On the connected components of a random permutation graph with a given number of edges

TL;DR

This paper investigates the connectivity of permutation graphs induced by permutations with a fixed number of inversions, establishing probabilistic thresholds for connectivity and component sizes using a Markov process and asymptotic analysis.

Contribution

It introduces a Markov process to analyze the probability of permutation graph connectivity and derives asymptotic thresholds for the emergence of a connected graph.

Findings

Probability of connectivity approaches 1 near a specific inversion count

Number of components follows a Poisson distribution asymptotically

Largest and smallest component sizes relate to random partitions of [0,1]

Abstract

A permutation of [n] induces a graph on [n] such that the edges of the graph correspond to inversion pairs of the permutation. This graph is connected if and only if the corresponding permutation is indecomposable. Let s(n,m) denote a permutation chosen uniformly at random among all permutations of [n] with exactly m inversions. Let p(n,m) be the common value for the probabilities that s(n,m) is indecomposable or the corresponding graph is connected. We prove that p(n,m) is non-decreasing with m by constructing a Markov process in which s(n,m+1) is obtained from s(n,m) by increasing one of the components of the inversion sequence of s(n,m) by one. We show that, with probability approaching 1, the graph corresponding to s(n,m) becomes connected for m asymptotic to (6/(\pi^2))nln(n). More precisely, for m=(6n/(\pi^2)) [ln(n)+ lnln(n)/2+ ln(12)- ln(\pi)- 12/(\pi^2)+x_n], where |x_n|=o(lnlnln(n)), the number of components of the random graph is shown to be asymptotically 1+Poisson(e^{-x_n}). When x_n goes to negative infinity, the sizes of the largest and the smallest components, scaled by n, are asymptotic to the lengths of the largest and the smallest subintervals in a partition of [0,1] by [e^{-x_n}] randomly, and independently, scattered points.