Graph Operations and Zipfian Degree Distributions

The probability distribution on a set S = { 1, 2, . . . , n } defined by Pr(k) = 1/(Hnk), where Hn in the nth harmonic number, is commonly called a Zipfian distribution. In this note we look at the degree distribution of graphs created from graphs with Zipfian degree distributions using some standard graph theoretical operations. Introduction: The probability distribution on a set S = { 1, 2, . . . , n } defined by Pr(k) = 1/(Hnk), where Hn in the nth harmonic number, is commonly called a Zipfian distribution, after George Zipf (1902 1950), who observed it in relation to word frequency in English. (See, for example, the discussion in [1]). Recently, it has been observed (see, e.g., [4] and [5]) that the worldwide web, modeled as a directed graph, has a degree sequence that is vaguely ”Zipf-like”. (Modeling the web as a graph is done in the obvious way, i.e., a website is a node and links between sites are edges.) In an earlier paper (see [2]) the question of the existence of graphs whose degree distributions are exactly Zipfian was addressed. Here we look at the degree distributions that result when graphs are constructed from Zipfian-distribution graphs using two graph theoretic operations. We consider undirected graphs with no multiple edges and no isolated vertices, and show that the degree distributions of the resulting graphs are have either a Zipfian distribution, or