Node ranking in labeled networks

The entities in directed networks arising from real-world interactions are often naturally organized under some hierarchical structure. Given a directed, weighted, graph with edges and node labels, we introduce ranking problem where the obtained hierarchy should be described using node labels. Such method has the advantage to not only rank the nodes but also provide an explanation for such ranking. To this end, we define a binary tree called label tree, where each leaf represents a rank and each non-leaf contains a single label, which is then used to partition, and consequently, rank the nodes in the input graph. We measure the quality of trees using agony score, a penalty score that penalizes the edges from higher ranks to lower ranks based on the severity of the violation. We show that the problem is NP-hard, and even inapproximable if we limit the size of the label tree. Therefore, we resort to heuristics, and design a divide-and-conquer algorithm which runs in $\bigO{(n + m) \log n + \ell R}$, where $R$ is the number of node-label pairs in the given graph, $\ell$ is the number of nodes in the resulting label tree, and $n$ and $m$ denote the number of nodes and edges respectively. We also report an experimental study that shows that our algorithm can be applied to large networks, that it can find ground truth in synthetic datasets, and can produce explainable hierarchies in real-world datasets.