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Small World Models

Small World Models. Shiwu Zhang Based on [M. Newman 2000, R. Albert 2000]. “Small World” phenomena. Experiment on delivering letters to a person from random selected people.(1967, Milgram) The result is “six degrees of seperation” Small world effect (Korte& Milgram. 1970)

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Small World Models

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  1. Small World Models Shiwu Zhang Based on [M. Newman 2000, R. Albert 2000]

  2. “Small World” phenomena • Experiment on delivering letters to a person from random selected people.(1967, Milgram) The result is “six degrees of seperation” • Small world effect (Korte& Milgram. 1970) • World Wide Web (R. Albert, et al. 1999) • Other experiments

  3. Motivation • Information spread on social networks • Disease spread social networks • Virus spread on internet • WWW and internet development • Culture formation and reservation

  4. Characteristics • Small world effect (Diameter d) • d: The average vertex-vertex distance • Clustering (Clustering coefficient C) • C: The average fraction of a node’s neighbor pairs that are also neighbors each other • Tolerance (d~f, C~f) • f: The fraction of nodes that is removed from network

  5. Models on Small world network • Random graphs (Taking N dots and drawing Nz/2 lines between random pairs) • Completely ordered lattice (a low dimension regular lattice) • Watts-Strogatz model (a low dimension regular lattice with some degrees of randomness) • Other models

  6. Scale-free network • Based on connectivity distribution P(k) • Exponential network • P(k) follows a Possion distribution • Random graph, Watts-Strogatz’s small world model • Scale-free network • P(k) follows a Power Law distribution • Barabasi and Albert’s model on internet • Real world models: Internet and WWW

  7. Applications • Cellular automata, prison dilemma and oscillator networks on small world network and regular network (Watts and Strogatz 1998, 1999) • Disease spread on small world graphs. (Newman and Watts(1999), Satorras and Vespignami(2001), Dezso and Barabasi(2002) )—epidemic point • Coherent and fast response(Fernandez et al. 2000) • Behavior of the B-S model oh species coevolution on small-world graph (Kulkarni 1999)

  8. Implications on MAS • Nodes->Agents • Dynamical links->Complex interactions • Small world effect->Cooperation • Tolerance->Robustness • Clustering->Species ……

  9. Related Papers • M.E.J. Newman (2000). Models of the Small World. • R. Albert et al. (2000). Error and attack tolerance of complex networks. Nature (406)378-382. • A.L. Barabasi and R. Albert, (1999). Emergence of scaling in random networks. Science (286)509-512. • A.L. Barabasi et al (2000). Scale-free characteristics of random networks: the topology of the World-wide web. Physica A (281)70-77.

  10. Related Papers (2) • R.P. Satorras and A.Vespignami, (2001). Epidemic spreading in scale-free networks. PRL, (86)14, p3200-3203 • Z. Dezso and A.L. Barabasi, (2002). Halting virus in scale-free networks. • D.J. Watts and S.H. Strogatz, (1998). Collective dynamics of ‘small world’ networks. Nature, (393), p440-442. • H, Jeong et al, (2000). The large-scale organization of metabolic networks, Nature, (407), p651-654.

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