Estimating the joint probability distribution of random vertices and arcs by means of second-order random graphs

Abstract

We review the approaches that model a set of Attributed Graphs (AGs) by extending the definition of AGs to include probabilistic information. As a main result, we present a quite general formulation for estimating the joint probability distribution of the random elements of a set of AGs, in which some degree of probabilistic independence between random elements is assumed, by considering only 2nd-order joint probabilities and marginal ones. We show that the two previously proposed approaches based on the random-graph representation (First-Order Random Graphs (FORGs) and Function- Described Graphs (FDGs)) can be seen as two different approximations of the general formulation presented. From this new representation, it is easy to derive that whereas FORGs contain some more semantic (partial) 2nd-order information, FDGs contain more structural 2nd-order information of the whole set of AGs. Most importantly, the presented formulation opens the door to the development of new and more powerful probabilistic representations of sets of AGs based on the 2ndorder random graph concept.