Estimating the Parameters of the Waxman Random Graph

TL;DR

This paper evaluates existing formal estimators for Waxman random graph parameters and introduces a new efficient maximum likelihood estimator that only requires link lengths, improving parameter estimation accuracy.

Contribution

It provides the first performance evaluation of formal estimators and proposes a novel, computationally efficient maximum likelihood estimator for Waxman graph parameters.

Findings

First evaluation of formal estimators' performance.

Introduction of a new $O(n)$ maximum likelihood estimator.

Estimator requires only link lengths as input.

Abstract

The Waxman random graph is a generalisation of the simple Erd\H{o}s-R\'enyi or Gilbert random graph. It is useful for modelling physical networks where the increased cost of longer links means they are less likely to be built, and thus less numerous than shorter links. The model has been in continuous use for over two decades with many attempts to select parameters which match real networks. In most the parameters have been arbitrarily selected, but there are a few cases where they have been calculated using a formal estimator. However, the performance of the estimator was not evaluated in any of these cases. This paper presents both the first evaluation of formal estimators for the parameters of these graphs, and a new Maximum Likelihood Estimator with $O(n)$ computational time complexity that requires only link lengths as input.