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23/4/21
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