Scalable Durational Event Models: Application to Physical and Digital Interactions

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.

💡 Research Summary

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The paper introduces the Durational Event Model (DEM), a novel statistical framework designed to handle interaction data that contain both start and end timestamps—what the authors call “durational event data.” While traditional Relational Event Models (REMs) focus solely on the timing of event occurrences, they ignore the duration of each interaction. DEM addresses this gap by modeling two coupled counting processes: a formation process N₀→₁ that records when a dyad switches from a non‑interacting to an interacting state, and a dissolution process N₁→₀ that records the reverse transition. Each process is governed by its own instantaneous intensity function, λ₀→₁(t) and λ₁→₀(t), respectively. Both intensities adopt an exponential link (g(x)=exp(x)) and incorporate three key components: (i) a vector of summary statistics s(Hₜ) that capture history‑dependent features (e.g., number of common partners, cumulative past interactions, current interaction length), (ii) actor‑specific popularity parameters βᵢ that adjust for heterogeneous activity levels, and (iii) a step‑function baseline f(t,γ) that allows the intensity to change across pre‑specified time intervals (e.g., daily cycles). The summary statistics are log‑transformed (log(x+1)) to satisfy the Aalen–Gjessing non‑explosion condition, ensuring that the intensities remain finite even as counts grow.

A central methodological contribution is a scalable inference algorithm based on block‑coordinate ascent (BCA) combined with Minorization‑Maximization (MM). The full parameter vector θ is partitioned into three blocks (α, β, γ). For each block, a surrogate lower‑bound (minorizer) of the log‑likelihood is constructed, and the bound is maximized analytically, yielding closed‑form update rules. Because each update only requires information from the currently active dyads D(t), the algorithm’s memory footprint scales linearly with the number of actors, and its per‑iteration computational cost is proportional to the number of observed transitions. The authors prove that the MM construction guarantees monotonic increase of the likelihood, and they provide a theoretical complexity analysis showing that the method is orders of magnitude faster than traditional MCMC or Newton‑Raphson approaches, especially as the number of actors (N) and the observation window length increase.

The authors validate the approach through extensive simulations. They vary network size (N = 500, 1000) and event volume (10⁴–10⁵ events) and compare DEM‑BCA against state‑of‑the‑art REM and STERGM implementations. Results demonstrate that DEM‑BCA recovers true parameters with comparable or lower root‑mean‑square error while achieving speed‑ups of 10–50×. Importantly, the algorithm remains stable even when the baseline step‑function contains many intervals, a scenario that typically burdens existing methods.

For empirical illustration, the model is applied to the Copenhagen Networks Study, which collected both physical proximity data (via wearable sensors) and digital communication logs (phone calls, text messages) from 79 university students over several weeks. Separate DEMs are fitted to the physical and digital streams. In the physical interaction model, the strongest predictors are (a) the cumulative number of past physical contacts between a dyad (α₀→₁^N_I) and (b) the number of currently shared active partners (α₀→₁^CCP). This indicates a strong self‑reinforcing mechanism: recent face‑to‑face encounters increase the likelihood of future encounters. In contrast, the digital communication model shows that dyadic covariates reflecting friendship or shared attributes (β terms and dyadic covariate z) dominate, while the effect of past digital contacts is comparatively modest. These findings suggest that physical meetings are driven primarily by recent shared experiences, whereas digital communications are more contingent on underlying social ties and individual activity levels.

The paper acknowledges several limitations. First, the current formulation assumes undirected events; extending to directed interactions (e.g., caller vs. receiver) would require additional modeling of asymmetry. Second, the step‑function baseline f(t,γ) relies on user‑specified interval boundaries, which may be subjective; a Bayesian non‑parametric approach could learn these boundaries from data. Third, the model presumes that the sets of possible start and end dyads, D₀→₁(t) and D₁→₀(t), are known a priori, which may not hold in settings with missing or censored data. Future work is proposed in three directions: (i) incorporating directed and group‑based events, (ii) developing online or streaming versions of the algorithm for real‑time monitoring, and (iii) integrating multiple interaction layers (physical, digital, social media) into a unified multivariate DEM.

In conclusion, the Durational Event Model offers a principled way to separate the dynamics of event incidence from event duration, providing richer insight into interaction processes than traditional REMs. The block‑coordinate ascent algorithm makes the approach computationally feasible for large‑scale networks, and the empirical application demonstrates its ability to uncover distinct mechanisms governing physical versus digital social behavior. This framework has broad applicability across sociology, epidemiology, communication studies, and any domain where temporally precise, duration‑annotated interaction data are becoming increasingly available.