T. Caelli et al. (Eds.): SSPR&SPR 2002, LNCS 2396, pp. 252-262, 2002.  Springer-Verlag Berlin Heidelberg 2002 Estimating the Joint Probability Distribution of Random Vertices and Arcs by Means of Second-Order Random Graphs Francesc Serratosa 1 , RenØ AlquØzar 2 , and Alberto Sanfeliu 3 1 Universitat Rovira i Virgili Dept. dEnginyeria Informtica i Matemtiques, Spain Francesc.Serratosa@etse.urv.es http://www.etse.urv.es/~fserrato 2 Universitat PolitLcnica de Catalunya Dept. de Llenguatges i Sistemes Informtics, Spain alquezar@lsi.upc.es 3 Universitat PolitLcnica de Catalunya Institut de Robtica i Informtica Industrial, Spain sanfeliu@iri.upc.es 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 2 nd -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) 2 nd -order information, FDGs contain more structural 2 nd -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 2 nd - order random graph concept. 1 Introduction There are two major problems that practical applications using structural pattern recognition are confronted with. The first problem is the computational complexity of comparing two AGs. The time required by any of the optimal algorithms may in the worst case become exponential in the size of the graphs. The approximate algorithms, on the other hand, have only polynomial time complexity, but do not guarantee to find