PageRank singular-gap conjecture

Let $G$ be a directed graph with $d_{\min}^{out}>0$. For a parameter $\alpha$ and associated PageRank random walk, let $\mathbf{R}$ be the probability transition matrix, given by

$$\mathbf{R}=\frac{\alpha}{n-1}(\mathbf{J}-\mathbf{I})+(1-\alpha)\mathbf{P},$$

where $\mathbf{J}$ is the all-one matrix and $\mathbf{P}$ is the transition matrix obtained by choosing an outgoing arc uniformly at random. PageRank singular-gap conjecture. There is a universal constant $k$ such that

$$\sigma_2(\mathbf{I}-\mathbf{R})^{-1}\leq \frac{k}{\alpha}.$$

The conjecture asserts that, like the spectral gap, the PageRank parameter controls the singular gap uniformly over all directed graphs with positive minimum out-degree. The authors note that examples based on Gambler's ruin imply $k\geq 1.65637$, while the general bound remains open.