Distinct Degrees and Their Distribution in Complex Networks

We investigate a variety of statistical properties associated with the number of distinct degrees that exist in a typical network for various classes of networks. For a single realization of a network with N nodes that is drawn from an ensemble in which the number of nodes of degree k has an algebraic tail, N_k ~ N/k^nu for k>>1, the number of distinct degrees grows as N^{1/nu}. Such an algebraic growth is also observed in scientific citation data. We also determine the N dependence of statistical quantities associated with the sparse, large-k range of the degree distribution, such as the location of the first hole (where N_k=0), the last doublet (two consecutive occupied degrees), triplet, dimer (N_k=2), trimer, etc.

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