Generalized PageRank on Directed Configuration Networks

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 attracting endogenous solution to a stochastic fixed-point equation of the form R (D)=Sigma(N)(i=1) CiRi + Q, where (Q, N, {C-i}) is a real-valued vector with N epsilon {0, 1, 2,...}, P(vertical bar Q vertical bar> 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. (C) 2016Wiley Periodicals, Inc.