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

TL;DR

This paper proves that in a directed preferential attachment network, the joint distribution of in-degree and out-degree exhibits non-standard regular variation, with different tail behaviors for each degree type.

Contribution

It establishes the joint regular variation of in-degree and out-degree in the model, revealing non-standard tail behavior not previously demonstrated.

Findings

Joint distribution has regularly varying tails

Marginal tails have different regular variation indices

Joint regular variation is non-standard

Abstract

For the directed edge preferential attachment network growth model studied by Bollobas 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.