PageRank on inhomogeneous random digraphs

Abstract

We study the typical behavior of a generalized version of Google's PageRank algorithm on a large family of inhomogeneous random digraphs. This family includes as special cases directed versions of classical models such as the Erd\"os-R\'enyi model, the Chung-Lu model, the Poissonian random graph and the generalized random graph, and is suitable for modeling scale-free directed complex networks where the number of neighbors a vertex has is related to its attributes. In particular, we show that the rank of a randomly chosen node in a graph from this family converges weakly to the attracting endogenous solution to the stochastic fixed-point equation $$\mathcal{R} \stackrel{\mathcal{D}}{=} \sum_{i=1}^{\mathcal{N}} \mathcal{C}_i \mathcal{R}_i + \mathcal{Q},$$ where $(\mathcal{N}, \mathcal{Q}, \{\mathcal{C}_i\}_{i \geq 1})$ is a real-valued vector with $\mathcal{N}\in \{0,1,2,...\}$, the $\{ \mathcal{R}_i\}$ are i.i.d.~copies of $\mathcal{R}$, independent of $(\mathcal{N}, \mathcal{Q}, \{\mathcal{C}_i\}_{i \geq 1})$, with $\{ \mathcal{C}_i\}$ i.i.d.~and independent of $(\mathcal{N}, \mathcal{Q})$; $\stackrel{\mathcal{D}}{=}$ denotes equality in distribution. This result can then be used to provide further evidence of the power-law behavior of PageRank on scale-free graphs.