  We develop a complete theory for the combinatorics of walk-counting on a
directed graph in the case where each backtracking step is downweighted by a
given factor. By deriving expressions for the associated generating functions,
we also obtain linear systems for computing centrality measures in this
setting. In particular, we show that backtrack-downweighted Katz-style network
centrality can be computed at the same cost as standard Katz. Studying the
limit of this centrality measure at its radius of convergence also leads to a
new expression for backtrack-downweighted eigenvector centrality that
generalizes previous work to the case where directed edges are present. The new
theory allows us to combine advantages of standard and nonbacktracking cases,
avoiding localization while accounting for tree-like structures. We illustrate
the behaviour of the backtrack-downweighted centrality measure on both
synthetic and real networks.
