Dynamic exponential random graph models with the fused lasso penalty

This paper investigates how the parameters governing network structure vary over time across a sequence of observed dynamic networks. We focus on ensembles of networks that exhibit sudden structural changes at certain time points but remain stable or undergo minimal changes in between. Many real-world networks, such as international trade networks and political co-voting networks, display this behaviour. Given a sequence of observed networks, our objective is to estimate the corresponding time-varying parameters of an exponential random graph model (ERGM) and identify periods of parameter stability as well as abrupt changes in the underlying network-generating mechanisms. To this end, we introduce a novel fused lasso exponential random graph model (FL-ERGM), which encourages similar parameter values across consecutive time points while allowing abrupt changes when supported by the data. To address the intractability of the ERGM likelihood, we adopt a pseudo-likelihood approximation and optimise the resulting penalised objective using the alternating direction method of multipliers (ADMM). We prove that the solution obtained by ADMM is equivalent to the optimal solution of the original objective function. Monte Carlo simulations demonstrate that FL-ERGM outperforms existing methods in estimating time-varying ERGM parameters and detecting structural changes. Applications to international trade and political co-voting networks reveal interpretable and contextually relevant patterns of temporal parameter variation.