1. 2005/11/3OPLAB, NTUIM1 Introduction to Small-World Networks and Scale-Free Networks Presented by Lillian Tseng

2. 2005/11/3 2 Agenda Introduction Terminologies Small-World Phenomenon Small-World Network Model Scale-Free Network Model Comparisons Application Conclusion

3. 2005/11/3OPLAB, NTUIM3 Introduction

4. 2005/11/3 4 Why is Network Interesting? Lots of important problems can be represented as networks. Any system comprising many individuals between which some relation can be defined can be mapped as a network. Interactions between individuals make the network complex. Networks are ubiquitous!!

5. 2005/11/3 5 Internet-Map

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10. 10 Categories of Complex Networks Complex Networks Social Networks Technological (Man-made) Networks Information (Knowledge) Networks Biological Networks Friendship Sexual contact Intermarriages Business Relationships Communication Records Collaboration (film actors) (company directors) (coauthor in academics) (co-appearance) Internet Software classes Airline routes Railway routes Roadways Telephone Delivery Electric power grids Electronic circuit WWW P2P Academic citations Patent citations Word classes Preference Metabolic pathways Protein interactions Genetic regulatory Neural Blood vessels Food web

11. 2005/11/3OPLAB, NTUIM11 Terminologies

12. 2005/11/3 12 Vertex and Edge Vertex (pl. Vertices) Node (computer science), Site (physics), Actor (sociology) Edge Link (computer science), Bond (physics), Tie (sociology) Directed: citations Undirected: committee membership Weighted: friendship

13. 2005/11/3 13 Degree and Component Degree The number of edges connected to a vertex. In-degree / Out-degree in a directed graph Component Set of vertices to be reached from a vertex by paths running along edges. In-component / Out-component in a directed graph Giant component

14. 2005/11/3 14 Diameter (d) Geodesic path (Shortest path) The shortest path from one vertex to another. Geodesic path length / Shortest path length / Distance Diameter (in number of edges) The longest geodesic path length between any two vertices.

15. 2005/11/3 15 Mean Path Length (L) Mean (geodesic) path length L – global property The shortest path between two vertices, averaged over all pairs of vertices. Definition I Definition II

16. 2005/11/3 16 Clustering Coefficient (C) Clustering coefficient C –local property The mean probability that two vertices that are network neighbors of the same other vertex will themselves be neighbors. Definition I (fraction of transitive triples, widely used in the sociology literature)

17. 2005/11/3 17 Clustering Coefficient (C) (cont.) Definition II (Watts and Strogatz proposed) Example Definition I: C = 3/8 Definition II: C = 13/30

18. 2005/11/3OPLAB, NTUIM18 Small-World Phenomenon

19. 2005/11/3 19 The Small World Problem / Effect First mentioned in a short story in 1929 by Hungarian writer Frigyes Karinthy. 30 years later, became a research problem “contact and influence”. In 1958, Pool and Kochen asked “what is the probability that two strangers will have a mutual friend?” (What is the structure of social networks?) i.e. the small world of cocktail parties Then asked a harder question: “What about when there is no mutual friend --- how long would the chain of intermediaries be?” Too hard…

20. 2005/11/3 20 The Small World Experiment In 1967, Stanley Milgram (and his student Jeffrey Travers) designed an experiment based on Pool and Kochen’s work. (How many intermediaries are needed to move a letter from person A to person B through a chain of acquaintances?) A single target in Boston. 300 initial senders in Boston (100) and Omaha (in Nebraska) (200). Each sender was asked to forward a packet to a friend who was closer to the target. The friends got the same instructions.

21. 2005/11/3 21 The Small World Experiment (cont.)

22. 2005/11/3 22 The Small World Experiment (cont.) Path Length Clustering Coefficient

23. 2005/11/3 23 “Six Degrees of Separation” Travers and Milgrams’ protocol generated 300 letter chains of which 44 (?) reached the target. Found that typical chain length was 6. “What a small-world!!” Led to the famous phrase: “Six Degrees of Separation.” Then not much happened for another 30 years. Theory was too hard to do with pencil and paper. Data was too hard to collect manually.

