  Four new centrality measures for directed networks based on unitary,
continuous-time quantum walks (CTQW) in $n$ dimensions -- where $n$ is the
number of nodes -- are presented, tested and discussed. The main idea behind
these methods consists in re-casting the classical HITS and PageRank algorithms
as eigenvector problems for symmetric matrices, and using these symmetric
matrices as Hamiltonians for CTQWs, in order to obtain a unitary evolution
operator. The choice of the initial state is also crucial. Two options were
tested: a vector with uniform occupation and a vector weighted w.r.t.~in- or
out-degrees (for authority and hub centrality, respectively). Two methods are
based on a HITS-derived Hamiltonian, and two use a PageRank-derived
Hamiltonian. Centrality scores for the nodes are defined as the average
occupation values. All the methods have been tested on a set of small, simple
graphs in order to spot possible evident drawbacks, and then on a larger number
of artificially generated larger-sized graphs, in order to draw a comparison
with classical HITS and PageRank. Numerical results show that, despite some
pathologies found in three of the methods when analyzing small graphs, all the
methods are effective in finding the first and top ten nodes in larger graphs.
We comment on the results and offer some insight into the good accordance
between classical and quantum approaches.
