The Universal Laws of Structural Dynamics in Large Graphs
description
Transcript of The Universal Laws of Structural Dynamics in Large Graphs
![Page 1: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/1.jpg)
The Universal Lawsof Structural Dynamics
in Large Graphs
Dmitri KrioukovUCSD/CAIDA
David Meyer & David RideoutUCSD/Math
M. Boguñá, M. Á. Serrano,F. Papadopoulos, M. Kitsak, kc claffy, A. Vahdat
DARPA’s GRAPHS, Chicago, July 2012
![Page 2: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/2.jpg)
High-level project description
• Motivation:– Predict network dynamics– Detect anomalies
• Goal:– Identify the universal laws of network dynamics
• Methods: random geometric graphs– Past work: static graphs– Future work: dynamic graphs
![Page 3: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/3.jpg)
Past workRandom geometric graphs in hyperbolic spaces
• Strengths:– Common structural properties of real networks– Optimality of their common functions
• Limitations:– Static graphs– Model real networks qualitatively
![Page 4: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/4.jpg)
Future workRandom geometric graphs on Lorentzian manifolds
• Dynamic graphs• Model real networks quantitatively
![Page 5: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/5.jpg)
Outline
• Introduction– Why geometric graphs?– Why fundamental laws?– Real networks– Network models
• Past work– Random hyperbolic graphs (RHGs)
• Future work– Random Lorentzian graphs (RLGs)
![Page 6: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/6.jpg)
Why geometric graphs?
• Graphs are not “geometry”• Yet real networks are navigable
– Efficient substrates for information propagation without global knowledge or central coordination
• How is this possible?
![Page 7: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/7.jpg)
![Page 8: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/8.jpg)
![Page 9: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/9.jpg)
Geometric graphs
• “Coarse approximations” of smooth manifolds– Riemann’s idea (Nature, v.7)
• “Sense of direction” (geometry) makes graphs navigable
• “Hyperbolicity”– Maximizes navigability (optimal Internet routing)– Reflects hierarchical (tree-like) organization– Explains common structural properties
![Page 10: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/10.jpg)
Why fundamental laws?
• Many different real networks have certain common structural properties
• Are there any common (fundamental) laws of network dynamics explaining the emergence of these common properties?– If “no”, then… too bad (each network is unique)– If “yes”, then one can utilize these laws to predict
network dynamics and detect anomalies
![Page 11: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/11.jpg)
Real networks• Technological
– Internet– Transportation– Power grid
• Social– Collaboration– Trust– Friendship
• Biological– Gene regulation– Protein interaction– Metabolic– Brain
• What can be common to all these networks???
• Naïve answer:– Nothing– Well, something: they all are
messy, complex, “very random”– And that’s it
![Page 12: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/12.jpg)
![Page 13: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/13.jpg)
![Page 14: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/14.jpg)
“Very random” graphs
• Classical random graphs (Erdős-Rényi)– Take N nodes– Connect each node pair with probability p
• Soft version of -regular graphs ( Np)• Maximum-entropy graphs of size N and
expected average degree
![Page 15: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/15.jpg)
Heterogeneity
• Distribution P(k)of node degrees k– Real: P(k) ~ k– Random: P(k) ~ k e /
k!
![Page 16: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/16.jpg)
“Less random” graphs• Random graphs with
expected node degree distributions– Take N nodes– Assign to each node a random variable
drawn from a desired distribution, e.g.,() ~
– Connect each node pair with probabilityp(,) ~ /N
• Soft version of the configuration model• Maximum-entropy graphs of size N and expected degree
distribution ()
![Page 17: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/17.jpg)
Clustering
• Distribution P(k)of node degrees k– Real: P(k) ~ k– Random: P(k) ~ k
• Average probability that node neighbors are connected– Real: 0.5– Random: 7104
![Page 18: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/18.jpg)
Random geometric graphs
• Take a compact region in a Euclidean space, e.g., a circle of radius R
• Sprinkle N nodes into it via the Poisson point process
• Connect each pair of nodes if the distance between them is d r R
![Page 19: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/19.jpg)
Heterogeneity lost
• Distribution P(k)of node degrees k– Real: P(k) ~ k– Random: P(k) ~ k e /
k!
