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Complex Dynamical Networks:Complex Dynamical Networks:Modeling, ControlModeling, Control

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Complex Networks:Complex Networks:

Some Typical ExamplesSome Typical Examples

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Complex Network Example: InternetComplex Network Example: Internet

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Complex Network Complex Network ExampleExample: : WWW WWW

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Complex Network Complex Network ExampleExample: HTTP: HTTP

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Complex Network Complex Network ExampleExample: : Telecomm NetworksTelecomm Networks

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Complex Network Complex Network Example: Routes of AirlinesExample: Routes of Airlines

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Complex Network Complex Network Example: UsenetExample: Usenet

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Complex Network Complex Network Example: VLSI Circuits, CNNExample: VLSI Circuits, CNN

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Complex Network Complex Network Example: Biological NetworksExample: Biological Networks

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Complex Network Complex Network Example: ArtsExample: Arts

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Complex Networks: Complex Networks: Topics for Topics for TodayToday

Network TopologyNetwork Topology

Three Approaches: Three Approaches: Random GraphsRandom Graphs

Small-World NetworksSmall-World Networks

Scale-Free NetworksScale-Free Networks

Network Control Network Control

Network SynchronizationNetwork Synchronization

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Network TopologyNetwork Topology

• A networknetwork is a set of nodesnodes interconnected via linkslinks

– ExamplesExamples: • Internet: Nodes – routers Links – optical fibers• WWW: Nodes – document files Links – hyperlinks • Scientific Citation Network: Nodes – papers Links – citation• Social Networks: Nodes – individuals Links – relations

– Nodes and LinksLinks can be anything depending on the context

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Network TopologyNetwork Topology

• Complex networks have been studied via Graph Theory - Erdös and Rényi (1960) – ER Random Graphs

• ER Random Graph model dominates for 40 some years• …… till today ……• Availability of huge databases and supper-computing

power have led to a rethinking of approach …rethinking of approach …• Two significant recent discoveries are:

– Small-World effect (Watts and Strogatz, Nature, 1998)

– Scale-Free feature (Barabási and Albert, Science, 1999)

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ER Random Graph ModelsER Random Graph Models

Features:Features:» Connectivity:

Poisson distribution » Homogeneous nature:

each node has roughly the same number of links

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Small-World NetworksSmall-World Networks

FeaturesFeatures: (Similar to ER Random

Graphs)

» Connectivity distribution: uniform but decays exponentially

» Homogeneous nature: each node has roughly the same number of links

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Scale-Free NetworksScale-Free Networks

Features:Features:» Connectivity:

in power-law form» Non-homogeneous

nature:

a few nodes have many links but most nodes have very few links

(Hawoong Jeong)

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Some Basic ConceptsSome Basic Concepts

• (Average) Distance• Clustering Coefficient• Degree and Degree Distribution

(Stephen G. Eick)

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Average DistanceAverage Distance

• DistanceDistance d(n,m) between two nodes n and m

= the number of links along the shortest path connecting them

• DiameterDiameter D = max{d(n,m)}• Average distanceAverage distance L = average over all d(n,m)

• Most large and complex networks have

small L small-world feature

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Clustering CoefficientClustering Coefficient

• Clustering CoefficientClustering Coefficient C of a network: 0 < C < 1

• C = 1 iff every pair of nodes are connected• C = 0 iff all nodes are isolated

• Most large and complex networks have

large C small-world feature

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Degree and Degree DistributionDegree and Degree Distribution

• Degree k(n) of node n = total number of its links

• The spread of node degrees over a network is characterized by a distribution function:

P(k) = probability that a randomly selected node has exactly k links

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Degree DistributionDegree Distribution

• Completely regular lattice:

P(k) ~ Delta distribution• Most networks: P(k) ~ k^{-γ} (power lawpower law)

– scale-free featurescale-free feature• Completely random networks:

P(k) ~ Poisson distribution

• (Regular) delta k^{-γ} Poisson (Random)

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ComparisonComparison

E-R random graph model

real-life complex networks

Ave. Distance

Clustering

Small / Large

Small

Small

Large(Small-world

feature)

Degree Distribution

Binomial / Poisson

Power-law (Scale-free feature)

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Three Typical ExamplesThree Typical Examples• World Wide WebWorld Wide Web

