Motivated by a series of examples (the U.S. Ph.D. exchange in mathematics, co-starring relationships among top-billed actors in Hollywood, and co-citations between U.S. Supreme Court decisions), we study the general problem of calculating eigenvector centralities as generalized to temporal directed network data. We investigate a parsimonious extension of hub and authority scores to "multilayer" situations where multiple slices of network data connect similar groups of agents in different settings. By varying the coupling between layers, we develop a family of centrality measures whose values are localized in time in one extreme (weakly coupled) and constant across time in the other (strongly coupled). By deriving the first terms of the singular perturbation expansion in the strongly-coupled singular limit of this framework, we derive expressions for principled specification of the time-averaged centrality and the biggest centrality movers over the time period.   This work is in collaboration with Sean Myers, Dane Taylor, Elizabeth Leicht, Aaron Clauset and Mason Porter.