Internet as a Complex Network. Guanrong Chen City University of Hong Kong

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1 Internet as a Complex Network Guanrong Chen City University of Hong Kong 1

2 Complex Network: Internet (K. C. Claffy) 2

3 Another View of the Internet 3

4 Complex Network: WWW (William R. Cheswick) 4

5 Complex Network: HTTP (Bradley Huffaker) 5

6 Topics for Today Mathematical Models of Networks Random-Graph Network Model Small-World Network Model Scale-Free Network Model Internet, a real-world example 6

7 Network Topology A network is a graph, with a set of nodes interconnected via edges Computer Networks: nodes PCs edges wires Internet: nodes routers edges optical fibres WWW: nodes web documents edges hyperlinks 7

8 Basic 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) 8

9 Random Graph Theory -- A revolution in the 1960s Paul Erdös Alfred Rényi The simplest model for the most complex networks 9

10 ER Random Graph Models Features: Connectivity node degree distribution - Poisson Homogeneity all nodes have about the same number of edges Non-growing 10

11 Random Graph and Poisson Degree Distribution 11

12 Small-World Networks Collective dynamics of 'small-world' networks --- Nature, 393: , 1998 D. J. Watts S. H. Strogatz Cornell University 12

13 Small-World Networks Features: (Similar to ER Random Graphs) Connectivity Poisson distribution Homogeneity all nodes have about the same number of edges Non-growing New: Small-World Property! 13

14 Scale-Free Networks Emergence of scaling in random networks Science, 286: 509 (1999) A.-L. Barabási R. Albert Norte Dame University 14

15 Scale-Free Networks (Barabasi-Albert, Science, 1999) (0) Start with a small connected network (initialization) (i) Add new nodes (incremental growth): At each step, one new node is added into the network (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 15

16 Scale-Free Networks Features: Connectivity: power-law form P ( k) ~ γ Non-homogeneity: very few nodes have many edges but most nodes have very few edges Growing k 16

Complex Networks and Mathematics International Congress of Mathematics (ICM) 22-28 August 2006, Madrid, Spain Jon M Kleinberg (Comp. Sci.

17 Complex Networks and Mathematics International Congress of Mathematics (ICM) 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 17

18 Internet: A Real-World Example World-Wide Web Internet (Bradley Huffaker) 18 (William R. Cheswick)

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20 World Wide Web Average distance Computed average distance L = 14 Diameter L = 19 at most 19 clicks to any webpage Degree distribution γ Outgoing edges: P ( k) ~ k = 2.38~2.72 γ Incoming edges: P ( k) ~ k γ = 2.1 γ 20

21 Internet (Computed in , at both domain level and router level) Average distance L = 4.0 (small) So, Internet is a small-world network Degree distribution γ Obey power law: P ( k) ~ k γ = 2.2 So, Internet is a scale-free network Small-world / Scale-free network is a good model for the Internet 21

22 Growth of the Internet 22

23 The Real Internet P ( k) ~ γ k γ = 2.2 At the Autonomous Systems level 23

24 Internet at the AS Level (Steve Uhlig) 24

25 Internet Hierarchical Structures AS-level Internet on 3 December 1998 (generated by Skitter) 25

26 Internet Hierarchical Structures AS on the Internet can be considered as some kind of Tier An AS at the highest Tier belongs to the Transit domain, called Tier-1 provider Those Transit and Stub domains at a lower Tier depend on the Transit nodes at a higher Tier to communicate with the other domains at their same level (Cai et al., 2004) 26

27 Geographic Layout of the Internet (a) Router density (b) Human population density (Yook et al., 2002) 27

28 Geographic Layout of the Internet Some real data 28

29 Overview Articles Main References Steven H. Strogatz, Exploring complex networks, Nature, 8 March 2001, Réka Albert and Albert-László Barabási, Statistical mechanics of complex networks, Review of Modern Physics, 2002, 74: Xiaofan Wang, Complex networks: Topology, dynamics and synchronization, Int. J. Bifurcation and Chaos, 2002, 12: Mark E. J. Newman, Models of the small world: A review, J. Stat. Phys., 2000, 101: Mark E. J. Newman, The structure and function of complex networks, SIMA Review, 2003, 45(2): Xiaofan Wang, Guanrong Chen, Complex Networks: Small-world, scale-free and beyond, IEEE Circuits and Systems Magazine, 2003, 3(1): 6-20 Stefono Boccaletti, et al. Complex networks: structure and dynamics. Physics Reports, 2006, 424: S. D. Dorogovtsev, A. V. Goltsev, Critical phenomena in complex networks, Reviews of Modern Physics, 2008, 80: A Arenas, A Diaz-Guilera, J Kurths, Y Moreno, C S Zhou, Synchronization in complex networks, Physics Reports, 2009, 469: 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,2006 Mark Newman, Albert-László Barabási, and Duncan J. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006 Stefan Bornhodt and Heinz G Schuster (eds.), Handbook of Graphs and Networks, Wiley-VCH,

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On Complex Dynamical Networks. G. Ron Chen Centre for Chaos Control and Synchronization City University of Hong Kong

On Complex Dynamical Networks. G. Ron Chen Centre for Chaos Control and Synchronization City University of Hong Kong On Complex Dynamical Networks G. Ron Chen Centre for Chaos Control and Synchronization City University of Hong Kong 1 Complex Networks: Some Typical Examples 2 Complex Network Example: Internet (William