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

Backhausz, Ágnes and Rozner, Bence (2019) Asymptotic degree distribution in preferential attachment graph models with multiple type edges. Stochastic Models. ISSN 1532-4214 (In Press)

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Abstract

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 d=(d_1,d_2,…,d_N), where d 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 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.

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