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Interactions in many real-world phenomena can be explained by a stronghierarchical structure. Typically, this structure or ranking is not known;instead we only have observed outcomes of the interactions, and the goal is toinfer the hierarchy from these observations. Discovering a hierarchy in thecontext of directed networks can be formulated as follows: given a graph,partition vertices into levels such that, ideally, there are only edges fromupper levels to lower levels. The ideal case can only happen if the graph isacyclic. Consequently, in practice we have to introduce a penalty function thatpenalizes edges violating the hierarchy. A practical variant for such penaltyis agony, where each violating edge is penalized based on the severity of theviolation. Hierarchy minimizing agony can be discovered in $O(m^2)$ time, andmuch faster in practice. In this paper we introduce several extensions toagony. We extend the definition for weighted graphs and allow a cardinalityconstraint that limits the number of levels. While, these are conceptuallytrivial extensions, current algorithms cannot handle them, nor they can beeasily extended. We solve the problem by showing the connection to thecapacitated circulation problem, and we demonstrate that we can compute theexact solution fast in practice for large datasets. We also introduce aprovably fast heuristic algorithm that produces rankings with competitivescores. In addition, we show that we can compute agony in polynomial time forany convex penalty, and, to complete the picture, we show that minimizinghierarchy with any concave penalty is an NP-hard problem.

Tiers for peers: a practical algorithm for discovering hierarchy in weighted networks