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 ¼ ðd 1 ; d 2 ;. .. ; d N Þ; where d k 2 Z þ 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 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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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 = (d1, d2, . . . , dN ), where d k ∈ 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 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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In this paper we introduce the perturbed version of the Barabási-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. one has to understand quantities like the number of vertices of a given type with a given number of edges. This kind of analysis is performed in the papers mentioned above. Now we consider models where it is not the vertices, but the edges that have different types. This can be used to model different kind of relationships between the individuals -in a social, biological or financial network, connections are usually not of the same nature, and this can be important from the point of view of contagion or epidemic spread. In our model, the type of an edge is an element of a fixed finite set, and it does not change with time. It is chosen randomly when the edge is born, with a distribution that depends on the current state of the graph and on the types of the edges going out from the neighbours of the new vertex. A general family of preferential attachment random graphs with multi-type edges has been examined in our previous work [3]. Growing networks with two different types of edges can be considered as directed graphs. In this case the type of an edge is its orientation. That is, when a new vertex is born, then it is attached to the graph with an edge directed from the new vertex to the already existing ones or directed from the existing vertices to the new one, and this corresponds to the two different types. Directed preferential attachment models were introduced and examined in [5,14], but with different dynamics than in [3]. In this paper we are interested in a version of robustness in preferential attachment graph models with multi-type edges. The goal is to compare (i) a model in which the probability of choosing a type is exactly the proportion of the current type among the edges going out from the endpoint of the new edge; and (ii) its modified version, when, after this step, types can change with certain probability. In particular, we introduce perturbation in the multi-type Barabási-Albert random graph, and prove that this shows different phenomena than the original version. That is, errors in the dynamics of multi-type random graphs can lead to essential changes in the asymptotic behaviour of the model. The multi-type Barabási-Albert random graph model has been described in [3], which is a generalization of the Barabási-Albert random graph model introduced in [4], specified in [6]. We prove the existence of the asymptotic degree distribution in the perturbed Barabási-Albert random graph, and we also provide recurrence equations for the asymptotic degree distribution. The main difference between the perturbed and...