Higher order clustering coefficients in Barabási–Albert networks

Higher order clustering coefficients C(x) are introduced for random networks. The coefficients express probabilities that the shortest distance between any two nearest neighbours of a certain vertex i equals x, when one neglects all paths crossing the node i. Using C(x) we found that in the Barabási–Albert (BA) model the average shortest path length in a node's neighbourhood is smaller than the equivalent quantity of the whole network and the remainder depends only on the network parameter m. Our results show that small values of the standard clustering coefficient in large BA networks are due to random character of the nearest neighbourhood of vertices in such networks.