# Concentration and consistency results for canonical and curved exponential-family models of random graphs

## TL;DR

This paper establishes non-asymptotic concentration and consistency results for maximum likelihood and M-estimators in exponential-family models of random graphs with local dependence, highlighting the role of additional structure.

## Contribution

It provides the first non-asymptotic concentration and consistency results for a broad class of exponential-family random graph models with local dependence.

## Key findings

- Non-asymptotic concentration results for estimators
- Consistency results for finite and infinite populations
- Facilitation of subgraph-to-graph estimation

## Abstract

Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and $M$-estimators of a wide range of canonical and curved exponential-family models of random graphs with local dependence. All results are non-asymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an application, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes.