Bollobás‐Riordan random pairing model of a preferential attachment graph is studied. Let {W_j}_{j ≤ mn + 1} be the process of sums of independent exponentials with mean 1. We prove that the degrees of the first vertices are jointly, and uniformly, asymptotic to , and that with high probability (whp) the smallest of these degrees is , at least. Next we bound the probability that there exists a pair of large vertex sets without connecting edges, and apply the bound to several special cases. We propose to measure an influence of a vertex v by the size of a maximal recursive tree (max‐tree) rooted at v. We show that whp the set of the first vertices does not contain a max‐tree, and the largest max‐tree has size of order n. We prove that, for m > 1, . We show that the distribution of scaled size of a generic max‐tree in converges to a mixture of two beta distributions.

中文翻译：

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关于优先依附图的Bollobás-Riordan随机配对模型

研究了优先依附图的Bollobás-Riordan随机配对模型。让{ w ^ _Ĵ } _{Ĵ  ≤  MN  + 1}是具有平均1独立指数的和的处理，我们证明度的第一顶点是共同，且均匀，渐近于，并且以高概率（WHP）的最小这些度数中的至少是。接下来，我们限制存在一对没有连接边的大顶点集的可能性，并将该限制应用于几种特殊情况。我们建议衡量一个顶点的影响v通过根的最大递归树（最大树）的大小v。我们表明，第一个顶点的集合不包含max-tree，最大的max-tree的大小为n。我们证明了，对米 > 1，。我们证明了通用max-tree的按比例缩放大小的分布收敛于两个beta分布的混合。