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We deal with a general preferential attachment graph model with multiple typeedges. The types are chosen randomly, in a way that depends on the evolution ofthe graph. In the $N$-type case, we define the (generalized) degree of a givenvertex 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 generalfamily 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 stepsgoes to infinity. We also provide recurrence equations for the asymptoticdegree distribution. Finally, we generalize the scale-free property of randomgraphs to the multi-type case.

Asymptotic degree distribution in preferential attachment graph models with multiple type edges