大型齐次二值乘法属性图模型中的节点隔离
Node isolation in large homogeneous binary multiplicative attribute graph models
摘要：
乘法属性图（MAG）模型是由Kim和Leskovec引入的，作为某些类别的现实世界网络的数学模型。它是隐藏图模型的一个实例，并实现了网络结构由与节点单独关联的属性共同塑造的合理想法。这些作者研究了这个模型的几个方面，包括它的连通性，巨分支的存在性，它的直径和度分布。这是在渐近制度时，节点的数量和节点属性的数量都增长无限大，后者与前者的自然容许条件下缩放。在相同的设置中，我们探索孤立节点的存在（或等价地，不存在），原始论文中没有讨论的属性。该论文的主要结果是一个{ \em零一}定律的孤立节点的情况下，这个零一定律与金和Leskovec得到的图的连通性（虽然在稍弱的假设下）。我们证明了这些结果，通过应用的方法，在一个非标准的方式，以多组计数随机变量与孤立节点的数量的第一和第二时刻。
英文摘要：
The multiplicative attribute graph (MAG) model was introduced by Kim and Leskovec as a mathematically tractable model of certain classes of real-world networks. It is an instance of hidden graph models, and implements the plausible idea that network structure is collectively shaped by attributes individually associated with nodes. These authors have studied several aspects of this model, including its connectivity, the existence of a giant component,its diameter and the degree distribution. This was done in the asymptotic regime when the number of nodes and the number of node attributes both grow unboundedly large, the latter scaling with the former under a natural admissibility condition. In the same setting, we explore the existence (or equivalently, absence) of isolated nodes, a property not discussed in the original paper. The main result of the paper is a {\em zero-one} law for the absence of isolated nodes; this zero-one law coincides with that obtained by Kim and Leskovec for graph connectivity (although under slightly weaker assumptions). We prove these results by applying the method of first and second moments in a non-standard way to multiple sets of counting random variables associated with the number of isolated nodes.
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