Some comments on equation (1):

Even with many details not making clear sense, I venture to throw some light on this equation. Equation (1) looks similar to Boltzmann factor. The terms inside the exponential correspond to something like an energy function with some parameters. Equation (1) is assigning some probability to a graph with its particular configuration determined by Y conditioned on X. 'Lower' the energy of the configuration, greater the probability assigned to this particular configuration.