Network comparison and the within-ensemble graph distance

TL;DR

This paper introduces a framework using well-understood random network ensembles to benchmark and understand graph distance measures, aiding in better network comparison and tool selection.

Contribution

It proposes using the expected distance between networks from the same model as a key property for evaluating graph distance measures.

Findings

Calculated within-ensemble graph distances for classic network models.

Demonstrated how these distances encapsulate key features of network models.

Provided a new framework for understanding and benchmarking graph distances.

Abstract

Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years a multitude of diverse, ad hoc solutions to this problem have been introduced. Here we propose that simple and well-understood ensembles of random networks (such as Erd\H{o}s-R\'{e}nyi graphs, random geometric graphs, Watts-Strogatz graphs, the configuration model, and preferential attachment networks) are natural benchmarks for network comparison methods. Moreover, we show that the expected distance between two networks independently sampled from a generative model is a useful property that encapsulates many key features of that model. To illustrate our results, we calculate this within-ensemble graph distance and related quantities for classic network models (and several parameterizations thereof) using 20 distance measures commonly used to compare graphs. The within-ensemble graph distance provides a new framework for developers of graph distances to better understand their creations and for practitioners to better choose an appropriate tool for their particular task.