1. Scotty Smith February 7, 2007 Small World Networks

2. Papers • M.E.J.Newman. Models of the Small World: A Review . J.Stat.Phys. Vol. 101, 2000, pp. 819-841. • M.E.J. Newman, C.Moore and D.J.Watts. Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201-3204 (2000). • M.E.J.Newman. The structure and function of networks.

3. 6 Degrees of Separation • Milgram Experiment • Kevin Bacon Game • http://www.oracleofbacon.org

4. Why Study Small World Networks • Social Networks • Spread of information, rumors • Disease Spread

5. Random Graphs • A graph with randomly placed edges between the N nodes of the graphs • z is the average number of connections per node (coordination number)‏ • .5*N*z connections in the graph

6. Random Graphs Continued • First Neighbors • z • Second Neighbors • z2 • D = Degree needed to reach the entire graph • D = log(N)/log(z)‏

7. Problems • No Clustering

8. Lattices

9. Benefits and Problems • Very specific clustering values • C = (3*(z-2))/(4*(z-1))‏ • No small-world effect

10. Rewiring • Take random links, and rewire them to a random location on the lattice • Gives small world path lengths

11. Analytical Problems • Rewiring connections could result in disconnected portions of the graph • For analysis, add shortcuts instead of rewiring

12. Important Results • Average Distance Scaling

13. Other models using Small Worlds • Density Classification • Iterated Prisoners Dilemma

14. Properties of Real World Networks • Small-World effect • Skewed degree of distribution • Clustering

15. Networks Studied • Regular Lattice • No small-world effect • Scales linearly • No skewed distribution • Fully connected • No skewed distribution • Very high clustering value • Random • Poissonian distribution • Very small clustering value

16. Fixing Random Graphs • The “stump” model • Growth model • Preferential attachment to nodes with larger degrees • Does not fix clustering

17. Bipartite Graphs • Explains how clustering arises • Analysis sometimes gives good estimates of clustering, but for others they do not

18. Growth Model Clustering • More specific preferential attachment • Higher probability of linking pairs of people who have common acquaintances • Very high clustering and development of communities

19. Mean Field Solution • Continuum Model • Treat the 1-d lattice ring as if it has an infinite number of points • Not the same as having an infinite number of locations • “Shortcuts” have 0 length • Consider neighborhoods of random points

20. Terminology • Neighborhood • Set of points which can be reached following paths of r or less.

21. Very Brief Trace of the Proof

22. Result