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Global Clustering Coefficient

The global clustering coefficient C of a graph G is the ratio of the number of closed trails of length 3 to the number of paths of length two in G.

Let A be the adjacency matrix of G. The number of closed trails of length 3 is equal to three times the number of triangles c_3 (i.e., graph cycles of length 3), given by

c_3=1/6Tr(A^3)
(1)

and the number of graph paths of length 2 is given by

p_2=1/2(A^2-sum_(ij)diag(A^2)),
(2)

so the global clustering coefficient is given by

C=(3c_3)/(p_2)=(Tr(A^3))/(A^2-sum_(ij)diag(A^2)).
(3)

It is implemented in the Wolfram Language as GlobalClusteringCoefficient[g].

See also

Local Clustering Coefficients, Mean Clustering Coefficient

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References

Luce, R. D. and Perry, A. D. "A Method of Matrix Analysis of Group Structure." Psychometrika 14, 95-116, 1949.Wang, Y.; Ghumare, E.; Vandenberghe, R.; and Dupont, P. "Comparison of Different Generalizations of Clustering Coefficient and Local Efficiency for Weighted Undirected Graphs." Neural Comput. 29, 313-331, 2017.Wasserman, S. and Faust, K. Social Network Analysis: Methods and Applications. Cambridge, England: Cambridge University Press, p. 243, 1994.

Cite this as:

Weisstein, Eric W. "Global Clustering Coefficient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GlobalClusteringCoefficient.html

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