Fast calculation of the variance of edge crossings
Lluís Alemany-Puig, Ramon Ferrer‐i‐Cancho · arXiv (Cornell University) · 2020
The crossing number, i.e. the minimum number of edge crossings arising when drawing a graph on a certain surface, is a very important problem of graph theory. The opposite problem, i.e. the maximum crossing number, is receiving growing attention. Here we consider a complementary problem of the distribution of the number of edge crossings, namely the variance of the number of crossings, when embedding the vertices of an arbitrary graph in some space at random. In his pioneering research, Moon derived that 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 algorithms to calculate the variance in arbitrary graphs in $o(nm^2)$-time, and in forests in $O(n)$-time. 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.