  We present a physically-inspired model and an efficient algorithm to infer
hierarchical rankings of nodes in directed networks. It assigns real-valued
ranks to nodes rather than simply ordinal ranks, and it formalizes the
assumption that interactions are more likely to occur between individuals with
similar ranks. It provides a natural statistical significance test for the
inferred hierarchy, and it can be used to perform inference tasks such as
predicting the existence or direction of edges. The ranking is obtained by
solving a linear system of equations, which is sparse if the network is; thus
the resulting algorithm is extremely efficient and scalable. We illustrate
these findings by analyzing real and synthetic data, including datasets from
animal behavior, faculty hiring, social support networks, and sports
tournaments. We show that our method often outperforms a variety of others, in
both speed and accuracy, in recovering the underlying ranks and predicting edge
directions.
