Minimum Weight Random Graphs with Edge Constraints

TL;DR

This paper investigates minimum weight random graphs with edge constraints, providing deviation estimates and behavior descriptions for subtrees and paths in complete graphs and integer lattices.

Contribution

It introduces new analytical techniques to estimate deviations and describe the scaled minimum weight in constrained random graph models.

Findings

Deviation estimates for subtree weights in complete graphs

Behavior characterization of constrained paths in integer lattices

Application of martingale difference techniques

Abstract

In this paper, we study two examples of minimum weight random graphs with edge constraints. First we consider the complete graph on ${n}$ vertices equipped with uniformly heavy edge weights and use iteration methods to obtain deviation estimates for the minimum weight of subtrees with a given number of edges. Next we analyze edge constrained minimum weight paths in the integer lattice ${\mathbb{Z}^d}$ and employ martingale difference techniques to describe the behaviour of the scaled minimum weight in terms of the edge constraint.