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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

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Small World Graphs

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Small world graphs
Small World Graphs

Amber Rice


Defining a small world graph
Defining a Small World Graph

  • Relatively HIGH Clustering Coefficient

  • Relatively LOW Characteristic Path Length


Clustering coefficient
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 coefficient1
Clustering Coefficient

  • C =

  • Where:

  • “triangles” are K graphs

  • “connected triples” are nonisomorphic paths of length two

3


Finding clustering coefficient
Finding Clustering Coefficient

One Triangle

8 Connected Triples

So the Clustering Coefficient is 3/8.


Characteristic path length
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
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
Conclusions

  • New topic

    • Not much information

  • Likely to be very important in the future

  • My honors project

    • Social networks on campus


References
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


Small world graphs

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


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