Analytic description of the phase transition of inhomogeneous multigraphs

We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the inhomogeneous random graph model of Söderberg (2002), extended by Bollobás, Janson and Riordan (2007). By means of analytic combinatorics, we then analyze the birth of complex components, which are components with at least two cycles. We apply those results to give a complete picture of the finite size scaling and the critical exponents associated to a rather broad family of decision problems. As applications, we derive new proofs of known results on the 2-colorability problem (Pittel and Yeum, 2010) and on the enumeration of properly q-colored multigraphs (Wright, 1972), and new results on the phase transition of the satisfiability of quantified  2-Xor-formulas (Daudé and Ravelomanana, 2011, Creignou et al., 2007).