Small World Models

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 nodes neighbor
  pairs that are also neighbors each other
• Tolerance (df, Cf)
• 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-Strogatzs small world model
• Scale-free network
• P(k) follows a Power Law distribution
• Barabasi and Alberts 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-gtAgents
• Dynamical links-gtComplex interactions
• Small world effect-gtCooperation
• Tolerance-gtRobustness
• Clustering-gtSpecies

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.