  The notions of subgraph centrality and communicability, based on the
exponential of the adjacency matrix of the underlying graph, have been
effectively used in the analysis of undirected networks. In this paper we
propose an extension of these measures to directed networks, and we apply them
to the problem of ranking hubs and authorities. The extension is achieved by
bipartization, i.e., the directed network is mapped onto a bipartite undirected
network with twice as many nodes in order to obtain a network with a symmetric
adjacency matrix. We explicitly determine the exponential of this adjacency
matrix in terms of the adjacency matrix of the original, directed network, and
we give an interpretation of centrality and communicability in this new
context, leading to a technique for ranking hubs and authorities. The matrix
exponential method for computing hubs and authorities is compared to the well
known HITS algorithm, both on small artificial examples and on more realistic
real-world networks. A few other ranking algorithms are also discussed and
compared with our technique. The use of Gaussian quadrature rules for
calculating hub and authority scores is discussed.
