Size Should not Matter: Scale-invariant Stress Metrics

TL;DR

This paper identifies issues with existing stress metrics in graph drawing, especially their sensitivity to scale, and proposes a scale-invariant stress measure with an efficient computation method.

Contribution

The paper introduces a scale-normalized stress metric that fairly compares graph layouts regardless of their size, addressing a key limitation of previous metrics.

Findings

Existing stress metrics are affected by layout scale.

Scale-normalized stress provides a fair comparison across different layouts.

An efficient computation method for scale-normalized stress is developed.

Abstract

The normalized stress metric measures how closely distances between vertices in a graph drawing match the graph-theoretic distances between those vertices. It is one of the most widely employed quality metrics for graph drawing, and is even the optimization goal of several popular graph layout algorithms. However, normalized stress can be misleading when used to compare the outputs of two or more algorithms, as it is sensitive to the size of the drawing compared to the graph-theoretic distances used. Uniformly scaling a layout will change the value of stress despite not meaningfully changing the drawing. In fact, the change in stress values can be so significant that a clearly better layout can appear to have a worse stress score than a random layout. In this paper, we study different variants for calculating stress used in the literature (raw stress, normalized stress, etc.) and show that many of them are affected by this problem, which threatens the validity of experiments that compare the quality of one algorithm to that of another. We then experimentally justify one of the stress calculation variants, scale-normalized stress, as one that fairly compares drawing outputs regardless of their size. We also describe an efficient computation for scale-normalized stress and provide an open source implementation.