• InternetInternet

• Scientific Collaboration NetworkScientific Collaboration Network

(Bradley Huffaker)

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1. 1. WWWWWW

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1. 1. World Wide WebWorld Wide Web

• Average distanceAverage distance– ComputedComputed Average distance L = 14– Diameter L = 19 at most 19 clicks to get anywhere

• Degree distributionDegree distribution– OutgoingOutgoing edges: P1(k) ~ k^{- γ1}

γ1 = 2.38~2.72

– IncomingIncoming edges: P2(k) ~ k^{- γ2}

γ2 = 2.1

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

• Average distanceAverage distance– L = 4.0– ER Random Graph model: L = 10 (too large)– So, Internet is a small-worldsmall-world network

• Degree distributionDegree distribution– Obey power law: P(k) ~ k^{-γ}, γ = 2.2 ~ 2.48 – So, Internet is a scale-freescale-free network

• Clustering coefficientClustering coefficient– C = 0.3 – ER Random Graph model: C = 0.001 (too small)

Small-world network is a better model for the Internet

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3. 3. Scientific Collaboration NetworkScientific Collaboration Network

• Pál Erdös (1913-1996)• Oliver Sacks: "A mathematical genius A mathematical genius

of the first orderof the first order, Paul Erdös was totally obsessed with his subject - he thought and wrote mathematics for nineteen hours a day until the day he died. He traveled constantly, living He traveled constantly, living out of a plastic bag, and had no out of a plastic bag, and had no interest in food, sex, companionship, interest in food, sex, companionship, artart - all that is usually indispensable

to a human life."

-- The Man Who Loved Only NumbersThe Man Who Loved Only Numbers (Paul

Hoffman, 1998)

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3. 3. Scientific Collaboration NetworkScientific Collaboration Network

• ErdErdöös Numbers Number:: – Erdös published > 1,6001,600 paperspapers with > 500500

coauthorscoauthors in his life time• Published 2 papers per month from 20-year old to die of age 83• Main contributionsMain contributions in modern mathematics: Ramsey

theory, graph theory, Diophantine analysis, additive number theory and prime number theory, …

– My Erdös Number is 22: P. Erdös – C. K. Chui – G. R. Chen

• Erdös had a (scale-freescale-free) small-world networksmall-world network of mathematical research collaboration

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3. 3. Scientific Collaboration NetworksScientific Collaboration Networks• Databases of Scientific ArticlesDatabases of Scientific Articles - showing

coauthors:– Los Alamos e-Print Archives:Los Alamos e-Print Archives: preprints (1992 - )– Medline:Medline: biomedical research articles (1961 - )– Stanford Public Information Retrieval System (SPIRES):Stanford Public Information Retrieval System (SPIRES):

high-energy physics articles (1974 - )– Network Computer Science Technical Reference Library Network Computer Science Technical Reference Library

(NCSTRL):(NCSTRL): computer science articles (10 years records)

• Computed for 10,000 to 2 million nodes (articles) over a few years They are all small-world small-world and scale-freescale-free (with power-law degree distributions)

- M.E.J.Newman (2001), A.L.Barabási et al (2001)

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Small-World Network Example:Small-World Network Example: LanguageLanguage

• Words in human language interact like a small-world networksmall-world network

• Human brain can memorize about 10^{4} ~10^{5} words (Romaine, 1992)

• Average distance between two words d = 2~3

• Degree distribution obeys a scale-free power-law: P(k) = k^{-γ},γ = 3

(Cancho and Sole)

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W W WW W W

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A model for scale-free network generationA model for scale-free network generation W. Aiello, F. Chung and L. Y. Lu (2001)

Start with no nodes and no links

At each time, a new node is added with probability p

With probability q, a random edge is added to the existing

nodes

Here, p + q = 1

Theorem: The degree distribution of the network so generated satisfies a power law with γ = 1 + 1/q

For q = 1, γ = 2; for q = ½, γ = 3; hence, 2 < γ < 3

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ControllingControlling Complex Dynamical NetworksComplex Dynamical Networks

• De-coupling control:

Make use of coupling / de-coupling • Pinning control: Only pinning a small fraction of nodes

Random / Specific pinning:

R: Pin a fraction of randomly selected nodes

S: First pin the most important node

Then select and pin the next important node Continue … till control goal is achieved

