  In many social networks it is a useful assumption that, regarding a given
quality, an underlying hierarchy of the connected individuals exists, and that
the outcome of interactions is to some extent determined by the relative
positions in the hierarchy. We consider a simple and broadly applicable method
of estimating individual positions in a linear hierarchy, and the corresponding
uncertainties. The method relies on solving a linear system characterized by a
modified Laplacian matrix of the underlying network of interactions, and is
equivalent to finding the equilibrium configuration of a system of directed
linear springs. We provide a simple first-order approximation to the exact
solution, which can be evaluated in linear time. The uncertainty of the
hierarchy estimate is determined by the network structure and the potantial
energy of the corresponding spring system in equilibrium. The method
generalizes straightforwardly to multidimensional hierarchies and higher-order,
non-pairwise interactions.
