Barabasi-Albert random graph with multiple type edges and perturbation

We introduce the perturbed version of the Barabasi-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.