On the formation of degree and cluster-degree correlations in scale-free networks

The cluster-degree of a vertex is the number of connections among the neighbors of this vertex. In this paper we study the cluster-degree of the generalized Barabási-Albert model (GBA model) whose exponent of degree distribution ranges from 2 to ∞. We present the mean-field rate equation for clustering and obtain analytically the degree-dependence of the cluster-degree. We study the distribution of the cluster-degree, which is size dependent but the tail is kept invariant for different degree exponents in the GBA model. In addition, for the degree dependence of the clustering coefficient, very different behaviors arise for different cases of the GBA model. The physical sense of the invariance property of cluster-degree is explained and more general cases are discussed. All the above theoretical results are verified by simulation.