Longitudinal Network Models and Permutation-Uniform Markov Chains

TL;DR

This paper develops a theoretical framework for longitudinal network models based on Markov chains, characterizing their distributions, and applying these ideas to exponential random graph models to improve analysis and inference.

Contribution

It introduces a characterization of exponential-family longitudinal network models, explores permutation-uniform subclasses, and applies these concepts to simplify analysis of existing models.

Findings

Joint distributions are exponential families with the same parameters under certain conditions.

Permutation-uniform chains can be interpreted as i.i.d. sequences on the same state space.

Provides closed-form maximum likelihood estimates for some network models.

Abstract

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.