• Average probability that node neighbors are connected– Real: 0.5– Random: 0.5
![Page 20: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/20.jpg)
Strong heterogeneity and clustering are common properties of large networks
Network Exponent of thedegree distribution
Average clustering
Internet 2.1 0.46
Air transportation 2.0 0.62
Actor collaboration 2.3 0.78
Protein interaction S. cerevisiae
2.4 0.09
Metabolic E. coli and S. cerevisiae
2.0 0.67
Gene regulation E. coli and S. cerevisiae
2.1 0.09
![Page 21: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/21.jpg)
Any other common properties?• No• Well, some randomness, of course• Real large networks appear to be quite different
and unique in all other respects• Any simple random graph model that can
reproduce these two universal properties?
![Page 22: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/22.jpg)
Random hyperbolic graphs
• Take a compact region in a hyperbolic space, e.g., a circle of radius R
• Sprinkle N nodes into it via the Poisson point process (R ln N)
• Connect each pair of nodes if the distance between them is x R
![Page 23: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/23.jpg)
Succeeded finally
• Distribution P(k)of node degrees k– Real: P(k) ~ k– Random: P(k) ~ k
• Average probability that node neighbors are connected– Real: 0.5– Random: 0.5
![Page 24: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/24.jpg)
![Page 25: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/25.jpg)
![Page 26: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/26.jpg)
![Page 27: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/27.jpg)
![Page 28: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/28.jpg)
![Page 29: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/29.jpg)
![Page 30: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/30.jpg)
![Page 31: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/31.jpg)
![Page 32: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/32.jpg)
![Page 33: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/33.jpg)
![Page 34: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/34.jpg)
![Page 35: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/35.jpg)
Rrer ~)(
Node density)(
21
~)(rR
erk
Node degree
kkP ~)(
Degree distribution
3 1~)( kkc
Clusteringmaximized
![Page 36: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/36.jpg)
Fermi-Dirac connection probability
• 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)(2
TRx
e
xp )(0 xRT
![Page 37: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/37.jpg)
Curvature and temperature
• Curvature 0 controls power-law exponent [2,]– Graphs are not geometric (node density is not uniform)
unless = 3• Temperature T 0 controls clusteringc [0,cmax]– Phase transition at T 1
![Page 38: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/38.jpg)
Limiting cases
Curvature \ Temperature T Finite T Infinite T
Finite Random hyperbolic graphs and real networks Classical random graphs
Infinite Random geometric graphs Configuration model
![Page 39: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/39.jpg)
Random graphswith hidden variables
• Definition:– Take N nodes– Assign to each node a random variable
drawn from distribution ()• can be a vector (of attributes or coordinates)
– Connect each node pair with probability p(,)• In random geometric graphs:
– ’s are node coordinates– () is the uniform distribution– p(,) is a step function of distance d(,)
![Page 40: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/40.jpg)
Exponential random graphs• Definition:
– Set of graphs G with probability measure , where , and H(G) is the graph Hamiltonian
• In soft configuration model:– , where
is the adjacency matrix and are the expected degrees
• In random hyperbolic graphs:– , where
, and
,
![Page 41: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/41.jpg)
Back to reality
• Infer/learn node coordinates in real networks using maximum-likelihood methods (MCMC)
• Compare the empirical probability of connections in the real network with the inferred coordinates against the theoretical prediction
• If the model is correct, the two should match
![Page 42: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/42.jpg)
![Page 43: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/43.jpg)
![Page 44: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/44.jpg)
![Page 45: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/45.jpg)
![Page 46: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/46.jpg)
![Page 47: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/47.jpg)
![Page 48: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/48.jpg)
![Page 49: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/49.jpg)
![Page 50: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/50.jpg)
Physical meaningof node coordinates
• Radial coordinates– Node degrees (popularity)
• Angular coordinates– Similarity
• Projections of a properly weighted combination of all the factors shaping the network structure
![Page 51: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/51.jpg)
Summary of RHG strengths• Explanation how the structure of complex networks maximizes the
efficiency of their transport function– Optimal Internet routing as a practical application
• Two universal structural properties of real networks(degree heterogeneity and strong clustering) emerge as simple consequences of the two basic properties of hyperbolic spaces (exponential expansion and metric property)
• Connections to– Similarity distances– Hidden variable models– Exponential random graphs
• interpreting auxiliary fields as linear function of hyperbolic distances– Fermionic systems– Self-similarity– Conformal invariance?– AdS/CFT correspondence?
