  The quantization of the PageRank algorithm is a promising tool for a future
quantum internet. Here we present a modification of the quantum PageRank
introducing arbitrary phase rotations (APR) in the underlying Szegedy's quantum
walk. We define three different APR schemes with only one phase as a degree of
freedom. We have analyzed the behavior of these algorithms in a small generic
graph, observing that a decrease of the phase reduces the standard deviation of
the instantaneous PageRank, so the nodes of the network can be distinguished
better. However, the algorithm takes more time to converge, so the phase can
not be decreased arbitrarily. With these results we choose a concrete value for
the phase to later apply the algorithm to complex scale-free graphs. In these
networks, the original quantum PageRank is able to break the degeneracy of the
residual nodes and detect secondary hubs that the classical algorithm
suppresses. Nevertheless, not all of the detected secondary hubs are real
according to the PageRank's definition. Some APR schemes can overcome this
problem, restoring the degeneration of the residual nodes and highlighting the
truly secondary hubs of the networks. Finally, we have studied the stability of
the new algorithms. The original quantum algorithm was known to be more stable
than the classical. We have found that one of our algorithms, whose PageRank
distribution resembles the classical one, has a stability similar to the
original quantum algorithm.
