Relational event models with global covariates

Bike sharing is an increasingly popular mobility choice as it is a sustainable, healthy and economically viable transportation mode. By interpreting rides between bike stations over time as temporal events connecting two bike stations, relational event models can provide important insights into this phenomenon. The focus of relational event models, as a typical event history model, is normally on dyadic or node-specific covariates, as global covariates are considered nuisance parameters in a partial likelihood approach. As full likelihood approaches are infeasible given the sheer size of the relational process, we propose an innovative sampling approach of temporally shifted non-events to recover important global drivers of the relational process. The method combines nested case-control sampling on a time-shifted version of the event process. This leads to a partial likelihood of the relational event process that is identical to that of a degenerate logistic additive model, enabling efficient estimation of both global and non-global covariate effects. The computational effectiveness of the method is demonstrated through a simulation study. The analysis of around 350,000 bike rides in the Washington D.C. area reveals significant influences of weather and time of day on bike sharing dynamics, besides a number of traditional node-specific and dyadic covariates.

💡 Research Summary

This paper addresses a notable limitation of relational event models (REMs) when applied to large‑scale dynamic networks: the inability to estimate global covariates—variables that vary over time but affect all dyads equally, such as weather conditions or time‑of‑day effects. Traditional REM inference relies on the Cox partial likelihood, in which the baseline hazard (the global effect) cancels out, rendering global covariates unidentifiable. While a full‑likelihood approach could, in principle, incorporate these effects, it requires integrating the linear predictor over time and scales quadratically with the number of nodes, making it infeasible for networks with thousands of entities.

The authors propose an innovative solution based on a “time‑shifted” version of the event process. For each observed event (a bike ride from station s to station r at time t), they generate synthetic non‑events by shifting the original event times forward or backward by a fixed lag. These shifted non‑events are then added to the risk set alongside the real events. Because the shifted observations are evaluated at different time points, the global covariates no longer cancel out in the partial likelihood; instead, they contribute distinct terms for each observation.

To keep computation tractable, the method incorporates nested case‑control (NCC) sampling: for every observed event, a single shifted non‑event is randomly selected as a control. Under this design, the resulting partial likelihood is mathematically equivalent to that of a degenerate logistic additive model. In practice, the binary outcome (event vs. control) can be modeled with a logistic regression that includes smooth functions (e.g., splines) for both global and non‑global covariates, allowing for non‑linear and time‑varying effects without sacrificing the consistency guarantees of the original partial likelihood.

The paper formalizes the intensity function as

λ_sr(t) = Y_sr(t) λ₀ exp{ Σ_{l=1}^q f_l