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Small World Graphs

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Small World Graphs

Amber 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) = 6/4

B – 1, 1, 2, 2mean(B) = 6/4

C – 1, 1, 1, 1mean(C) = 4/4

D – 1, 2, 2, 2mean(D) = 7/4

E – 1, 2, 2, 2mean(E) = 7/4

Next, take the median of the averages.

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

A

D

C

E

B

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

Conclusions

- New topic
- Not much information

- Likely to be very important in the future
- My honors project
- Social 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

http://www.bordalierinstitute.com/images/worldwideweb.jpeg

http://images.google.com/imgres?imgurl=http://cmore.soest.hawaii.edu/cruises/operex/images/terrestrial_food_web

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