Probabilistic Properties of GIG Digraphs

TL;DR

This paper analyzes the probabilistic characteristics of GIG digraphs, including edge connection probabilities, sink distributions, and component sizes, providing insights into their structural properties.

Contribution

It introduces new probabilistic analyses of GIG digraphs, including joint sink probabilities and convergence of maximum component size.

Findings

Probability of specific directed edge sequences computed

Joint probability of vertices being sinks derived

Expected maximum component size shown to converge

Abstract

We study the probabilistic properties of the Greatest Increase Grid (GIG) digraph. We compute the probability of a particular sequence of directed edges connecting two random vertices. We compute the joint probability that a set of vertices are all sinks, and derive the mean and variance in the number of sinks in a randomly labeled GIG digraph. Finally, we show that the expected size of the maximum component of vertices converges.