A general probabilistic formulation referred to as ‘Conditional Harmonic Mixing’ is provided, in which links between classification nodes are directed, a conditional probability matrix is associated with each link, and where the numbers of classes can vary from node to node. A posterior class probability at each node is updated by minimizing a divergence between its distribution and that predicted by its neighbors. For arbitrary graphs, as long as each unlabeled point is reachable from at least one training point, a solution generally always exists, is unique, and can be found by solving a sparse linear system iteratively. In one aspect, an automated data classification system is provided. The system includes a data set having at least one labeled category node in the data set. A semi-supervised learning component employs directed arcs to determine the label of at least one other unlabeled category node in the data set.