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 where 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.

中文翻译：

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具有多类型边的优先附着图模型中的渐近度分布

摘要 我们处理具有多种类型边的一般优先附件图模型。类型是随机选择的，方式取决于图的演变。在 N 类型的情况下，我们将给定顶点的（广义）度定义为 其中 是连接到它的类型 k 边的数量。我们证明了具有多类型边的优先附着随机图模型的一般族的渐近度分布的存在。更准确地说，我们表明随着步数趋于无穷大，具有（广义）度数为 d 的顶点的比例趋向于某个随机变量。我们还提供渐近度分布的递推方程。最后，我们将随机图的无标度特性推广到多类型情况。