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 [18], extended by Bollobás, Janson and Riordan [6]. 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 [16] and on the enumeration of properly q-colored multigraphs [19]. We also obtain new results on the phase transition of the satisfiability of quantified 2-Xor-formulas [8, 7].