  The quantum SearchRank algorithm is a promising tool for a future quantum
search engine based on PageRank quantization. However, this algorithm loses its
functionality when the $N/M$ ratio between the network size $N$ and the number
of marked nodes $M$ is sufficiently large. We propose a modification of the
algorithm, replacing the underlying Szegedy quantum walk with a semiclassical
walk. To maintain the same time complexity as the quantum SearchRank algorithm
we propose a simplification of the algorithm. This new algorithm is called
Randomized SearchRank, since it corresponds to a quantum walk over a randomized
mixed state. The performance of the SearchRank algorithms is first analyzed on
an example network, and then statistically on a set of different networks of
increasing size and different number of marked nodes. On the one hand, to test
the search ability of the algorithms, it is computed how the probability of
measuring the marked nodes decreases with $N/M$ for the quantum SearchRank, but
remarkably it remains at a high value around $0.9$ for our semiclassical
algorithms, solving the quantum SearchRank problem. The time complexity of the
algorithms is also analyzed, obtaining a quadratic speedup with respect to the
classical ones. On the other hand, the ranking functionality of the algorithms
has been investigated, obtaining a good agreement with the classical PageRank
distribution. Finally, the dependence of these algorithms on the intrinsic
PageRank damping parameter has been clarified. Our results suggest that this
parameter should be below a threshold so that the execution time does not
increase drastically.
