  We deal with a general preferential attachment graph model with multiple type
edges. The types are chosen randomly, in a way that depends on the evolution of
the graph. In the $N$-type case, we define the (generalized) degree of a given
vertex as $\boldsymbol{d}=(d_{1},d_{2},\dots,d_{N})$, where
$d_{k}\in\mathbb{Z}_{0}^{+}$ is the number of type $k$ edges connected to it.
We prove the existence of an a.s.\ asymptotic degree distribution for a general
family of preferential attachment random graph models with multi-type edges.
More precisely, we show that the proportion of vertices with (generalized)
degree $\boldsymbol{d}$ tends to some random variable as the number of steps
goes to infinity. We also provide recurrence equations for the asymptotic
degree distribution. Finally, we generalize the scale-free property of random
graphs to the multi-type case.
