The crossing number of a graph $G$, $\mathrm{cr}(G)$, is the minimum number of edge crossings arising when drawing a graph on a certain surface. Determining $\mathrm{cr}(G)$ is a problem of great importance in Graph Theory. Its maximum variant, i.e. the maximum crossing number, $\mathrm{max-cr}(G)$, is receiving growing attention. Instead of an optimization problem on the number of crossings, here we consider the variance of the number of edge crossings, when embedding the vertices of an arbitrary graph uniformly at random in some space. In his pioneering research, Moon derived this variance on random linear arrangements of complete unipartite and bipartite graphs. Given the need of efficient algorithms to support this sort of research and given also the growing interest of the number of edge crossings in spatial networks, networks where vertices are embedded in some space, here we derive an algorithm to calculate the variance in arbitrary graphs in time $o(nm^2)$ that we transform into one that runs in time $O(nm)$ by reusing computations. We also derive one for forests that runs in time $O(n)$. These algorithms work on a wide range of random layouts (not only on Moon's) and are based on novel arithmetic expressions for the calculation of the variance that we develop from previous theoretical work. This paves the way for many applications that rely on a fast but exact calculation of the variance.

翻译：图$G$的交叉数$\mathrm{cr}(G)$是在特定曲面上绘制图时产生的最小边缘交叉数量。确定$\mathrm{cr}(G)$是图论中一个极其重要的问题。其最大值变体，即最大交叉数$\mathrm{max-cr}(G)$，正受到越来越多的关注。本文我们不将其视为关于交叉数的优化问题，而是考虑当将任意图的顶点均匀随机嵌入到某个空间时，边缘交叉数量的方差。在开创性研究中，Moon推导了完全单部图和完全二部图在随机线性排列中的这一方差。考虑到支撑此类研究的高效算法的需求，以及空间网络（顶点嵌入某些空间的网络）中边缘交叉数量日益增长的关注度，本文我们提出一种在任意图中计算方差的算法，其时间复杂度为$o(nm^2)$，并通过重用计算将其转化为运行时间为$O(nm)$的算法。我们还针对森林推导了一种运行时间为$O(n)$的算法。这些算法适用于广泛的随机布局（不仅限于Moon的布局），并基于我们从先前理论工作中推导出的新颖方差计算算术表达式。这为许多依赖快速而精确方差计算的应用铺平了道路。