Empirical Reference Distributions for Networks of Different Size

TL;DR

This paper introduces a new method for comparing network statistics across networks of different sizes by using a mixture model-based reference distribution, improving comparability and interpretability.

Contribution

It proposes a novel adjustment technique using mixture models for reference distributions, enhancing the comparison of network measures across varying network sizes.

Findings

Mixture model-based adjustments improve comparability of network statistics.

Normalized statistics relative to Erdos-Renyi graphs are better but still limited.

Application to real co-location networks demonstrates practical utility.

Abstract

Network analysis has become an increasingly prevalent research tool across a vast range of scientific fields. Here, we focus on the particular issue of comparing network statistics, i.e. graph-level measures of network structural features, across multiple networks that differ in size. Although "normalized" versions of some network statistics exist, we demonstrate via simulation why direct comparison of raw and normalized statistics is often inappropriate. We examine a recent suggestion to normalize network statistics relative to Erdos-Renyi random graphs and demonstrate via simulation how this is an improvement over direct comparison, but still sometimes problematic. We propose a new adjustment method based on a reference distribution constructed as a mixture model of random graphs which reflect the dependence structure exhibited in the observed networks. We show that using simple Bernoulli models as mixture components in this reference distribution can provide adjusted network statistics that are relatively comparable across different network sizes but still describe interesting features of networks, and that this can be accomplished at relatively low computational expense. Finally, we apply this methodology to a collection of co-location networks derived from the Los Angeles Family and Neighborhood Survey activity location data.