24. 2005/11/3 24 “Six Degrees of Separation” (cont.) Duncan Watts et al. did it again via e-mails (384 out of 60,000) in 2003.

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26. 2005/11/3 26 Six Degrees of Bacon Kevin Bacon has acted creditedly in 56 movies so far Any body who has acted in a film with Bacon has a bacon number of 1. Anybody who does not have a bacon number 1 but has worked with somebody who does, they have bacon number 2, and so on. Most people in American movies have a number 4 or less. Given that there are about 630,000 such people, and this is remarkable. The Oracle of Bacon http://www.cs.virginia.edu/oracle

27. 2005/11/3 27 Kevin Bacon & Harrison Ford A Few Good Men Top Gun Witness Star Wars

28. 2005/11/3 28 What is “Six Degree”? “Six degrees of separation between us and everyone else on this planet.” A play : John Guare, 1990. An urban myth? (“Six handshakes to the President”) The Weak Version There exists a short path from anybody to anybody else. The Strong Version There is a path that can be found using local information only.

29. 2005/11/3 29 The Caveman World Many caves, and people know only others in their caves, and know all of them. Clearly, there is no way to get a letter across to somebody in another cave. If we change things so that the head-person of a cave is likely to know other head-people, letters might be got across, but still slowly. There is too much “acquaintance-overlap.”

30. 2005/11/3 30 The World of Chatting People meet others over the net. In these over-the-net-only interactions, there is almost no common friends. Again, if a message needed to be sent across, it would be hard to figure out how to route it.

31. 2005/11/3 31 Small Worlds Are Between These Extremes When there is some, but not very high, overlap between acquaintances of two people who are acquainted, small worlds results. If somebody knows people in different groups (caves?), they can act as linchpins that connect the small world. For example, cognitive scientists are lynchpins that connect philosophers, linguists, computer scientists etc. Bruce Lee is a linchpin who connects Hollywood to its Chinese counterpart.

32. 2005/11/3OPLAB, NTUIM32 Small-World Network Model

33. 2005/11/3 33 The “New” Science of Networks Mid 90’s, Duncan Watts and Steve Strogatz worked on another problem altogether. Decided to think about the urban myth. They had three advantages. They did not know anything. They had many faster computers. Their background in physics and mathematics caused them to think about the problem somewhat differently.

34. 2005/11/3 34 The “New” Science of Networks (cont.) Instead of asking “How small is the actual world?”, they asked “What would it take for any world at all to be small?” As it turned out, the answer was not much. Some source of “order” and “regularity” The tiniest amount of “randomness” Small World Networks should be everywhere.

35. 2005/11/3 35 Small-World Networks fraction p of the links is converted into shortcuts. Randomly rewire each edge with probability p to introduce increased amount of disorder. high clustering high distance high clustering low distance low clustering low distance

36. 2005/11/3 36 Small-World Networks (cont.)

37. 2005/11/3 37 Small-World Networks (cont.) Low mean path length High clustering coefficient

38. 2005/11/3 38 Power Grid NW USA-Canada |V| = 4,941  max = 19  aver = 2.67 L = 18.7 (12.4) C = 0.08 (0.005)

39. 2005/11/3OPLAB, NTUIM39 Scale-Free Network Model

40. 2005/11/3 40 What is Scale-Free? The term “scale-free” refers to any distribution functional form f(x) that remains unchanged to within a multiplicative factor under a rescaling of the independent variable x. In effect, this means power-law forms f(x) =x - , since these are the only solutions to f(ax) = bf(x), and hence “power-law” and “scale-free” are, for some purposes, synonymous.