• RHGs subsume– Classical random graphs– Random geometric graphs– Soft configuration model
as limiting cases with degenerate geometry
![Page 52: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/52.jpg)
Summary of RHG limitations• Exponential random graphs are intrinsically
static graphs (equilibrium ensembles)– Real networks are growing (far from equilibrium)
• or node density is not uniform(graphs are not coarse approximations of smooth manifolds)– in real networks
![Page 53: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/53.jpg)
Proposal• These observations suggest that the geometry
of real networks is actually different• Question:
– What is it?• Proposal:
– Lorentzian geometry• Why:
– Lorentzian geometry explicitly models time– Indications that in some RLGs
• Challenge:– Dynamic exponential random graphs
![Page 54: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/54.jpg)
Lorentzian manifolds
• Pseudo-Riemannian manifold is a manifold with a non-degenerate metric tensor– Distances can be positive, zero, or negative
• Lorentzian manifold is a manifold with signature – Coordinate corresponding to the minus sign is
called time– Negative distance are time-like– Positive distance are space-like
![Page 55: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/55.jpg)
Causal structure
• For each point , the set of points at time-like distances from p can be split in two subsets:– ’s future– ’s past
• If , then is called the Alexandrov set of
![Page 56: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/56.jpg)
![Page 57: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/57.jpg)
![Page 58: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/58.jpg)
Alexandrov sets
• Form a base of the manifold topology– Similar to open balls in Riemannian case
![Page 59: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/59.jpg)
Random Lorentzian graphs
• Take a compact region of a Lorentzian manifold, e.g., a patch – Similar to a circle in the Riemannian case
• Sprinkle N nodes into it via the Poisson point process
• Connect each pair of nodes if the distance between them is x 0– Because Alexandrov sets are analogous to balls
![Page 60: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/60.jpg)
![Page 61: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/61.jpg)
Summary of RLGs
• Random geometric graphs by constructions– Are they also exponential random graphs?
• Growing T explicitly models graph growth• Not to sacrifice the strengths of RHGs,
find Lorentzian manifold M such that:– There exist a map between M and hyperbolic space
H satisfying certain duality properties– The groups of isometries of M and H are isomorphic
• Lorentz group
![Page 62: The Universal Laws of Structural Dynamics in Large Graphs](https://reader035.fdocuments.net/reader035/viewer/2022070423/5681667a550346895dda1a93/html5/thumbnails/62.jpg)
• M. Boguñá, F. Papadopoulos, and D. Krioukov,Sustaining the Internet with Hyperbolic Mapping,Nature Communications, v.1, 62, 2010
• D. Krioukov, F. Papadopoulos, A. Vahdat, and M. Boguñá,Hyperbolic Geometry of Complex Networks,Physical Review E, v.82, 036106, 2010,Physical Review E, v.80, 035101(R), 2009
• F. Papadopoulos, D. Krioukov, M. Boguñá, and A. Vahdat,Greedy Forwarding in Scale-Free NetworksEmbedded in Hyperbolic Metric Spaces,INFOCOM 2010,SIGMETRICS MAMA 2009
• M. Boguñá, D. Krioukov, and kc claffy,Navigability of Complex Networks,Physical Review Letters, v.102, 058701, 2009Nature Physics, v.5, p.74-80, 2009
• M. Á. Serrano, M. Boguñá, and D. Krioukov,Self-Similarity of Complex Networks,Physical Review Letters, v.106, 048701, 2011Physical Review Letters, v.100, 078701, 2008