随机网络图的Buckley-Osthus模型中给定度节点间的度分布和边数

Q3 Mathematics
E. Grechnikov
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引用次数: 16

摘要

摘要本文研究了Buckley-Osthus模型中随机图H (t) a,k的一些重要统计量,其中t为节点数,kt为边数(使),>0为节点的初始吸引度。这个模型是对著名的Bollobás-Riordan模型的修改。首先,我们在该模型中找到了图中给定阶数d的节点数R(d, t)的期望的一个新的渐近公式。这样的公式为和d≤t 1/100(a+1)。从理论和实践的角度来看,这两种限制都不能令人满意。我们完全去除它们。然后,我们计算任意两个量R(d1, t)和R(d2, t)之间的协方差,并使用第二矩方法证明R(d, t)对于d和t的所有可能值都紧密集中在其平均值附近。此外,我们研究了网络图的一个更复杂的统计量:X(d1, d2, t)是度分别等于d1和d2的节点之间的边的总数。我们还找到了X(d1, d2, t)期望的渐近公式,并证明了一个紧密集中的结果。同样,我们没有对d1 d2和t的值施加任何实质性的限制。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Degree Distribution and Number of Edges between Nodes of Given Degrees in the Buckley–Osthus Model of a Random Web Graph
Abstract In this paper, we study some important statistics of the random graph H (t) a,k in the Buckley–Osthus model, where t is the number of nodes, kt is the number of edges (so that ), and a>0 is the so-called initial attractiveness of a node. This model is a modification of the well-known Bollobás–Riordan model. First, we find a new asymptotic formula for the expectation of the number R(d, t) of nodes of a given degree d in a graph in this model. Such a formula is known for and d⩽t 1/100(a+1). Both restrictions are unsatisfactory from theoretical and practical points of view. We completely remove them. Then we calculate the covariances between any two quantities R(d 1, t) and R(d 2, t), and using the second-moment method we show that R(d, t) is tightly concentrated around its mean for all possible values of d and t. Furthermore, we study a more complicated statistic of the web graph: X(d 1, d 2, t) is the total number of edges between nodes whose degrees are equal to d 1 and d 2 respectively. We also find an asymptotic formula for the expectation of X(d 1, d 2, t) and prove a tight concentration result. Again, we do not impose any substantial restrictions on the values of d 1, d 2, and t.
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来源期刊
Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
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