PageRank Asymptotics on Directed Preferential Attachment Networks

TL;DR

This paper analyzes the tail behavior of PageRank in directed preferential attachment networks, revealing a heavier power law tail than the in-degree distribution and contrasting with static models, with implications for network structure understanding.

Contribution

It provides the first explicit characterization of PageRank tail behavior in directed preferential attachment graphs, highlighting deviations from the power law hypothesis.

Findings

PageRank tail follows a power law with an explicit exponent.

The PageRank tail is heavier than the in-degree distribution tail.

The growth rate of PageRank for the oldest vertex is characterized.

Abstract

We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted {\em power law hypothesis}. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.