Power-law hypothesis for PageRank on undirected graphs

TL;DR

This paper investigates the distribution of PageRank in undirected graphs, proving it is bounded by node degree and generally has a lighter tail than the degree distribution, contrasting with directed models.

Contribution

It provides the first rigorous bounds on PageRank distribution in undirected graphs and compares these findings with directed models, highlighting fundamental differences.

Findings

PageRank in undirected graphs is upper bounded by node degree.

PageRank has a lighter tail than the degree distribution in undirected networks.

Contrasts the behavior of PageRank in undirected graphs with directed preferential attachment models.

Abstract

Based on observations in the web-graph, the power-law hypothesis states that PageRank has a power-law distribution with the same exponent as the in-degree. While this hypothesis has been analytically verified for many random graph models, such as directed configuration models and generalized random graphs, surprisingly it has recently been disproven for the directed preferential attachment model. In this paper, we prove that in undirected networks, the graph-normalized PageRank is always upper bounded by the degree. Furthermore, we prove that the corresponding (asymptotic) lower bound holds true under reasonable assumptions on the local weak limit, but not in general, and we provide a counterexample. Our result shows that PageRank always has a lighter tail than the degree, which contrasts the case of the directed preferential attachment model, where PageRank has a heavier tail instead. We end the paper with a discussion, where we extend our results to directed networks with a bounded ratio of in- and out-degrees, and reflect on our methods by contrasting the undirected and directed preferential attachment model.