# Section 7.8: Groups of vertices - Naval Postgraduate School Section 8.6: Clustering Coefficients By: Ralucca Gera, NPS Most pictures are from Newmans textbook 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:

1. Clustering coeff. of G: 2. Local Clustering coefficient: 3. Avg. clustering coeff. of G: An example 3 Clustering coeff distrrribution example in Gephi h =

1 5 2 () One triangle paths, only 5 displayed Statistics for real networks = clustering coefficient

= ave clustering coefficient 5 Observed vs. expected values for 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 ws ws C = .84 6 Source: N. Przulj. Graph theory analysis of protein-protein interactions. 2005. 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 triadic closure 7 as a function of the network size Source: R. Albert and A. L. Barabasi. Statistical mechanics of

complex networks. Reviews of Modern Physics, 74:4797, 2002 8 as a function of degree PPI: protein-protein interaction netw. SF = scale free synthetic network Source: N. Przulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio.

MN/0404017, 2004. 9 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 : , where 1 Thoughts on why this occurs? 10 Section 8.6.1:Local clustering coefficient Possible explanations for the decrease in 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 larger Communities are generally connected by large degree nodes, and being a connector will decrease its value of of these large degree 11 nodes. Extensions Clustering coefficient measures the density of

in networks The density of other small groups of vertices can be studied as well (density of motifs) 12 Graphlet frequency in Scale Free netw Source: N. Przulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/0404017, 2004. 13