1. Small World Networks
Scotty Smith
February 7, 2007

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

3. 6 Degrees of Separation Milgram Experiment Kevin Bacon Game

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
*Y How much the average distance changes from standard
*X = Number of shortcuts

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 Fully connected Random
No small-world effect
Scales linearly
No skewed distribution
Fully connected
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