Edge crossings in random linear arrangements

Abstract

The interest in spatial networks where vertices are embedded in a one-dimensional space is growing. Remarkable examples of these networks are syntactic dependency trees and RNA structures. In this setup, the vertices of the network are arranged linearly and then edges may cross when drawn above the sequence of vertices. Recently, two aspects of the distribution of the number of crossings in uniformly random linear arrangements have been investigated: the expectation and the variance. While the computation of the expectation is straightforward, that of the variance is not. Here we present fast algorithms to calculate that variance in arbitrary graphs and forests. As for the latter, the algorithm calculates variance in linear time with respect to the number of vertices. This paves the way for many applications that rely on an exact but fast calculation of that variance. These algorithms are based on novel arithmetic expressions for the calculation of the variance that we develop from previous theoretical work.

Bounds of the sum of edge lengths in linear arrangements of trees

“…Here we are interested in two distinct representatives: quasistar trees (quasi), where one of the original stars has only two vertices (figure 1 (c)) and balanced bistar trees (b-bistar), where the two original stars have the same size or differ in one vertex (figure 1 (d)). Quasistar trees are important for the theory of edge crossings in linear arrangements [23,24]. In this article, we will unveil that balanced bistar trees maximize D t max over trees of n vertices.…”

Reappraising the distribution of the number of edge crossings of graphs on a sphere

“…Statistical properties of C, the number of edge crossings of a graph G, have been studied in generic embeddings, denoted as * , that meet three mathematical conditions [2]: (1) only independent edges can cross (edges that do not share vertices), (2) two independent edges can cross in at most one point, and (3) if several edges of the graph, say e edges, cross at exactly the same point then the amount of crossings equals e 2 = e(e − 1)/2. In our view, generic embeddings are two-fold: a space and a statistical distribution of the vertices in such space.…”

“…In [16], the space is the surface of a sphere while the distribution of the vertices on that surface is uniformly random. Compact formulae for the expectation and the variance have been obtained [2]. Here we apply such a framework to revise the problem of calculating the distribution of C in arrangements of vertices on the surface of a sphere.…”

“…His derivations of E rsa [C] are straightforward. Borrowing the notation in [2], Moon obtained that the expectation of C is…”

“…where q is the number of pairs of independent edges [18] and δ rsa is the probability that two independent edges cross. Indeed, q is a handle for the size of the set Q, consisting of the pairs of independent edges of a graph G [18,2]. Thanks to…”