Many real transportation and mobility networks have their vertices placed on the surface of the Earth. In such embeddings, the edges laid on that surface may cross. In his pioneering research, Moon analyzed the distribution of the number of crossings on complete graphs and complete bipartite graphs whose vertices are located uniformly at random on the surface of a sphere assuming that vertex placements are independent from each other. Here we revise his derivation of that variance in the light of recent theoretical developments on the variance of crossings and computer simulations. We show that Moon's formulae are inaccurate in predicting the true variance and provide exact formulae.

中文翻译：

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重新评估球面上图的边缘交叉数的分布

许多真实的交通和移动网络的顶点都位于地球表面。在这种嵌入中，放置在该表面上的边缘可能会交叉。在他的开创性研究中，Moon 分析了完全图和完全二部图的交叉数分布，这些图的顶点在球面上随机均匀分布，假设顶点位置彼此独立。在这里，我们根据最近关于交叉和计算机模拟方差的理论发展来修改他对该方差的推导。我们表明，Moon 的公式在预测真实方差方面是不准确的，并提供了准确的公式。