41. 2005/11/3 41 Degree Distribution Power-law distribution Scale-free Network Poisson distribution Exponential Network

42. 2005/11/3 42 Degree distribution (cont.) Continuous hierarchy of vertices Smooth transition from biggest hub over several more slightly less big hubs to even more even smaller vertices…down to the huge mass of tiny vertices

43. 2005/11/3 43 World Wide Web Nodes: WWW documents Links: URL links Based on 800 million web pages  Finite size scaling: create a network with N nodes with P in (k) and P out (k) = 0.35 + 2.06 log(N) nd.edu 19 degrees of separation

44. 2005/11/3 44  k  ~ 6 P( k=500 ) ~ 10 -99 N WWW ~ 10 9  N(k=500)~10 -90 What did we expect? In fact, we find: P out (k) ~ k -  out P( k=500 ) ~ 10 -6  out = 2.45  in = 2.1 P in (k) ~ k -  in N WWW ~ 10 9  N(k=500) ~ 10 3

45. 2005/11/3 45 INTERNET BACKBONE (Faloutsos, Faloutsos and Faloutsos, 1999) Nodes: computers, routers Links: physical lines

46. 2005/11/3 46 ACTOR CONNECTIVITIES Nodes: actors Links: cast jointly N = 212,250 actors  k  = 28.78 P(k) ~k -  Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999)  =2.3

47. 2005/11/3 47 SCIENCE CITATION INDEX (  = 3) Nodes: papers Links: citations (S. Redner, 1998) P(k) ~k -  2212 25 1736 PRL papers (1988) Witten-Sander PRL 1981

48. 2005/11/3 48 Nodes: scientist (authors) Links: write paper together (Newman, 2000, H. Jeong et al 2001) SCIENCE COAUTHORSHIP

49. 2005/11/3 49 SEX WEB Nodes: people (females, males) Links: sexual relationships Liljeros et al. Nature 2001 4781 Swedes; 18-74; 59% response rate.

50. 2005/11/3 50 Food Web Nodes: trophic species Links: trophic interactions R. Sole (cond-mat/0011195)

51. 2005/11/3 51 Metabolic Network Nodes : chemicals (substrates) Links : bio-chemical reactions

52. 2005/11/3 52 Metabolic network Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) ArchaeaBacteriaEukaryotes

53. 2005/11/3 53 Characteristics of Scale-Free Networks The number of vertices N is not fixed. Networks continuously expand by the addition of new vertices. The attachment is not uniform. A vertex is linked with higher probability to a vertex that already has a large number of edges.

54. 2005/11/3 54 Characteristics of Scale-Free Networks (cont.) Growth Start with few linked-up vertices and, at each time step, a new vertex with m edges is added. Potential for imbalance Preferential Attachment Each edge connects with a vertex in the network according to a probability  i proportional to the connectivity k i of the vertex. Emergence of hubs The result is a network with degree distribution P(k)  k - .

55. 2005/11/3 55 Creation of Scale-Free Networks

56. 2005/11/3 56 Small-World Networks v.s. Scale- Free Networks Small-world networks Properties: mean path length / clustering coefficient Democratic (homogeneous vertices) Egalitarian (single- scale) Scale-free networks Property: degree distribution Undemocratic (heterogeneous vertices) Aristocratic (scale- free) A subset of small- world networks (?)

57. 2005/11/3 57 Single-Scale Networks Proc Nat Acad Sci USA 97, 11149 (2000)

58. 2005/11/3 58 Scale-Free Networks Nature 411, 907 (2001) Phys Rev Lett 88, 138701 (2002)

59. 2005/11/3 59 Survivability of Small-World Networks and Scale-Free Networks d=the diameter of the network

60. 2005/11/3 60 Survivability of Small-World Networks and Scale-Free Networks (cont.)

61. 2005/11/3 61 Short Summary The numerical simulations indicate There is a strong correlation between robustness and network topology. Scale-free networks are more robust than random networks against random vertex failures (error tolerance) because of their heterogeneous topology, but are more fragile when the most connected vertices are targeted (attack vulnerability / low attack survivability) with the same reason.

62. 2005/11/3OPLAB, NTUIM62 Application

63. 2005/11/3 63 Applications Social search / Network navigation Decision making Mobile ad hoc networks Peer-to-peer networks.

64. 2005/11/3 64 Social Search Find jobs. We tend to use “weak ties” (Granovetter) and also “friends of friends”. It is true that at any point in time, someone who is six degrees away is probably impossible to find and would not help you if you could find them. But, social networks are not static, and they can be altered strategically. Over time, we can navigate out to six degrees. Search process is just like Milgram’s experiment.

