  This paper shows that pairwise PageRank orders emerge from two-hop walks. The
main tool used here refers to a specially designed sign-mirror function and a
parameter curve, whose low-order derivative information implies pairwise
PageRank orders with high probability. We study the pairwise correct rate by
placing the Google matrix $\textbf{G}$ in a probabilistic framework, where
$\textbf{G}$ may be equipped with different random ensembles for
model-generated or real-world networks with sparse, small-world, scale-free
features, the proof of which is mixed by mathematical and numerical evidence.
We believe that the underlying spectral distribution of aforementioned networks
is responsible for the high pairwise correct rate. Moreover, the perspective of
this paper naturally leads to an $O(1)$ algorithm for any single pairwise
PageRank comparison if assuming both $\textbf{A}=\textbf{G}-\textbf{I}_n$,
where $\textbf{I}_n$ denotes the identity matrix of order $n$, and
$\textbf{A}^2$ are ready on hand (e.g., constructed offline in an incremental
manner), based on which it is easy to extract the top $k$ list in $O(kn)$, thus
making it possible for PageRank algorithm to deal with super large-scale
datasets in real time.
