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Introduction to Small-World Networks and Scale-Free Networks Presented by Lillian Tseng 2005/11/3 OPLAB, NTUIM 1 Agenda Introduction Terminologies Small-World Phenomenon Small-World Network Model Scale-Free Network Model Comparisons Application Conclusion 2005/11/3 2 Introduction 2005/11/3 OPLAB, NTUIM 3 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!! 2005/11/3 4 2005/11/3 5 2005/11/3 6 2005/11/3 7 2005/11/3 8 2005/11/3 9 Categories of Complex Networks Complex Networks Technological Information Social Biological (Man-made) (Knowledge) Networks Networks Networks Networks Friendship Internet Sexual contact Software classes WWW Metabolic pathways Intermarriages Airline routes P2P Protein interactions Business Relationships Railway routes Academic citations Genetic regulatory Communication Records Roadways Patent citations Neural Collaboration Telephone Word classes Blood vessels (film actors) Delivery Preference Food web (company directors) Electric power grids (coauthor in academics) Electronic circuit (co-appearance) 2005/11/3 10 Terminologies 2005/11/3 OPLAB, NTUIM 11 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 2005/11/3 12 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 2005/11/3 13 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. 2005/11/3 14 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 2005/11/3 15 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) 2005/11/3 16 Clustering Coefficient (C) (cont.) Definition II (Watts and Strogatz proposed) Example Definition I: C = 3/8 Definition II: C = 13/30 2005/11/3 17 Small-World Phenomenon 2005/11/3 OPLAB, NTUIM 18 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… 2005/11/3 19 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. 2005/11/3 20 The Small World Experiment (cont.) 2005/11/3 21 The Small World Experiment (cont.) Path Length Clustering Coefficient 2005/11/3 22 “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. 2005/11/3 23 “Six Degrees of Separation” (cont.) Duncan Watts et al. did it again via e-mails (384 out of 60,000) in 2003. 2005/11/3 24 2005/11/3 25 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 2005/11/3 26 Kevin Bacon & Harrison Ford Top Gun Witness A Few Good Men Star Wars 2005/11/3 27 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. 2005/11/3 28 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.” 2005/11/3 29 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. 2005/11/3 30 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. 2005/11/3 31 Small-World Network Model 2005/11/3 OPLAB, NTUIM 32 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. 2005/11/3 33 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. 2005/11/3 34 Small-World Networks high clustering high clustering low clustering high distance low distance low distance • fraction p of the links is converted into shortcuts. • Randomly rewire each edge with probability p to introduce increased amount of disorder. 2005/11/3 35 Small-World Networks (cont.) 2005/11/3 36 Small-World Networks (cont.) • Low mean path length • High clustering coefficient 2005/11/3 37 Power Grid NW USA-Canada |V| = 4,941 max = 19 aver = 2.67 L = 18.7 (12.4) C = 0.08 (0.005) 2005/11/3 38 Scale-Free Network Model 2005/11/3 OPLAB, NTUIM 39 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. 2005/11/3 40 Degree Distribution Poisson distribution Power-law distribution Exponential Network 2005/11/3 Scale-free Network 41 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 2005/11/3 42 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 Pin(k) and Pout(k) < l > = 0.35 + 2.06 log(N) 19 degrees of separation nd.edu <l> 2005/11/3 43 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 2005/11/3 Pin(k) ~ k- in 44 INTERNET BACKBONE Nodes: computers, routers Links: physical lines 2005/11/3 45 (Faloutsos, Faloutsos and Faloutsos, 1999) 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 2005/11/3 46 SCIENCE CITATION INDEX 25 Nodes: papers Links: citations Witten-Sander PRL 1981 1736 PRL papers (1988) 2212 P(k) ~k- ( = 3) 2005/11/3 47 (S. Redner, 1998) SCIENCE COAUTHORSHIP Nodes: scientist (authors) Links: write paper together 2005/11/3 (Newman, 2000, H. Jeong et al 2001)48 SEX WEB Nodes: people (females, males) Links: sexual relationships 4781 Swedes; 18-74; 59% response rate. 2005/11/3 49 Liljeros et al. Nature 2001 Food Web Nodes: trophic species Links: trophic interactions 2005/11/3 R. Sole (cond-mat/0011195) 50 Metabolic Network Nodes: chemicals (substrates) Links: bio-chemical reactions 2005/11/3 51 Metabolic network Archaea Bacteria Eukaryotes Organisms from all three domains of life are scale-free networks! 52 H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) 2005/11/3 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. 2005/11/3 53 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 ki of the vertex. Emergence of hubs k (ki ) i The result is a network with degree distribution j k j k -. P(k) 2005/11/3 54 Creation of Scale-Free Networks 2005/11/3 55 Small-World Networks v.s. Scale- Free Networks Small-world networks Scale-free networks Properties: mean path Property: degree length / clustering distribution coefficient Undemocratic Democratic (heterogeneous (homogeneous vertices) vertices) Aristocratic (scale- Egalitarian (single- free) scale) A subset of small- world networks (?) 2005/11/3 56 Single-Scale Networks 2005/11/3 Proc Nat Acad Sci USA 97, 11149 (2000) 57 Scale-Free Networks Nature 411, 907 (2001) 2005/11/3 Phys Rev Lett 88, 138701 (2002) 58 Survivability of Small-World Networks and Scale-Free Networks d=the diameter of the network 2005/11/3 59 Survivability of Small-World Networks and Scale-Free Networks (cont.) 2005/11/3 60 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. 2005/11/3 61 Application 2005/11/3 OPLAB, NTUIM 62 Applications Social search / Network navigation Decision making Mobile ad hoc networks Peer-to-peer networks. 2005/11/3 63 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. 2005/11/3 64 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! 2005/11/3 65 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). 2005/11/3 66 Conclusion 2005/11/3 OPLAB, NTUIM 71 Conclusion What’s small-world phenomenon 10 0 Six degrees of separation -5 10 Shortcuts -10 Networks with small-world 10 property P(k) -15 10 Small-world networks -20 10 High clustering coefficient Low mean path length 10 -25 P(k) ~ Poisson -3 P(k) ~ k Scale-free networks -30 10 Power-law distribution 10 0 10 1 k 10 2 10 3 2005/11/3 72 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. 2005/11/3 73 Q&A Thanks for your listening ^_^ 2005/11/3 OPLAB, NTUIM 74