# Ranking algorithms on directed configuration networks

Ningyuan Chen, Nelli Litvak, Mariana Olvera-Cravioto

Research output: Book/ReportReportOther research output

## Abstract

This paper studies the distribution of a family of rankings, which includes Google’s PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable $R^*$ that can be written as a linear combination of i.i.d. copies of the endogenous solution to a stochastic fixed point equation of the form $R \stackrel {D}{=} \sum^N _{i=1} C_iR_i + Q,$ where $(Q,N, \{C_i\})$ is a real-valued vector with $N \in \{0, 1, 2, ... \}$, $P(|Q| > 0) > 0$, and the $\{R_i\}$ are i.i.d. copies of $R^*$, independent of $(Q,N, \{C_i\})$. Moreover, we provide precise asymptotics for the limit $R^*$, which when the in-degree distribution in the directed configuration model has a power law imply a power law distribution for $R^*$ with the same exponent.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 39 Published - 16 Jun 2015

### Publication series

Name Memorandum of the Department of Applied Mathematics 2046 1874-4850

## Keywords

• PageRank
• Stochastic fixed-point equations
• Ranking algorithms
• Weighted branching processes
• Directed configuration model
• Complex networks
• IR-96268
• METIS-312642
• MSC-68P20
• MSC-60J80
• MSC-05C80
• EWI-26087
• Power laws