Time-varying $\beta$-model for dynamic directed networks

Abstract

We extend the well-known $\beta$-model for directed graphs to dynamic network setting, where we observe snapshots of adjacency matrices at different time points. We propose a kernel-smoothed likelihood approach for estimating $2n$ time-varying parameters in a network with $n$ nodes, from $N$ snapshots. We establish consistency and asymptotic normality properties of our kernel-smoothed estimators as either $n$ or $N$ diverges. Our results contrast their counterparts in single-network analyses, where $n\to \infty$ is invariantly required in asymptotic studies. We conduct comprehensive simulation studies that confirm our theory's prediction and illustrate the performance of our method from various angles. We apply our method to an email data set and obtain meaningful results.

References in corpus (5)

• Motifs in Temporal Networks
• Random graphs with a given degree sequence
• Maximum lilkelihood estimation in the $\beta$-model
• A central limit theorem in the $\beta$-model for undirected random graphs with a diverging number of vertices
• Asymptotics in directed exponential random graph models with an increasing bi-degree sequence