Cayley graphs as models of deterministic small-world networks
Information Processing Letters
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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 P驴(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 P驴(h)C(h) ≤ clogN/ N, where cis a small constant that seldom exceeds 1. This result is significant because it shows that the product P驴(h)C(h) is upper-bounded by a value that is considerably smaller than the product of maximum values for P驴(h) and C(h).