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Internet as a Complex NetworkInternet as a Complex NetworkGuanrongGuanrong ChenChen

City University of Hong Kong

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Complex Network: Complex Network: InternetInternet(K. C. Claffy)

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Another Another View of the InternetView of the Internet

http://www.caida.org/analysis/topology/as_core_network/

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Complex Network: Complex Network: WWWWWW(William R. Cheswick)

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Complex Network: Complex Network: HTTPHTTP

(Bradley Huffaker)

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Topics Topics for TodayTodayMathematical Models of Networks

Random-Graph Network Model

Small-World Network Model

Scale-Free Network Model

InternetInternet, a reala real--world exampleworld example

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Network TopologyNetwork Topology

A network is a graph, with a set of nodesinterconnected via edges

Computer Networks: nodes – PCs edges – wiresInternet: nodes – routers edges – optical fibresWWW: nodes – web documents edges – hyperlinks… …

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Basic Network ModelsBasic Network Models

Random Graph Theory - Erdös and Rényi (1960)

ER Random Graph model dominates for 50 years ……until recently

Small-World Networks (Watts and Strogatz, Nature, 1998)

Scale-Free Networks (Barabási and Albert, Science, 1999)

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Random Graph TheoryRandom Graph Theory-- A revolution in the 1960s

The simplest model for the most complex networks

Paul Erdös Alfred Rényi

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ER Random Graph ModelsER Random Graph Models

Features:Connectivity node degree distribution - PoissonHomogeneityall nodes have about the same number of edgesNon-growing

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Random Graph and Poisson Degree Distribution

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SmallSmall--World NetworksWorld Networks“Collective dynamics of 'small-world' networks”

--- Nature, 393: 440-442, 1998

D. J. Watts S. H. Strogatz

Cornell University

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SmallSmall--World NetworksWorld NetworksFeatures:(Similar to ER Random Graphs)

Connectivity Poisson distributionHomogeneityall nodes have about the same number of edgesNon-growing

New:New: Small-World Property !

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ScaleScale--Free NetworksFree Networks“Emergence of scaling in random networks”Science, 286: 509 (1999)

A.-L. Barabási R. Albert

Norte Dame University

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Scale-Free Networks(Barabasi-Albert, Science, 1999)

(ii) Add new links (preferential attachment): The probability p of the new node connect to an existing node is proportional to the degree of the existing node

(i) Add new nodes (incremental growth): At each step, one new node is added into the network

(0) Start with a small connected network (initialization)

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ScaleScale--Free NetworksFree Networks

Features:Connectivity:power-law form

Non-homogeneity:very few nodes have many edges but most nodes have very few edgesGrowing

γ−kkP ~)(

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Complex Networks and MathematicsInternational Congress of Mathematics (ICM)

22-28 August 2006, Madrid, Spain

Jon M Kleinberg (Comp. Sci.) received the Nevanlinna Prize for Applied Mathematics

He gave a 45-minute talk -“Complex Networks and Decentralized Search Algorithms”

J M Kleinberg, “Navigation in a small world,”Nature, 2000

Cornell University

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Internet: A Real-World Example

World-Wide WebInternet

(Bradley Huffaker)

(William R. Cheswick)

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World Wide WebWorld Wide Web

Average distanceComputed average distance L = 14Diameter L = 19 at most 19 clicks to any webpage

Degree distributionOutgoing edges: = 2.38~2.72Incoming edges: = 2.1

γ−kkP ~)(γ−kkP ~)(

γγ

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InternetInternet(Computed in 1995-1999, at both domain level and router level)

Average distanceL = 4.0 (small)

So, Internet is a small-world networkDegree distribution

Obey power law: = 2.2So, Internet is a scale-free network

Small-world / Scale-free network is a good model for the Internet

γγ−kkP ~)(

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Growth of the Internet

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The Real Internet

γ−kkP ~)(

At the Autonomous Systems level

γ = 2.2

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Internet at the AS Level

(Steve Uhlig)

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Internet Hierarchical Structures

AS-level Internet on 3 December 1998 (generated by Skitter)

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Internet Hierarchical StructuresAS on the Internet can be considered as some kind of TierAn AS at the highest Tier belongs to the Transit domain, called Tier-1 provider

Those Transit and Stub domains at a lower Tierdepend on the Transit nodes at a higher Tier to communicate with the other domains at their same level

(Cai et al., 2004)

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Geographic Layout of the Internet

(a) Router density (b) Human population density(Yook et al., 2002)

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Geographic Layout of the Internet

Some real data http://www.ixiacom.com/products/

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Main ReferencesOverview ArticlesSteven H. Strogatz, Exploring complex networks, Nature, 8 March 2001, 268-276Réka Albert and Albert-László Barabási, Statistical mechanics of complex networks, Review of Modern Physics, 2002, 74: 47-97Xiaofan Wang, Complex networks: Topology, dynamics and synchronization, Int. J. Bifurcation and Chaos, 2002, 12: 885-916Mark E. J. Newman, Models of the small world: A review, J. Stat. Phys., 2000, 101: 819-841Mark E. J. Newman, The structure and function of complex networks, SIMA Review, 2003, 45(2): 167-256Xiaofan Wang, Guanrong Chen, Complex Networks: Small-world, scale-free and beyond, IEEE Circuits and Systems Magazine, 2003, 3(1): 6-20Stefono Boccaletti, et al. Complex networks: structure and dynamics. Physics Reports, 2006, 424: 175-308S. D. Dorogovtsev, A. V. Goltsev, Critical phenomena in complex networks, Reviews of Modern Physics, 2008, 80: 1275-2335 A Arenas, A Diaz-Guilera, J Kurths, Y Moreno, C S Zhou, Synchronization in complex networks, Physics Reports, 2009, 469: 93-153

Technical Books汪小帆，李翔，陈关荣，复杂网络理论及其应用，清华大学出版社，2006Mark Newman, Albert-László Barabási, and Duncan J. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006Stefan Bornhodt and Heinz G Schuster (eds.), Handbook of Graphs and Networks, Wiley-VCH, 2003

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