Introduction to Small-World Networks and Scale-Free Networks

1. Introduction to Small-World Networks and Scale-Free Networks Presentedby Lillian Tseng OPLAB, NTUIM

2. Agenda • Introduction • Terminologies • Small-World Phenomenon • Small-World Network Model • Scale-Free Network Model • Comparisons • Application • Conclusion

3. Introduction OPLAB, NTUIM

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. Internet-Map

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. Terminologies OPLAB, NTUIM

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

18. Small-World Phenomenon OPLAB, NTUIM

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. 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. The Small World Experiment (cont.)

22. The Small World Experiment (cont.) Path Length Clustering Coefficient

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

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. Kevin Bacon & Harrison Ford Top Gun Witness A Few Good Men Star Wars

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 informationonly.

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. 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. 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. Small-World Network Model OPLAB, NTUIM

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

36. Small-World Networks (cont.)

37. Small-World Networks (cont.) • Low mean path length • High clustering coefficient

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. Scale-Free Network Model OPLAB, NTUIM

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. Power-law distribution Poisson distribution Exponential Network Scale-free Network Degree Distribution

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.  Finite size scaling: create a network with N nodes with Pin(k) and Pout(k) < l > = 0.35 + 2.06 log(N) 19 degrees of separation nd.edu < l > World Wide Web Nodes: WWW documents Links: URL links Based on 800 million web pages

44. What did we expect? k ~ 6 P(k=500) ~ 10-99 NWWW ~ 109  N(k=500)~10-90 In fact, we find: out= 2.45 in = 2.1 P(k=500) ~ 10-6 NWWW ~ 109  N(k=500) ~ 103 Pout(k) ~ k-out Pin(k) ~ k- in

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

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

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

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

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

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