65. 2005/11/3 65 Social Search (Experiment) Identical protocol to Travers and Milgrams’, but conducted via the Internet. http://smallworld.sociology.columbia.edu 60,000 participants from 170 countries attempting to reach 18 different targets Important results: Median true chain length 5 < L < 7. Geography and Occupation most important. Weak ties help, but medium-strength ties typical. Professional ties lead to success. Hubs don’t seem to matter. Participation and Perception matter most!

66. 2005/11/3 66 Collective Problem Solving Small-world problem is an example of “social search.” Individuals search for remote targets by forwarding message to acquaintance. Social networks turn out to be searchable. But search process is collective in that chain knows more about the network than any individual. Not possible in all networks. Social search is relevant not only to finding jobs and locating answers / resources (i.e. individual problem solving) but also collective problem solving (innovation / recovery from catastrophe).

67. 2005/11/3 67 Network Navigation Two fundamental components in the small-world Short chains are ubiquitous. Individuals operating with purely local information are easy at finding these chains. Agents are on a grid. Everybody is connected to their neighbors. But they are also connected to k other agents randomly.

68. 2005/11/3 68 Network Navigation (cont.) The distribution could be uniform, or biased towards closer agents. It could be inversely proportional to the distance d from us to that agent, or inversely proportional to the square of the distance (sort of like gravitation). These can be represented as inversely proportional to d to the power r (clustering exponent), where r is 0, 1, 2 or above.

69. 2005/11/3 69 Network Navigation (cont.) If r is 0, the neighbor is chosen randomly, and the world is like the solarium world. If r is very high, you only know your immediate neighbors: and the world is like the caveman world. For intermediate values, we get more and more small-world-like behavior. There is always a findable path whose length is not too big only when r is 2!!

70. 2005/11/3 70 Network Navigation (cont.) For any other R (smaller or bigger than 2), the expected length of a findable path is larger. Efficient navigability is a fundamental property of only some small-world structures (???). The correlation between local structure and long- range connections provides critical cues for finding paths through networks.

71. 2005/11/3OPLAB, NTUIM71 Conclusion

72. 2005/11/3 72 Conclusion What’s small-world phenomenon Six degrees of separation Shortcuts Networks with small-world property Small-world networks High clustering coefficient Low mean path length Scale-free networks Power-law distribution

73. 2005/11/3 73 Conclusion (cont.) All complex networks in nature seems to have power-law degree distribution. It is far from being the case!! Some networks have degree distribution with exponential tail. They do not belong to random graph because of evolving property. Evolving networks can have both power-law and exponential degree distributions.

74. 2005/11/3OPLAB, NTUIM74 Q & A Thanks for your listening ^_^

75. 2005/11/3 75 References (Papers) [] S. Milgram, The small world problem, Psych. Today, 2 (1967), pp. 60–67. [] D. J. Watts and S. H. Strogatz, Collective dynamics of “small-world” networks, Nature, 393 (1998), pp. 440–442. [] M. Barth´el´emy and L. A. N. Amaral, Small-world networks: Evidence for a crossover picture, Phys. Rev. Lett., 82 (1999), pp. 3180–3183. [] S. Lawrence and C. L. Giles, Accessibility of information on the web, Nature, 400 (1999), pp. 107–109. [] R. Albert, H. Jeong, and A.-L. Barab´asi, Diameter of the world-wide web, Nature, 401 (1999), pp. 130–131. [] A.-L. Barab´asi and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), pp. 509–512. A.-L. Barab´asi, R. Albert, and H. Jeong, Mean-field theory for scale-free random networks, Phys. A, 272 (1999), pp. 173–187. [] A.-L. Barab´asi, R. Albert, H. Jeong, and G. Bianconi, Power-law distribution of the World Wide Web, Science, 287 (2000), p. 2115a. [] S. H. Yook, H. Jeong, and A.-L. Barabasi, Modeling the internet’s large-scale topology, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 13382–13386. [] J. M. Kleinberg, Navigation in a small world, Nature, 406 (2000), p. 845.

