Clustering Coefficients

Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California [email protected] 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 the graph: # # 2. Local Clustering coefficient: 3 # C # 3. Avg. clustering coeff. of G: ∑ ∈ An example

3 Clustering coeff distrrribution example in Gephi 1 3 # C #

One triangle

Observed: Networks are globally sparse, but locally dense paths, only 5 displayed Statistics for real networks

= clustering coefficient = ave clustering coefficient

Newman, “The Structure and Function of Complex Networks” http://epubs.siam.org/doi/pdf/10.1137/S003614450342480 Observed vs. random

Network Observed Expected value based on random graphsws of similar wssize Collaboration of C = .45 C= .0023 physicists Food webs C = . 16 (or .12) similar Internet C = .012 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 Dependency to other parameters as a function of the network size (N)

Recall: ∑ ∈ Analytical prediction

BA model

ER model

Source: L. Barab´asi. http://barabasi.com/networksciencebook/chapter/5#diameter as a function of degree (k)

Recall: ∑ ∈ PPI: protein-protein interaction netw. SF = scale free synthetic network.

10 Source: N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/0404017, 2004. Local clustering coefficient

# Recall: C # Notice: higher degree nodes exhibit lower local clustering coeff (with larger variance as well) Thoughts on why this occurs?

Internet network.

For nodes of degree , the best fit is: α , where .75 α 1

11 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 may be connected by large degree nodes, and being a connector will decrease its value of of these large degree nodes. 12 Extensions • Clustering coefficient measures the density of in networks • Count the density of: Enumeration of all graphs of fixed n: – other motifs, – clique expansion (clique + edge). Frequency of cliques in clique+edge occurrences:

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

Benson, Yin, Leskovec, “Higher-order clustering coefficient” (2017) https://www.slideshare.net/arbenson/higherorder-clustering-coefficients-80864022 Graphlet frequency in Scale Free netw

14 Source: N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/0404017, 2004. Higher order clustering in WS networks

Benson, Yin, Leskovec, “Higher-order clustering coefficient” (2017) https://www.slideshare.net/arbenson/higherorder-clustering-coefficients-80864022 References

• Newman, “The Structure and Function of Complex Networks” http://epubs.siam.org/doi/pdf/10.1137/S003614450342480 • Source: L. Barabasi. http://barabasi.com/networksciencebook/chapter/5#diameter • Benson, Yin, Leskovec, “Higher-order clustering coefficient” (2017) https://www.slideshare.net/arbenson/higherorder- clustering-coefficients-80864022 • N. Prˆzulj, D. G. Corneil, and I. Jurisica. Modeling interactome: Scale free or geometric? arXiv:qbio. MN/0404017, 2004.