Rankings in directed configuration models with heavy tailed in-degrees

TL;DR

This paper investigates the extremal values of stationary distributions and PageRank scores in directed configuration models with heavy-tailed in-degree distributions, establishing power-law behaviors and extending previous bounded-degree results.

Contribution

It provides the first analysis of extremal stationary distribution and PageRank behaviors in heavy-tailed directed random graphs, confirming the power-law hypothesis.

Findings

Maximum stationary distribution aligns with maximum in-degree.

Stationary distribution and PageRank follow power-law with same index.

Results extend previous bounded-degree models to heavy-tailed in-degrees.

Abstract

We consider the extremal values of the stationary distribution of sparse directed random graphs with given degree sequences and their relation to the extremal values of the in-degree sequence. The graphs are generated by the directed configuration model. Under the assumption of bounded $(2+\eta)$-moments on the in-degrees and of bounded out-degrees, we obtain tight comparisons between the maximum value of the stationary distribution and the maximum in-degree. Under the further assumption that the order statistics of the in-degrees have a power-law behavior, we show that the extremal values of the stationary distribution also have a power-law behavior with the same index. In the same setting, we prove that these results extend to the PageRank scores of the random digraph, thus confirming a version of the so-called power-law hypothesis. Along the way, we establish several facts about the model, including the mixing time cutoff and the characterization of the typical values of the stationary distribution, which were previously obtained under the assumption of bounded in-degrees.