Directed Probabilistic Watershed

The Probabilistic Watershed is a semi-supervised learning algorithm applied on undirected graphs. Given a set of labeled nodes (seeds), it defines a Gibbs probability distribution over all possible spanning forests disconnecting the seeds. It calculates, for every node, the probability of sampling a forest connecting a certain seed with the considered node. We propose the "Directed Probabilistic Watershed", an extension of the Probabilistic Watershed algorithm to directed graphs. Building on the Probabilistic Watershed, we apply the Matrix Tree Theorem for directed graphs and define a Gibbs probability distribution over all incoming directed forests rooted at the seeds.