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Pinning ControlPinning Control ExampleExample

A network with 10-nodes generated by the B-A scale-free model (N=10, m=m0=3)

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Network SynchronizationNetwork Synchronization

• Network SynchronizationNetwork Synchronization

Complex DynamicsComplex Dynamics

• Network SynchronizationNetwork Synchronization– Synchronization in Small-World Networks– Synchronization in Scale-Free Networks

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Synchronization inSynchronization inGlobally Connected NetworksGlobally Connected Networks

Observation:Observation: No matter how large the network

is, a global coupled network will synchronize if its coupling strength is sufficiently strong

Good – if synchronization is useful

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Synchronization in Synchronization in Locally Connected NetworksLocally Connected Networks

Observation:Observation:No matter how strong the coupling strength is, a locally coupled network will not synchronize if its size is sufficiently large

Good - if synchronization is harmful

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SynchronizationSynchronization in Small-World Networks

Start from a nearest-neighbor coupled network

Add a link, with probability p, between a pair of nodes

Small-World ModelSmall-World Model

Good news: : A small-world network is easy to synchronize ! !

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Synchronization Synchronization in Scale-FreeScale-Free NetworksNetworks

• RobustRobust against random attacks and random failures

• FragileFragile to intentional attacks and purposeful removals of ‘big’ nodes

Both are due to the extremely inhomogeneous

connectivity distribution of scale-free networks

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SCI papers: Complex NetworksSCI papers: Complex Networks

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EI papersEI papers ：： Complex NetworksComplex Networks

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SCI papers: SCI papers: Small-World NetworksSmall-World Networks

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EI papersEI papers ： ： Small-World NetworksSmall-World Networks

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SCI papers: Scale-Free NetworksSCI papers: Scale-Free Networks

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EI papersEI papers ： ： Scale-Free NetworksScale-Free Networks

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So much for today …So much for today …

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Main ReferencesMain References

• Overview ArticlesOverview Articles• Steven H. Strogatz, Exploring complex networks, Nature, 8 March 2001, 268-276• Réka Albert and Albert-László Barabási, Statistical mechanics of complex networks, Review of Modern

Physics, Vol. 74, 47-97, 2002.• Xiaofan Wang, Complex networks: Topology, dynamics and synchronization, Int. J. Bifurcation & Chaos, Vol.

12, 885-916, 2002.• Mark E. J. Newman, Models of the small world: A review, J. Stat. Phys., Vol. 101, 2000, 819-841. • Mark E. J. Newman, The structure and function of complex networks, SIMA Review, Vol. 45, No. 2, 2003, 167-

256.• Xiaofan Wang and Guanrong Chen, Complex Networks: Small-world, scale-free and beyond, IEEE Circuits and

Systems Magazine, Vol. 3, No. 1, 2003, 6-20. • Some Related Technical PapersSome Related Technical Papers• Xiaofan Wang and Guanrong Chen, Synchronization in scale-free dynamical networks: Robustness and

Fragility, IEEE Transactions on Circuits and Systems, Part I, Vol. 49, 2002, 54-62.• Xiaofan Wang and Guanrong Chen, Synchronization in small-world dynamical networks, Int. J. of Bifurcation

& Chaos, Vol. 12, 2002, 187-192.• Xiang Li and Guanrong Chen, Synchronization and desynchronization of complex dynamical networks: An

engineering viewpoint, IEEE Trans. on Circuits and Systems - I, Vol. 50, 2003, 1381-1390.• Jinhu Lu, Xinghuo Yu and Guanrong Chen, Chaos synchronization of general complex dynamical networks,

Physica A, Vol. 334, 2004, 281-302.• Xiang Li, Xiaofan Wang and Guanrong Chen, Pinning a complex dynamical network to its equilibrium, IEEE

Trans. Circ. Sys. –I, Vol. 51, 2004, 2074-2087.• Chunguang Li and Guanrong Chen, A comprehensive weighted evolving network model, Physica A, Vol. 343,

2004, 288-294.• Guanrong Chen, Zhengping Fan and Xiang Li, Modeling the complex Internet topology, in Complex Dynamics

in Communication Networks, G. Vattay and L. Kocarev (eds.), Springer-Verlag, 2005, in press.• http:www.ee.cityu.edu.hk/~gchen/Internet.htm