  In this paper we introduce the perturbed version of the Barab\'asi-Albert
random graph with multiple type edges and prove the existence of the
(generalized) asymptotic degree distribution. Similarly to the non-perturbed
case, the asymptotic degree distribution depends on the almost sure limit of
the proportion of edges of different types. However, if there is perturbation
then the resulting degree distribution will be deterministic, which is a major
difference compared to the non-perturbed case.
