complex networks
Dmitri KrioukovCAIDA/UCSD
F. Papadopoulos, M. Kitsak, A. Vahdat, M. Boguñá
MMDS 2010Stanford, June 2010
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Rrer ~)(
Node density
R – disk radius],0[ Rr
)(2~)(
rRerk
Node degree
KK – disk curvature
kkP ~)(
Degree distribution
12
1~)( kkc
Clusteringmaximized
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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
)(
1
1)( 0
2
xR
e
xp T
T
Rx
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Chemical potential Ris a solution of
• number of links M – number of particles • number of node pairs N(N1)/2 – number of energy states• distance distribution g(x) – degeneracy of state x• connection probability p(x) – Fermi-Dirac distribution
dxxpxg
NM )()(
2
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Cold regime 0T1
• Chemical potential R(2/)ln(N/)– Constant controls the average node degree
• Clustering decreases from its maximum at T0 to zero at T1
• Power law exponent does not depend on T, (2/)1
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Phase transition T1
• Chemical potential R diverges as ln(|T1|)
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Hot regime T1
• Chemical potential RT(2/)ln(N/)• Clustering is zero
• Power law exponent does depend on T, T(2/)1
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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
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From structureTo function
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Navigation efficiency(2.1; T0)
• Percentage of successful greedy paths 99.99%
• Percentage of shortest greedy paths 100%
• Percentage of successful greedy paths after removal of x% of links or nodes– x10% 99%– x30% 95%
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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(Rr)/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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Metropolis-Hastings
• Compute current likelihood Lc
• Select a random node
• Move it to a new random angular coordinate
• Compute new likelihood Ln
• If Ln > Lc, accept the move
• If not, accept it with probability Ln / Lc
• Repeat
ji
aij
aij
ijij xpxpL 1)](1[)(
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Navigation efficiency (2.1; T0.69)
• Percentage of successful greedy paths 97%
• Average stretch 1.1
• Percentage of successful greedy paths after removal of x% of links or nodes:
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Conclusions• We have discovered a new geometric framework to study the structure and function of complex networks
• The framework provides a natural (no “enforcement” on our side) explanation of two basic structural properties of complex networks (scale-free degree distribution and strong clustering) as consequences of underlying hyperbolic geometry
• The framework explains how the hierarchical structure of complex networks ensures the maximum efficiency of their function – efficient navigation without global information
• The framework subsumes the configuration model and classical random graphs as two limiting cases with degenerate geometric structures
• The framework provides a basis for mapping real complex networks to their hidden/underlying hyperbolic spaces
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Applications of network mapping
• Other networks– discover hidden distances between nodes (e.g.,
similarity distances between people in social networks)• “soft” communities become areas in the underlying space with higher node densities
– tell what drives signaling in networks, and what network perturbations drive it to failures (e.g., in the brain, regulatory, or metabolic networks)
• Internet– optimal (maximally efficient/most scalable) routing– routing table sizes, stretch, and communication overhead approach
their theoretical lower bounds
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• D. Krioukov, F. Papadopoulos, A. Vahdat, and M. Boguñá,Curvature and Temperature of Complex Networks,Physical Review E, v.80, 035101(R), 2009
• F. Papadopoulos, D. Krioukov, M. Boguñá, and A. Vahdat,Greedy Forwarding in Dynamic Scale-Free NetworksEmbedded in Hyperbolic Metric Spaces,INFOCOM 2010
• M. Boguñá, F. Papadopoulos, and D. Krioukov,Sustaining the Internet with Hyperbolic Mapping,Nature Communications (to appear), 2010
• M. Boguñá, D. Krioukov, and kc claffy,Navigability of Complex Networks,Nature Physics, v.5, p.74-80, 2009
• M. Boguñá and D. Krioukov,Navigating Ultrasmall Worlds in Ultrashort Time,Physical Review Letters, v.102, 058701, 2009
• M. Ángeles Serrano, D. Krioukov, and M. Boguñá,Self-Similarity of Complex Networksand Hidden Metric Spaces,Physical Review Letters, v.100, 078701, 2008