Extended Clustering Coefficients : Generalization of Clustering Coefficients in Small - World Networks

The clustering coefficient C of a network, which is a measure of direct connectivity between neighbors of the various nodes, ranges from 0 (for no connectivity) to 1 (for full connectivity). We define extended clustering coefficients C(h) of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employed in computing the ordinary clustering coefficient C = C(1). Based on known results about the distance distribution Pδ (h) in a network, that is, the probability that a randomly chosen pair of vertices have distance h, we derive and experimentally validate the law Pδ (h)C(h) ≤ c log N / N, where c is a small constant that seldom exceeds 1. This result is significant because it shows that the product Pδ (h)C(h) is upper-bounded by a value that is considerably smaller than the product of maximum values for Pδ (h) and C(h). Extended clustering coefficients and laws that govern them offer new insights into the structure of small-world networks and open up avenues for further exploration of their properties.