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Section 8.6: Clustering Coefficients
By: Ralucca Gera, NPS Most pictures are from Newmanβs textbook
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Clustering coefficients for real networks
The clustering coefficients measure the average probability that two neighbors of a vertex are themselves neighbors (a measure of the density of triangles in a network). There are three versions: Clustering coeff. of G: πΆ πΊ = 3 # ππ πΎ 3 # ππ πππππππ‘ππ π‘ππππππ Local Clustering coefficient: C i = 3 # ππ πΎ 3 π‘βππ‘ ππππ’ππ π # ππ πππππππ‘ππ π‘ππππππ ππππ‘ππππ ππ‘ π Avg. clustering coeff. of G: πΆ π€π πΊ = 1 π πβπ(πΊ) πΆ π Mention that if we use the second definition of clustering coefficient (Eq. 7.44) we get different results β usually overestimate of C Also some studies instead of using the expected clustering coefficient given from the equation on this slide, they use the one for the Poissonian random graph (i.e., edge density), which again will give discrepancies in the results.
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An example
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Clustering coeff distrrribution example in Gephi
πΆ β = One triangle
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Statistics for real networks
πΆ= clustering coefficient πΆ π€π = ave clustering coefficient
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Observed vs. expected values for πͺ ππ
Network Observed Expected value based on random graphs Collaboration of physicists C = .45 C= .0023 Food webs C = . 16 (or .12) similar Internet C = .012 C = .84 ws ws Source: N. Przulj. Graph theory analysis of protein-protein interactions
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Explanations? The exact reason for this phenomenon is not well understood, but it may be connected with The structure of the graph (since the random one lacks it) The formation of groups or communities E.g., in social networks ο triadic closure
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πΆ π€π as a function of the network size
πΆ π€π π΅π΄ ~ π β3/4 πΆ π€π ππππππ ~ π β1 Source: R. Albert and A. L. BarabΒ΄asi. Statistical mechanics of complex networks. Reviews of Modern Physics, 74:47β97, 2002
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πͺ ππ as a function of degree
PPI: protein-protein interaction netw. SF = scale free synthetic network Source: N. PrΛzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/ , 2004.
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Section 8.6.1:Local clustering coefficient
If we calculate the local clustering coefficient of each vertex in a network an interesting pattern occurs On average, vertices of higher degree exhibit lower local clustering Internet network. For nodes of degree π : πΆ π π =π βΞ± , where .75 β€Ξ± β€ 1 Thoughts on why this occurs?
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Section 8.6.1:Local clustering coefficient
Possible explanations for the decrease in πΆ π as degree increases: Vertices tend to group in communities, sharing mostly neighbors within the same community Thus some vertices have small/large degree based on the size of the community Smaller communities are denser ο larger πΆ π Communities are generally connected by large degree nodes, and being a connector will decrease its value of πΆ π of these large degree nodes.
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Extensions Clustering coefficient measures the density of πΎ 3 in networks The density of other small groups of vertices can be studied as well (density of motifs)
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Graphlet frequency in Scale Free netw
Source: N. PrΛzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/ , 2004.
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