New consistent and asymptotically normal estimators for random graph mixture models

TL;DR

This paper introduces new estimators for random graph mixture models that are proven to be consistent and asymptotically normal, improving the theoretical understanding and practical estimation of network structures.

Contribution

The paper provides the first rigorous proof of consistency and asymptotic normality for estimators in random graph mixture models, using moment equations and composite likelihood methods.

Findings

Estimators are strongly consistent and $ oot n$-convergent.

Network structure can be captured by triads and edges.

Method outperforms existing procedures on simulated and real data.

Abstract

Random graph mixture models are now very popular for modeling real data networks. In these setups, parameter estimation procedures usually rely on variational approximations, either combined with the expectation-maximisation (\textsc{em}) algorithm or with Bayesian approaches. Despite good results on synthetic data, the validity of the variational approximation is however not established. Moreover, the behavior of the maximum likelihood or of the maximum a posteriori estimators approximated by these procedures is not known in these models, due to the dependency structure on the variables. In this work, we show that in many different affiliation contexts (for binary or weighted graphs), estimators based either on moment equations or on the maximization of some composite likelihood are strongly consistent and $\sqrt{n}$-convergent, where $n$ is the number of nodes. As a consequence, our result establishes that the overall structure of an affiliation model can be caught by the description of the network in terms of its number of triads (order 3 structures) and edges (order 2 structures). We illustrate the efficiency of our method on simulated data and compare its performances with other existing procedures. A data set of cross-citations among economics journals is also analyzed.