NETWORKS. David J Hill Research School of Information Sciences and Engineering The Australian National University

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1 Lab Net Con 网络控制 Short-course: Complex Systems Beyond the Metaphor UNSW, February 2007 NETWORKS David J Hill Research School of Information Sciences and Engineering The Australian National University 8/2/2007 David J Hill The Australian National University Networks 1

2 Lectures 1. Introduction and Basics 2. Synchronization Fundamentals 3. Dynamic Networks and Control 4. Research Directions 8/2/2007 David J Hill The Australian National University Networks 2

3 Lecture 1 Introduction and Basics Introduction Basic concepts Basic models Acknowledgement: Guanrong Chen, Xiaofan Wang Ref: X.F.Wang and G.Chen, Complex networks: Small-World, Scale-free and Beyond, IEEE CAS Magazine, X.F.Wang, X.Li and G.Chen, Complex Networks Theory and Applications, in preparation. 8/2/2007 David J Hill The Australian National University Networks 3

4 What is a Complex Network? Almost everything, everywhere 8/2/2007 David J Hill The Australian National University Networks 4

5 Kanyini 8/2/2007 David J Hill The Australian National University Networks 5

6 Modern Home 8/2/2007 David J Hill The Australian National University Networks 6

7 Social networks Relationships Spreading, e.g. disease, rumours etc 8/2/2007 David J Hill The Australian National University Networks 7

8 Complex Network Example: Routes of Airlines 8/2/2007 David J Hill The Australian National University Networks 8

9 Complex Network Example: Internet (William R. Cheswick) 8/2/2007 David J Hill The Australian National University Networks 9

10 Complex Network Example: WWW (K. C. Claffy) 8/2/2007 David J Hill The Australian National University Networks 10

11 Complex Network Example: Telecommunication Networks (Stephen G. Eick) 8/2/2007 David J Hill The Australian National University Networks 11

12 8/2/2007 David J Hill The Australian National University Networks 12

13 Will it Happen Again? 1959, 1961, 1965, 1977, /2/2007 David J Hill The Australian National University Networks 13

14 Complex Network Example: Biological Networks 8/2/2007 David J Hill The Australian National University Networks 14

15 Complex Systems Aren t we just looking at the hard problems in all areas? Complex Social Economics Other Engi neer ing Physics Mathem atics Biology problems What are the unifying ideas? 8/2/2007 David J Hill The Australian National University Networks 15

16 CS models PDEs Messy nonlinear ODEs Large sets of ODEs Statistical Complex Systems and Networks Networks Arise naturally, but Also as a way to underpin and/or approximate these models Explicitly model connections Enable study of the impact of structure. 8/2/2007 David J Hill The Australian National University Networks 16

17 Road Map vs Airline Map Poisson distribution Power law distribution Exponential Network Scale-free Network (nodes: cities links: highways) (nodes: airports links: flights) Each major city has at least one link to the highway system. No cities served by hundreds of highways, uniform Vast majority of airports are tiny nodes, connected by a few hubs 8/2/2007 David J Hill The Australian National University Networks 17

18 Basic Concepts Network description Graphs Three models Typical examples 8/2/2007 David J Hill The Australian National University Networks 18

19 Network Topology A network is a set of nodes interconnected via links Examples: 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 Links can be anything depending on the context 8/2/2007 David J Hill The Australian National University Networks 19

20 Network 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 super-computing power have led to a rethinking of approach Two significant recent discoveries are: Small-World effect (Watts and Strogatz, Nature, 1998) Scale-Free feature (Barabási and Albert, Science, 1999) 8/2/2007 David J Hill The Australian National University Networks 20

21 Graph Theory Problem of Seven Bridges of Königsberg Is it possible to walk with a route that crosses each bridge exactly once returning to the start point? 8/2/2007 David J Hill The Australian National University Networks 21

22 Graph Theory Problem of Seven Bridges of Königsberg In 1736, Leonhard Euler proved that it was not possible. In proving the result, Euler formulated the problem in terms of graph theory, by abstracting the case of Königsberg first, by eliminating all features except the landmasses and the bridges connecting them; second, by replacing each landmass with a dot, called a vertex or node, and each bridge with a line, called an edge or link. The resulting mathematical structure is called a graph. Ref: Wikipedia 8/2/2007 David J Hill The Australian National University Networks 22

23 Graph Theory Theorem (Handshaking Lemma) The total node degree of a graph is always an even number. Lots of counting results for nodes, edges. Lots on trees, cutsets some of us studied in circuit theory. Theorem A connected graph is Eulerian iff the degree of every node of the graph is an even number. Corollary The Seven Bridges of Königsberg problem has no solution. 8/2/2007 David J Hill The Australian National University Networks 23

24 Some Basic Concepts (Average) Path Length L Clustering Coefficient C Degree and Degree Distribution P(k) (Stephen G. Eick) 8/2/2007 David J Hill The Australian National University Networks 24

