Clustering Coefficients

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: Local Clustering coefficient of node 𝑖 : C i = 3 # 𝑜𝑓 𝐾 3 𝑡ℎ𝑎𝑡 𝑖𝑛𝑙𝑢𝑑𝑒 𝑖 # 𝑜𝑓 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑟𝑖𝑝𝑙𝑒𝑠 𝑐𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑎𝑡 𝑖 Avg. clustering coeff. of G: 𝐶 𝑤𝑠 𝐺 = 1 𝑛 𝑖∈𝑉(𝐺) 𝐶 𝑖 (Global) Clustering coeff. of G: 𝐶 𝐺 = 3 # 𝑜𝑓 𝐾 3 # 𝑜𝑓 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑟𝑖𝑝𝑙𝑒𝑠 Mention that if we use the second definition of clustering coefficient (Eq. 7.44) we get different results – 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.

An example

Clustering coeff distribution example in Gephi 𝐶 ℎ = 1 5 2 One triangle

Statistics for real networks 𝐶= clustering coefficient 𝐶 𝑤𝑠 = ave clustering coefficient

Observed vs. expected values for 𝑪 𝒘𝒔 Network Observed Expected value based on random graphs with the same number of vertices and edges 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. 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

𝐶 𝑤𝑠 as a function of the network size 𝐶 𝑤𝑠 𝐵𝐴 ~ 𝑁 −3/4 𝐶 𝑤𝑠 𝑟𝑎𝑛𝑑𝑜𝑚 ~ 𝑁 −1 𝑪 𝒘𝒔 : average clustering coeff Source: R. Albert and A. L. Barab´asi. Statistical mechanics of complex networks. Reviews of Modern Physics, 74:47–97, 2002

𝑪 𝒘𝒔 as a function of degree 𝑪 𝒘𝒔 : average clustering coeff PPI: protein-protein interaction netw. SF = scale free synthetic network Source: N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/0404017, 2004.

Local clustering coefficient If we calculate the local clustering coefficient 𝐶 𝑖 of each vertex 𝑖 in a network: on average, vertices of higher degree exhibit lower local clustering. Internet network. For nodes of degree 𝑘 : 𝐶 𝑖 𝑘 =𝑘 −α , where .75 ≤α ≤ 1 Thoughts on why this occurs?

Local clustering coefficient Possible explanations for the decrease in 𝐶 𝑖 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 𝐶 𝑖 Communities are generally connected by large degree nodes, and being a connector will decrease its value of 𝐶 𝑖 of these large degree nodes.

Extensions Clustering coefficient measures the density of 𝐾 3 in networks The density of other small groups of vertices can be studied as well (density of motifs)

Graphlet frequency in Scale Free netw Source: N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv: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 Calculation Count all paths of length two, count how many of these are closed, and take their ratio i j k closed triad Probability that two vertices with a common adjacent vertex are themselves adjacent Slide courtesy of Dr. Tim Chung