Local weak convergence for PageRank

TL;DR

This paper investigates the asymptotic behavior of PageRank scores in large networks using local weak convergence, extending the theory to directed graphs and applying it to various network models.

Contribution

It introduces a framework for analyzing the limiting distribution of PageRank in large directed graphs via local weak convergence, broadening understanding beyond specific models.

Findings

Proves existence of an asymptotic PageRank distribution in large graphs.

Provides a method to compute the limiting distribution from the local weak limit.

Applies the theory to configuration models, branching processes, and preferential attachment networks.

Abstract

PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called `power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees. Up to date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in- and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as preferential attachment models.