25 Average Distance Distance d(n,m) between two nodes n and m = the number of links along the shortest path connecting them Diameter D = max{d(n,m)} Average distance L = average over all d(n,m) Most large and complex networks have small L small-world feature 8/2/2007 David J Hill The Australian National University Networks 25

26 Clustering Coefficient Clustering Coefficient C of a network: C= #links #possible links e.g. 4 nodes, 4 links C=0.66 Always satisfies 0 < C < 1 Note 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 8/2/2007 David J Hill The Australian National University Networks 26

27 Fully Connected Network Theorem A fully connected network has the average path length L=1 and the largest clustering coefficient C=1. Moreover, a globally connected network with N nodes has a total of N(N-1)/2 edges. 8/2/2007 David J Hill The Australian National University Networks 27

28 Star Network Theorem For a star-shaped network of size N, counting the central node, the average path length is L=2-2/N 2 as N and the clustering coefficient is C=0. Similar results for rings, random graphs, small-world and the BA scale-free networks 8/2/2007 David J Hill The Australian National University Networks 28

29 Degree 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 Often some lack of rigour in its use. 8/2/2007 David J Hill The Australian National University Networks 29

30 Degree Distribution Completely regular lattice: P(k) ~ Delta distribution Most networks: P(k) ~ k -γ scale-free feature (power law) Completely random networks: P(k) = μ k e -μ Poisson distribution k! μ=pn k (Regular) delta k -γ Poisson (Random) 8/2/2007 David J Hill The Australian National University Networks 30

31 Model 1: ER Random Graph Features:» Connectivity: Poisson distribution» Homogeneous nature: each node has roughly the same number of links 8/2/2007 David J Hill The Australian National University Networks 31

32 Model 2: Small-World Networks Features: (Similar to ER Random Graphs)» Connectivity distribution: uniform but decays exponentially» Homogeneous nature: each node has roughly the same number of links 8/2/2007 David J Hill The Australian National University Networks 32

33 Model 3: Scale-Free Networks Features:» Connectivity: in power-law form» Non-homogeneous nature: a few nodes have many links but most nodes have very few links (yeast proteome, protein-protein interactions by Hawoong Jeong) 8/2/2007 David J Hill The Australian National University Networks 33

34 Random Graph Model and Small-World Network Model Common Features: Connectivity Distribution Poisson/binomial or near uniform Homogeneous Nature Each node has about the same number of links Fixed Size Network does not grow 8/2/2007 David J Hill The Australian National University Networks 34

35 Scale-free Network Model (Barabasi and Albert,1999) Features: Connectivity Distribution: power-law distribution k -r with r = 3 Non-homogeneous Nature: Few nodes have many links Many nodes have few links Fixed Size: Network is growing Extended BA (EBA) Model (allow r < 3) (Albert and Barabasi, 2000) 8/2/2007 David J Hill The Australian National University Networks 35

36 Comparison Ave. Distance Clustering E-R random graph model Small / Large Small Real-life complex networks Small Large (Small-world feature) Degree Distribution Binomial / Poisson Power-law (Scale-free feature) 8/2/2007 David J Hill The Australian National University Networks 36

37 Examples World Wide Web Internet Scientific Collaboration Network Power Grids (Stephen G. Eick) 8/2/2007 David J Hill The Australian National University Networks 37

38 World Wide Web Average distance Computed Average distance L = 14 Diameter L = 19 at most 19 clicks to get anywhere Degree distribution Outgoing edges: P 1 (k) ~ k -γ1 γ 1 = 2.38~2.72 Incoming edges: P 2 (k) ~ k -γ2 γ 2 = 2.1 8/2/2007 David J Hill The Australian National University Networks 38

39 Internet (Computed in , at both domain level and router level) Average distance L = 4.0 So, Internet is a small-world network Degree distribution Obey power law: P(k) ~ k -γ, γ = 2.2 ~ 2.48 So, Internet is a scale-free network Clustering coefficient C = 0.3 Small-world network is a better model for the Internet 8/2/2007 David J Hill The Australian National University Networks 39

40 Erdös Number: Scientific Collaboration Network Erdös published > 1,600 papers with > 500 coauthors in his life time - Published 2 papers per month from 20-years old to his death at age 83 My Erdös Number is 3: P. Erdös C. K. Chui G. R. Chen D.J.Hill Erdös had a (scale-free) small-world network of mathematical research collaboration 8/2/2007 David J Hill The Australian National University Networks 40

41 Scientific Collaboration Networks Databases of Scientific Articles - showing coauthors: Los Alamos e-print e Archives: preprints ( ) Medline: biomedical research articles ( ) Stanford Public Information Retrieval System (SPIRES): highenergy physics articles ( ) Network Computer Science Technical Reference Library (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 and scale-free (with power-law degree distributions) - M.E.J.Newman (2001) 8/2/2007 David J Hill The Australian National University Networks 41

42 Limitation of BA and EBA Models Global preferential attachments: Π i = k j Simple rich get richer model i k j or ( k i ) = ki + 1 ( kl + 1) l However, many real complex networks have local preferential attachment probabilities 8/2/2007 David J Hill The Australian National University Networks 42

