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