  The main results of our work is determining the differences between limiting
properties in various models of quantum stochastic walks. In particular, we
prove that in the case of strongly connected and a class of weakly connected
directed graphs, local environment interaction evolution is relaxing, and in
the case of undirected graphs, global environment interaction evolution is
convergent. For other classes of directed graphs we show, that the character of
connectivity large influence on the limiting properties. We also study the
limiting properties for the non-moralizing global interaction case. We
demonstrate that the digraph observance is recovered in this case.
