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