On Exponential Random Graph Models with Dyadic Independence

TL;DR

This paper characterizes exponential random graph models with dyadic independence and permutation-equivariant parameters, showing they are essentially the ta model or the additive stochastic block model, including directed networks.

Contribution

It provides a complete characterization of exponential random graph models under specific independence and symmetry assumptions, identifying the ta model and additive stochastic block model as unique solutions.

Findings

The ta model is the only dyadic independent ERGM with permutation-equivariant nodal parameters.

ERGM with fewer block parameters corresponds to the additive stochastic block model.

Results extend to directed networks, characterizing their ERGMs under similar assumptions.

Abstract

We show that the only exponential random graph model with n nodal parameters, dyads being independent, and the natural assumption of permutation-equivariant nodal parametrization is the \b{eta} model. In addition, we show that an exponential random graph model with similar assumptions but with fewer than n block parameters is the additive stochastic block model. We also provide similar results for directed networks