# Consistency of the maximum likelihood and variational estimators in a dynamic stochastic block model

## TL;DR

This paper proves the consistency and convergence rates of maximum likelihood and variational estimators in a dynamic stochastic block model with evolving node memberships modeled by a hidden Markov chain.

## Contribution

It establishes the theoretical consistency and convergence rates of estimators in a dynamic stochastic block model with temporal evolution of node classes.

## Key findings

- Proves consistency of estimators as nodes and time steps increase
- Provides upper bounds on convergence rates of estimators
- Analyzes a case with fixed time steps and varying connectivity parameters

## Abstract

We consider a dynamic version of the stochastic block model, in which the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution is modeled through a hidden Markov chain on the nodes memberships. We prove the consistency (as the number of nodes and time steps increase) of the maximum likelihood and variational estimators of the model parameters, and obtain upper bounds on the rates of convergence of these estimators. We also explore the particular case where the number of time steps is fixed and connectivity parameters are allowed to vary.