Distinct Degrees and Their Distribution in Complex Networks

TL;DR

This paper analyzes the distribution of distinct degrees in complex networks, revealing algebraic growth patterns and statistical properties related to degree distribution, with implications for understanding network structure and citation data.

Contribution

It introduces a detailed analysis of the growth and distribution of distinct degrees in networks, including algebraic growth laws and statistical measures for large-degree ranges.

Findings

Number of distinct degrees grows as N^{1/nu} in networks with algebraic tail distributions.

Identifies statistical properties of large-degree ranges, such as the location of the first hole and last doublet.

Observes similar algebraic growth patterns in scientific citation data.

Abstract

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.