Simple estimation of hierarchical positions and uncertainty in networks of asymmetric interactions

Abstract: 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.