Small World Graphs
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Transcript of Small World Graphs
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Small World GraphsAmber Rice
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Defining a Small World Graph
Relatively HIGH Clustering Coefficient
Relatively LOW Characteristic Path Length
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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.
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Clustering Coefficient
C =
Where:
• “triangles” are K graphs
• “connected triples” are nonisomorphic paths of length two
3
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Finding Clustering Coefficient
One Triangle
8 Connected Triples
So the Clustering Coefficient is 3/8.
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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
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Finding Characteristic Path Length
First, find the distances between all the vertices and each average length.
A – 1, 1, 2, 2 mean(A) = 6/4B – 1, 1, 2, 2 mean(B) = 6/4C – 1, 1, 1, 1 mean(C) = 4/4D – 1, 2, 2, 2 mean(D) = 7/4E – 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
C
D
B E
So, the Characteristic Path Length of this graph is 6/4.
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Information Networks
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Biological Networks
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Technological Networks
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Social Networks
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Six Degrees of Separation
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Conclusions
New topicNot much information
Likely to be very important in the future
My honors projectSocial networks on campus
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References
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45(2):167-256.
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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
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http://www.amazon.com/Small-Worlds-Duncan-J-Watts/dp/0691005419