small world graphs

Small World GraphsAmber Rice

Defining a Small World Graph

Relatively HIGH Clustering Coefficient

Relatively LOW Characteristic Path Length

Clustering Coefficient

Measure of degree to which vertices in a graph tend to cluster together

If A is connected to B and B is connected to C, then there’s a heightened probability that A is connected to C.

Clustering Coefficient

C =

Where:

• “triangles” are K graphs

• “connected triples” are nonisomorphic paths of length two

3

Finding Clustering Coefficient

One Triangle

8 Connected Triples

So the Clustering Coefficient is 3/8.

Characteristic Path Length

The average number of “steps” along the shortest paths for all possible pairs of vertices in the graph

The median of the means of shortest distances between all pairs of vertices

Finding Characteristic Path Length

First, find the distances between all the vertices and each average length.

A – 1, 1, 2, 2 mean(A) = 6/4B – 1, 1, 2, 2 mean(B) = 6/4C – 1, 1, 1, 1 mean(C) = 4/4D – 1, 2, 2, 2 mean(D) = 7/4E – 1, 2, 2, 2 mean(E) = 7/4

Next, take the median of the averages.

Median ( 4/4, 6/4, 6/4, 7/4, 7/4 ) = 6/4

A

C

D

B E

So, the Characteristic Path Length of this graph is 6/4.

Information Networks

Biological Networks

Technological Networks

Social Networks

Six Degrees of Separation

Conclusions

New topicNot much information

Likely to be very important in the future

My honors projectSocial networks on campus

References

Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45(2):167-256.

http://www.cmth.bnl.gov/~maslov/citerank/images/CitationNetworkDiagram1.gif

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http://onlineaikido.com/blog_resources/pictures/neural_network_3.jpg

http://www.technologyreview.com/articlefiles/fairley80701.jpg

http://www.barnabu.co.uk/wp-content/uploads/usa-air-routes-google-earth.JPG

http://polymer.bu.edu/~amaral/Sex_partners/idahlia_web.jpg

http://film-buff.tripod.com/kevinbacon.jpg

http://insanityoverrated.files.wordpress.com/2009/02/six-degrees1.jpg

http://en.wikipedia.org/wiki/Small_world_experiment

http://en.wikipedia.org/wiki/Small-world_network

http://en.wikipedia.org/wiki/Clustering_coefficient

http://getoutfoxed.com/files/small-world-ring-with-rando.png

http://www.amazon.com/Small-Worlds-Duncan-J-Watts/dp/0691005419