Nonstandard regular variation of in-degree and out-degree in the preferential attachment model

Abstract

Abstract. For the directed edge preferential attachment network growth model studied by Bollobás et al. (2003) and Krapivsky and Redner (2001), we prove that the joint distribution of in-degree and out-degree has jointly regularly varying tails. Typically the marginal tails of the in-degree distribution and the out-degree distribution have different regular variation indices and so the joint regular variation is non-standard. Only marginal regular variation has been previously established for this distribution in the cases where the marginal tail indices are different.

Connectivity of a general class of inhomogeneous random digraphs

“…constitutes a series of classical results within the random graph literature. Closely related to the results presented here for directed graphs, are the existence of a giant strongly connected component and giant weak-component in the directed configuration model [11,18,19], the existence of a giant strongly connected component in the deterministic directed kernel model with a finite number of types [3], the scale-free property on a directed preferential attachment model [28,31], and the limiting degree distributions in the directed configuration model [7]. 1 From a computational point of view, the work in [33] provides numerical algorithms to identify secondary structures on directed graphs.…”

On a Minimum Distance Procedure for Threshold Selection in Tail Analysis

“…Limit behavior of the degree counts in this linear PA model is studied in [26,35,36,42,43]. Two parametric estimation methods for this directed linear PA model are derived in [41], giving estimates of α in and α out by simply plugging in the estimated parameters into (4.5) and (4.6), respectively.…”

Tauberian theory for multivariate regularly varying distributions with application to preferential attachment networks

“…In fact, the limiting degree distribution (p ij ) in (4.2) generates a distribution that has jointly nonstandard regularly varying tails and the limit measure of regular variation has a density as shown in Samorodnitsky et al (2014). 4.2.…”

“…Notation and results summary. We summarize results and notation for the preferential attachment model from Samorodnitsky et al (2014).…”

“…The explicit structure and form in (4.8),(4.9) and (4.10) allowed analysis of the asymptotic multivariate power law structure performed in Samorodnitsky et al (2014). Absent such structure, if all one has is the joint generating function, one would have to rely on Tauberian analysis of the transform.…”