76. 2005/11/3 76 References (Surveys) [] R. Albertand A.-L. Barab´asi, Statistical mechanics of complex networks, Rev. Modern Phys., 74 (2002), pp. 47–97. [] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks, Adv. in Phys., 51 (2002), pp. 1079–1187. Long survey (the above two): focus on physical literature, and devote the larger part of attention to the models of growing network models. [] M. E. J. Newman, The structure and function of complex networks, Rev. Society for Industrial and Applied Mathematics, Vol. 45, No. 2 (2003), pp. 167-256. Long survey: focus on all aspects of literature. [] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), pp. 268–276. Short Survey: devote discussion of the behavior of dynamical systems on networks. [] B. Hayes, Graph theory in practice: Part I, Amer. Sci., 88 (2000), pp. 9–13. [] B. Hayes, Graph theory in practice: Part II, Amer. Sci., 88 (2000), pp. 104–109. Short Survey (the above two): devote discussion on the small-world model.

77. 2005/11/3 77 References (Books) [] A.-L. Barab´asi, Linked: The New Science of Networks, Perseus, Cambridge, MA, 2002. Focus on scale-free network [] M. Buchanan, Nexus: Small Worlds and the Groundbreaking Science of Networks, Norton, New York, 2002. From the point of view of a science journalist [] D. J. Watts, Six Degrees: The Science of a Connected Age, Norton, New York, 2003. From the point of view of a sociologist

78. 2005/11/3 78 Reference [1] R. Albert, H. Jeong, and A.-L. Barab´asi, Attack and error tolerance of complex networks, Nature, 406 (2000), pp. 378–382. [2] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener, Graph structure in the web, Computer Networks, 33 (2000), pp. 309–320. [3] R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Resilience of the Internet to random breakdowns, Phys. Rev. Lett., 85 (2000), pp. 4626–4628. [4] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Network robustness and fragility: Percolation on random graphs, Phys. Rev. Lett., 85 (2000), pp. 5468–5471. [5] R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Breakdown of the Internet under intentional attack, Phys. Rev. Lett., 86 (2001), pp. 3682–3685. [6] S. N. Dorogovtsev and J. F. F. Mendes, Comment on “Breakdown of the Internet under intentional attack,” Phys. Rev. Lett., 87 (2001), art. no. 219801. [7] P. Holme, B. J. Kim, C. N. Yoon, and S. K. Han, Attack vulnerability of complex networks, Phys. Rev. E, 65 (2002), art. no. 056109. [8] V. Latora and M. Marchiori, Efficient behavior of small-world networks, Phys. Rev. Lett., 87 (2001), art. no. 198701. [9] V. Latora and M. Marchiori, Economic Small-World Behavior in Weighted Networks, Preprint 0204089 (2002); available from http://arxiv.org/abs/cond-mat/.

79. 2005/11/3 79 Reference (cont.) [] J. O. Kephartand S. R. White, Directed-graph epidemiological models of computer viruses, in Proceedings of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, IEEE Computer Society, Los Alamitos, CA, 1991, pp. 343– 359. [] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), pp. 3200–3203. [] A. L. Lloyd and R. M. May, How viruses spread among computers and people, Science, 292 (2001), pp. 1316–1317. [] R. M. May and A. L. Lloyd, Infection dynamics on scale-free networks, Phys. Rev. E, 64 (2001), art. no. 066112. [] M. E. J. Newman, S. Forrest, and J. Balthrop, Email networks and the spread of computer viruses, Phys. Rev. E, 66 (2002), art. no. 035101. [] R. Cohen, D. ben-Avraham, and S. Havlin, Efficient Immunization of Populations and Computers, Preprint 0207387 (2002); available from http://arxiv.org/abs/cond- mat/.

80. 2005/11/3 80 Reference (cont.) For food webs [] J. A. Dunne, R. J. Williams, and N. D. Martinez, Food-webstructur e and network theory: The role of connectance and size, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 12917–12922. [] J. A. Dunne, R. J. Williams, and N. D. Martinez, Network structure and biodiversity loss in food webs: Robustness increases with connectance, Ecology Lett., 5 (2002), pp. 558–567. For metabolic networks [] H. Jeong, S. Mason, A.-L. Barab´asi, and Z. N. Oltvai, Lethality and centrality in protein networks, Nature, 411 (2001), pp. 41–42.