具有多个邻接约束的相交随机图和网络：一个简单的例子
Intersecting random graphs and networks with multiple adjacency constraints: A simple example
摘要：
在使用随机图模型研究网络时，有时会遇到节点之间的邻接概念反映了多个约束的情况。传统的随机图模型不足以处理这种情况。考虑多个约束的一个简单想法是取随机图的交集。在本文中，我们开始研究随机图，从而通过一个简单的例子。我们研究了Erdos-Renyi图和一维几何随机图的交集。我们研究了零-一定律的性质，没有孤立的节点。当几何分量定义在单位圆上时，我们建立了一个完整的零-一定律，并确定了它的临界尺度。当几何分量位于单位区间时，存在一个缺口，因为所得到的零和一定律被发现表示偏离不同的临界标度。特别是，一阶矩法需要一个更大的临界尺度比单位圆的情况下，以获得一个法律。这种差异有点令人惊讶，因为在单位区间和单位圆上，没有孤立节点的零-一定律在几何随机图中是相同的。
英文摘要：
When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations. A simple idea to account for multiple constraints consists in taking the intersection of random graphs. In this paper we initiate the study of random graphs so obtained through a simple example. We examine the intersection of an Erdos-Renyi graph and of one-dimensional geometric random graphs. We investigate the zero-one laws for the property that there are no isolated nodes. When the geometric component is defined on the unit circle, a full zero-one law is established and we determine its critical scaling. When the geometric component lies in the unit interval, there is a gap in that the obtained zero and one laws are found to express deviations from different critical scalings. In particular, the first moment method requires a larger critical scaling than in the unit circle case in order to obtain the one law. This discrepancy is somewhat surprising given that the zero-one laws for the absence of isolated nodes are identical in the geometric random graphs on both the unit interval and unit circle.
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