  We introduce a new model of random multigraphs with colored vertices and
weighted edges. It is similar to the "inhomogeneous random graph model" of
S\"oderberg (2002), extended by Bollob\'as, 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, already investigated by Pittel and Yeum (2010), and on
the enumeration of properly q-colored multigraphs, analyzed by Wright (1972).
We also obtain new results on the phase transition of the satisfiability of
quantified 2-Xor-formulas, a problem introduced by Creignou, Daud\'e and Egly
(2007).
