Global Clustering Coefficient

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

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

(1)

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

(2)

so the global clustering coefficient is given by

(3)

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

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.