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Small World Graphs

This overview explores the concept of small world graphs, illustrated through the clustering coefficient and characteristic path length measures. The clustering coefficient indicates the degree to which vertices cluster together, enhancing connectivity. For example, if A is connected to B, and B to C, the likelihood of A connecting to C increases significantly. The characteristic path length reflects the average steps required to traverse the shortest paths between vertex pairs. As social, biological, and technological networks grow in importance, understanding these metrics becomes increasingly vital for analyzing complex systems.

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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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