Open Access
February 2020 Local weak convergence for PageRank
Alessandro Garavaglia, Remco van der Hofstad, Nelly Litvak
Ann. Appl. Probab. 30(1): 40-79 (February 2020). DOI: 10.1214/19-AAP1494


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. 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 to preferential attachment models.


Download Citation

Alessandro Garavaglia. Remco van der Hofstad. Nelly Litvak. "Local weak convergence for PageRank." Ann. Appl. Probab. 30 (1) 40 - 79, February 2020.


Received: 1 March 2018; Revised: 1 February 2019; Published: February 2020
First available in Project Euclid: 25 February 2020

zbMATH: 07200523
MathSciNet: MR4068306
Digital Object Identifier: 10.1214/19-AAP1494

Primary: 60B20
Secondary: 05C80

Keywords: Directed random graphs , Local weak convergence , PageRank

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.30 • No. 1 • February 2020
Back to Top