  Various quantum-walk based algorithms have been proposed to analyse and rank
the centrality of graph vertices. However, issues arise when working with
directed graphs --- the resulting non-Hermitian Hamiltonian leads to
non-unitary dynamics, and the total probability of the quantum walker is no
longer conserved. In this paper, we discuss a method for simulating directed
graphs using PT-symmetric quantum walks, allowing probability conserving
non-unitary evolution. This method is equivalent to mapping the directed graph
to an undirected, yet weighted, complete graph over the same vertex set, and
can be extended to cover interdependent networks of directed graphs. Previous
work has shown centrality measures based on the CTQW provide an
eigenvector-like quantum centrality; using the PT-symmetric framework, we
extend these centrality algorithms to directed graphs with a significantly
reduced Hilbert space compared to previous proposals. In certain cases, this
centrality measure provides an advantage over classical algorithms used in
network analysis, for example by breaking vertex rank degeneracy. Finally, we
perform a statistical analysis over ensembles of random graphs, and show strong
agreement with the classical PageRank measure on directed acyclic graphs.