43 Limitation of BA and EBA models Example: Internet 8/2/2007 David J Hill The Australian National University Networks 43

44 Multi-Local-World (MLW) Model Many local-worlds Inter-connections are sparse Inner-connections are dense Ref: Li and Chen, /2/2007 David J Hill The Australian National University Networks 44

45 Real Internet Data (Ref: Faloutsos x3, Com Com Review, 1999) P ( k) k γ The exponent r approx.= 2.2 (in the AS level; also SF at router level) 8/2/2007 David J Hill The Australian National University Networks 45

46 The Real Internet Statistical Data of the Internet ( , AS) #nodes #links Av degree Av CC Av distance 8/2/2007 David J Hill The Australian National University Networks 46

47 Comparisons BA Model Extended BA (EBA) Model MLW Model Real Internet (2005, AS) N k C d γ /2/2007 David J Hill The Australian National University Networks 47

48 General comparisons Structural Features Locality Hierarch y Random Graph No No No BA-EBA Models No No Statistic al Features Yes MLW Model Yes Yes Yes So, MLW is the best model for the Internet 8/2/2007 David J Hill The Australian National University Networks 48

49 Further Work Better and More Realistic Models Rigorous Mathematical Theory Recent work: L. Li, D. Alderson, R. Tanaka, J. C. Doyle, and W. Willinger, Towards a theory of scale-free graphs: definition, properties, and implications, preprint, arxiv 2005) 8/2/2007 David J Hill The Australian National University Networks 49

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51 Achilles Heel An interesting phenomenon of complex networks is their Achilles heel robustness versus fragility. Robustness relates to the vulnerability to random removal. Fragility relates to the vulnerability to random removal. The ARPANET early design was focussed on the latter and had a lattice type structure. 8/2/2007 David J Hill The Australian National University Networks 51

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53 Scale-free questioned Recent criticism of BA type scale-free models goes beyond the Internet Obsession with power laws Need degree, betweeness, centrality measures at least Duplication, divergence in the model for protein models Ref: E.Fox Keller, Revisiting scale-free networks, /2/2007 David J Hill The Australian National University Networks 53

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56 Some Reading A-L.Barabási, Linked, Plume Books, 2003 good general introduction M.Newman, A-L.Barabási and D.J.Watts (Eds.), The Structure and Dynamics of Networks, Princeton University Press, 2006 collection of key papers X.F.Wang and G.Chen, Complex networks: Small-World, Scale-free and Beyond, IEEE CAS Magazine, 2003 reading for these lectures 8/2/2007 David J Hill The Australian National University Networks 56

57 Some References W.Aiello, F. Chung and L. Y. Lu, A random graph model for massive graphs, in Proc.32 nd Annual ACM Symp, 2000; see NBW pp A-L.Barabasi and R.Albert, Emergence of scaling in random networks, Science, 286, 1999; see NBW pp R.Albert and A_L.Barabási, Topology of evolving networks: Local events and universality, Physics Review Letters, 85, 24, R.Albert, H.Jeong and A_L.Barabási, Diameter of the world wide web, Nature, 401, pp , 1999; see NBW p.182. R.Ferrer i Cancho and R.V.Sole, The small world of human language, Proc.R.Soc.London B, 268, pp , P.Erdös and Rényi, On the evolution of random graphs, Publ Math Inst Hung Acad Sci, 5, 17-61, 1960; see NBW pp M.,P.and C.Faloutsos, On power-law relationships of the internet technology, Comp Com Review, 29, pp ; see NBW pp (1999). E.F.Keller, Revisiting scale-free networks, BioEssays, 27.10, 2005 another view on universality Hoffman,P., The Man Who Loved Only Numbers, Hyperion, S.Lawrence and Giles,C.L., Accessibility of information on the web, Nature, 400, pp , L. Li, D. Alderson, R. Tanaka, J. C. Doyle, and W. Willinger, Towards a theory of scale-free graphs: definition, properties, and implications, preprint, arxiv 2005 more rigorous approach. X. Li and G. Chen, A local-world evolving network model, Physica A, Vol. 328, pp , Oct M.E.J.Newman, The structure of scientific collaboration networks, Proc Natl Acad Sci USA, 2001; see NBW pp G. Peng, K.-T. Ko, L. S. Tan and G. Chen, Router-level Internet as a local-world weighted evolving network, Dynamics of Continuous, Discrete and Impulsive Systems, Vol. 13, pp , Dec S.Redner, How popular is your paper? An empirical study of the citation distribution, Euro. Physics J, M-J Shi, X Li, X Wang, The Complex Software Networks Topology of Java Evolution, 2006, WCICA. D.J.Watts and S.H.Strogatz, Collective dynamics of small-world networks, Nature, 393, 1998; see NBW pp /2/2007 David J Hill The Australian National University Networks 57

58 What Next Dynamic networks Issues Static design Dynamic design Synchronization Robustness Vulnerability Collapse Control Restoration Oscillating steel balls suspended by variable length and strength elastic 8/2/2007 David J Hill The Australian National University Networks 58