Small World Networks
description
Transcript of Small World Networks
Small World Networks
Scotty Smith
February 7, 2007
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.
6 Degrees of Separation
Milgram Experiment
Kevin Bacon Game
http://www.oracleofbacon.org
Why Study Small World Networks
Social Networks
Spread of information, rumors
Disease Spread
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
Random Graphs Continued
First Neighbors
z
Second Neighbors
z2
D = Degree needed to reach the entire graph
D = log(N)/log(z)
Problems
No Clustering
Network N l CMovie Actor 225226 3.65 0.79 .0003Neural 282 2.65 0.28 0.05Power Grid 4941 18.7 0.08 .0005
Crand
Lattices
Benefits and Problems
Very specific clustering values
C = (3*(z-2))/(4*(z-1))
No small-world effect
Rewiring
Take random links, and rewire them to a
random location on the lattice
Gives small world path lengths
Analytical Problems
Rewiring connections could result in
disconnected portions of the graph
For analysis, add shortcuts instead of rewiring
Important Results
Average Distance Scaling
Other models using Small Worlds
Density Classification
Iterated Prisoners Dilemma
Properties of Real World Networks
Small-World effect
Skewed degree of distribution
Clustering
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
Fixing Random Graphs
The “stump” model
Growth model
Preferential attachment to nodes with larger
degrees
Does not fix clustering
Bipartite Graphs
Explains how
clustering arises
Analysis sometimes
gives good estimates
of clustering, but for
others they do not
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
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
Terminology
Neighborhood
Set of points which can be reached following paths
of r or less.
Very Brief Trace of the Proof
Result