Scalable Durational Event Models: Application to Physical and Digital Interactions

TL;DR

The paper tackles modeling durational interactions in large networks by introducing the Durational Event Model (DEM), which separates incidence (start) and duration (end) processes into two dependent counting processes with history- and covariate-driven intensities. A scalable block-coordinate ascent algorithm estimates incidence and duration parameters separately, achieving substantial improvements in time and memory compared to Newton-Raphson while guaranteeing ascent and convergence. The DEM is validated via simulations and applied to the Copenhagen Networks Study, revealing that physical interactions are mainly driven by triadic and prior contact dynamics, whereas digital interactions are more influenced by friendship and prior dyadic contact, with baseline temporal patterns evident in both. The work provides a practical, interpretable, and scalable tool for understanding the dynamics of durational interactions in large-scale social systems, along with an open-source implementation in the DEM R package.

Abstract

Durable interactions are ubiquitous in social network analysis and are increasingly observed with precise time stamps. Phone and video calls, for example, are events to which a specific duration can be assigned. We term data encoding interactions with the start and end times ``durational event data''. Recent advances in data collection have enabled the observation of such data over extended periods of time and between large populations of actors. Methodologically, we propose the Durational Event Model, an extension of Relational Event Models that decouples the modeling of event incidence from event duration. Computationally, we derive a fast, memory-efficient, and exact block-coordinate ascent algorithm to facilitate large-scale inference. Theoretical complexity analysis and numerical simulations demonstrate computational superiority of this approach over state-of-the-art methods. We apply the model to physical and digital interactions among college students in Copenhagen. Our empirical findings reveal that past interactions drive physical interactions, whereas digital interactions are influenced predominantly by friendship ties and prior dyadic contact.

Figures (13)

• Figure 1: Graphs illustrating the proposed summary statistics. Dashed lines ( ) refer to transitions from $0 \rightarrow 1$, while the wiggly line ( ) relates to $1 \rightarrow 0$. Other observed interactions are drawn as solid lines, in black if they are currently interacting actors ( ) and gray if the event occurred sometime in the past ( ).
• Figure 2: Simulation Study 2: (a) RMSE of $\hat{\bm{\alpha}}$ (blue), $\hat{\bm{\beta}}$ (yellow), and $\hat{\bm{\gamma}}$ (orange) pooled over incidence and duration model. Simulation Study 3: Comparison of computational time (b) and memory needed for estimation (c) for different number of actors of our proposed three-step estimator and the state-of-the-art Newton-Raphson method.
• Figure 3: (a) Comparison of empirical and theoretical survival functions. (b) and (c): Top-$k$ recall curves applied to physical (b) and digital (c) interaction data for three model specifications: Full (blue), Reduced (yellow), and No Popularity Models (orange).
• Figure 4: Simulation Study 1: Quantile-quantile plots for the incidence (top row) and duration model (bottom row) comparing the theoretical quantiles of a standard normal distribution with the sample quantiles of $z_{(s)} = \bm{\Lambda}(\hat{\bm{\theta}}_{(s)})^{1/2}(\hat{\bm{\alpha}}_{(s)} - \bm{\alpha})$ for $s = 1, ..., \hbox{1,000}$.
• Figure 5: Simulation Study 4: Comparison of computational time (left) and memory needed for estimation (right) for different numbers of events $(M)$ of our proposed three-step estimator.