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 similar to 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.