Universal Emergence of PageRank

TL;DR

This paper investigates the universal behavior of PageRank as the damping parameter approaches 1, revealing power law distributions and eigenvalue properties that govern convergence on large web networks.

Contribution

It analytically and numerically characterizes the universal features of PageRank emergence at high damping, highlighting the network's core-invariant subspace structure.

Findings

PageRank converges to a universal power law distribution on invariant subspaces.

The size distribution of invariant subspaces also follows a power law.

Eigenvalues near unity cause slow convergence, similar to spin glass systems.

Abstract

The PageRank algorithm enables to rank the nodes of a network through a specific eigenvector of the Google matrix, using a damping parameter $\alpha \in ]0,1[$. Using extensive numerical simulations of large web networks, with a special accent on British University networks, we determine numerically and analytically the universal features of PageRank vector at its emergence when $\alpha \rightarrow 1$. The whole network can be divided into a core part and a group of invariant subspaces. For $ \alpha \rightarrow 1$ the PageRank converges to a universal power law distribution on the invariant subspaces whose size distribution also follows a universal power law. The convergence of PageRank at $ \alpha \rightarrow 1$ is controlled by eigenvalues of the core part of the Google matrix which are extremely close to unity leading to large relaxation times as for example in spin glasses.