Null models for network data

TL;DR

This paper unifies two common null models for undirected network data under a broader class, showing they produce similar estimates in sparse graphs, which simplifies analysis and helps choose models based on context.

Contribution

It demonstrates that the logistic-linear and implicit log-linear models are special cases of a broader null model class, streamlining likelihood-based inference for network data.

Findings

Both models yield similar link probability estimates in sparse graphs.

The broader class simplifies likelihood computation and model selection.

Empirical comparisons show practical advantages of the unified approach.

Abstract

The analysis of datasets taking the form of simple, undirected graphs continues to gain in importance across a variety of disciplines. Two choices of null model, the logistic-linear model and the implicit log-linear model, have come into common use for analyzing such network data, in part because each accounts for the heterogeneity of network node degrees typically observed in practice. Here we show how these both may be viewed as instances of a broader class of null models, with the property that all members of this class give rise to essentially the same likelihood-based estimates of link probabilities in sparse graph regimes. This facilitates likelihood-based computation and inference, and enables practitioners to choose the most appropriate null model from this family based on application context. Comparative model fits for a variety of network datasets demonstrate the practical implications of our results.