Searching for a source of difference in graphical models

We look at a two-sample problem within the framework of decomposable graphical models. When the global hypothesis of equality of two distributions is rejected, the interest is usually in localizing the source of difference. Motivated by the idea that diseases can be seen as system perturbations, and by the need to distinguish between the origin of perturbation and components affected by the perturbation, we introduce the concept of a minimal seed set, and its graphical counterpart a graphical seed set. They intuitively consist of variables driving the difference between the two conditions. We propose a simple testing procedure, linear in the number of nodes, to estimate the graphical seed set from data. We illustrate our approach in the context of gene set analysis, where we show that is possible to zoom in on the origin of perturbation in a gene network.