Ranking algorithms on directed configuration networks

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 $\mathcal{R}^*$ 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 $$\mathcal{R} \stackrel{\mathcal{D}}{=} \sum_{i=1}^{\mathcal{N}} \mathcal{C}_i \mathcal{R}_i + \mathcal{Q},$$ where $(\mathcal{Q}, \mathcal{N}, \{ \mathcal{C}_i\})$ is a real-valued vector with $\mathcal{N} \in \{0,1,2,\dots\}$, $P(|\mathcal{Q}| > 0) > 0$, and the $\{\mathcal{R}_i\}$ are i.i.d. copies of $\mathcal{R}$, independent of $(\mathcal{Q}, \mathcal{N}, \{ \mathcal{C}_i\})$. Moreover, we provide precise asymptotics for the limit $\mathcal{R}^*$, which when the in-degree distribution in the directed configuration model has a power law imply a power law distribution for $\mathcal{R}^*$ with the same exponent.