Small World Graphs

1 / 15

# Small World Graphs - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Small World Graphs' - cayla

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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, 2 mean(B) = 6/4

C – 1, 1, 1, 1 mean(C) = 4/4

D – 1, 2, 2, 2 mean(D) = 7/4

E – 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

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

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