Conditionally Poissonian random digraphs

In this short note we define a Poissonian model of directed random graphs which generalises the undirected Poissonian random graph process introduced in [Norros, I.; Reittu, H. "On a conditionally Poissonian graph process." Adv. in Appl. Probab. 38 (2006), no. 1, 59--75]. We discuss the relation of our model to the Norros-Reittu model, characterise the limiting distribution of the degree of a typical vertex and discuss the component structure of the model in some special cases which are relevant to the theory of infection processes.