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Small World Graphs

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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  1. Small World Graphs Amber Rice

  2. Defining a Small World Graph • Relatively HIGH Clustering Coefficient • Relatively LOW Characteristic Path Length

  3. 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.

  4. Clustering Coefficient • C = • Where: • “triangles” are K graphs • “connected triples” are nonisomorphic paths of length two 3

  5. Finding Clustering Coefficient One Triangle 8 Connected Triples So the Clustering Coefficient is 3/8.

  6. 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

  7. 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.

  8. Information Networks

  9. Biological Networks

  10. Technological Networks

  11. Social Networks

  12. Six Degrees of Separation

  13. Conclusions • New topic • Not much information • Likely to be very important in the future • My honors project • Social networks on campus

  14. 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

  15. 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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