### Abstract

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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 astochastic fixed-point equation of the form

R=^{D}Σ_{i=1}^{N} C_{i}R_{i}+ Q ,

where (Q,N, {C_{i}}) is a real-valued vector with N ∈ {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 | English |
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Pages (from-to) | 237-274 |

Number of pages | 38 |

Journal | Random Structures and Algorithms |

Volume | 51 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Sep 2017 |

### Keywords

- PageRank
- ranking algorithms
- irected conﬁguration model
- complex networks
- stochastic ﬁxed-point equations
- Weighted branching processes
- power laws

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## Cite this

*Random Structures and Algorithms*,

*51*(2), 237-274. https://doi.org/10.1002/rsa.20700