### 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 |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 39 |

Publication status | Published - 16 Jun 2015 |

### Publication series

Name | Memorandum of the Department of Applied Mathematics |
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No. | 2046 |

ISSN (Print) | 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

## Cite this

Chen, N., Litvak, N., & Olvera-Cravioto, M. (2015).

*Ranking algorithms on directed configuration networks*. (Memorandum of the Department of Applied Mathematics; No. 2046). Enschede: University of Twente, Department of Applied Mathematics.