Section 8 6 of Newmans book Clustering Coefficients

Section 8. 6 of Newman’s book: Clustering Coefficients By: Ralucca Gera, NPS Most pictures are from Newman’s textbook

Clustering coefficients for real networks •

An example 3

Clustering coeff distribution example in Gephi One triangle

Statistics for real networks 5

Network Observed Expected value based on random graphs ws with the same numberwsof vertices and edges Collaboration of physicists C = . 45 C=. 0023 Food webs C =. 16 (or. 12) similar Internet C = . 012 C =. 84 Source: N. Przulj. Graph theory analysis of protein-protein interactions. 2005. 6

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

Source: R. Albert and A. L. Barab´asi. Statistical mechanics of complex networks. Reviews of Modern Physics, 74: 47– 97, 2002 8

Source: N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? ar. Xiv: qbio. MN/0404017, 2004. 9

Section 8. 6. 1: Local clustering coefficient • Thoughts on why this occurs? 10

Section 8. 6. 1: Local clustering coefficient • 11

Extensions • 12

Graphlet frequency in Scale Free netw Source: N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? ar. Xiv: qbio. MN/0404017, 2004.

Global Clustering Coefficient • Fraction of the paths of length two in the graph that are closed – A “closed triad” is a closed path of length three through vertices i, j, and k i closed triad j k Probability that two vertices with a common adjacent vertex are themselves adjacent • Calculation – Count all paths of length two, count how many of these are closed, and take their ratio Slide courtesy of Dr. Tim Chung