Internet as a Complex Network
Transcript of Internet as a Complex Network
1
Internet as a Complex NetworkInternet as a Complex NetworkGuanrongGuanrong ChenChen
City University of Hong Kong
3
Another Another View of the InternetView of the Internet
http://www.caida.org/analysis/topology/as_core_network/
6
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
7
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… …
8
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)
9
Random Graph TheoryRandom Graph Theory-- A revolution in the 1960s
The simplest model for the most complex networks
Paul Erdös Alfred Rényi
10
ER Random Graph ModelsER Random Graph Models
Features:Connectivity node degree distribution - PoissonHomogeneityall nodes have about the same number of edgesNon-growing
12
SmallSmall--World NetworksWorld Networks“Collective dynamics of 'small-world' networks”
--- Nature, 393: 440-442, 1998
D. J. Watts S. H. Strogatz
Cornell University
13
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 !
14
ScaleScale--Free NetworksFree Networks“Emergence of scaling in random networks”Science, 286: 509 (1999)
A.-L. Barabási R. Albert
Norte Dame University
15
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)
16
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 ~)(
17
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
20
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
21
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 ~)(
26
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)
27
Geographic Layout of the Internet
(a) Router density (b) Human population density(Yook et al., 2002)
29
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