Introduction to Relational Event Modelling

Abstract: Interactions and time shape many aspects of life. Everyday activities -- like conversations, emails, money transfers, citations, and even acts of violence -- are relational events: interactions between a sender and a receiver at a specific moment. At the intersection of event-history analysis and network modelling, relational event models (REMs) offer a powerful framework for studying when and why these events occur. Recent advances have made it possible to express REMs as generalized additive models, allowing researchers to capture complex, non-linear patterns over time. While an essay and a comprehensive review exist, a hands-on tutorial paper on REMs is still missing. This work fills that gap. It provides a practical introduction to REMs, incorporating the latest developments in the field. It demonstrates how to simulate synthetic relational-event data and walks through several empirical applications, comparing different modelling and inference strategies. By bringing together theory, simulation, and application, this tutorial lowers the barrier to entry and makes REMs a more accessible and practical tool.

• The paper presents relational event modelling as a point-process framework for analysing directed, time-stamped interactions without losing sequence information through static network aggregation.
• The tutorial shows how case-control likelihoods, generalized additive models, splines, random effects, latent spaces, and hyper event models extend REMs to large, nonlinear, and heterogeneous interaction data.
• The paper provides reproducible workflows for estimation and validation, including corrected AIC and martingale-residual goodness-of-fit tests, with applications to radio communications, species invasions, and workplace email.

Purpose and scope

This paper by Boschi and Wit is a hands-on tutorial on relational event models (REMs), positioned at the intersection of event-history analysis and dynamic network modelling. While a conceptual essay and a comprehensive review of REMs already exist, the authors identify the absence of a practical introduction that incorporates recent methodological developments — notably the expression of REMs as generalized additive models (GAMs) with time-varying, non-linear, and random effects. The tutorial covers data structures, covariate construction, likelihood-based inference, model selection and goodness-of-fit testing, simulation-based validation, and three empirical applications, all supported by reproducible R code.

The relational event framework

A relational event is a tuple $(s, r, t)$ denoting a directed interaction from sender $s$ to receiver $r$ at time $t$. The authors distinguish these instantaneous events from relational states, arguing that aggregating events into repeated static network snapshots loses information both through arbitrary choices of aggregation windows and by collapsing distinct event sequences into identical structures. The statistical foundation is a marked point process $\mathbb{P}$ with an associated counting process $N_{sr}(t)$, which under standard regularity assumptions admits a Doob–Meyer decomposition into a predictable compensator $$\Lambda_{sr}(t) = \int_0^t \lambda_{sr}(u)\,\mathrm{d}u$$ and a zero-mean martingale. The hazard $\lambda_{sr}(t)$ is modelled multiplicatively as a risk indicator $I_{sr}(t)$ times a baseline rate $\lambda_0(t)$ times an exponential link function of edge-specific covariates. The baseline itself may depend on global covariates and calendar-time smooths, generalizing Butts' original constant-baseline formulation.

Covariates are divided into exogenous processes (node attributes, dyadic relational covariates, external events that modify the risk set) and endogenous statistics computed from the event history. A useful contribution is a systematic taxonomy of endogenous building blocks — indicator, volume, exponential decay, and interarrival-time transforms — applied uniformly to nodal (sender activity, receiver popularity), dyadic (reciprocity, repetition), and triadic (transitive and cyclic closure) mechanisms, with attention to whether event ordering matters and to the hierarchy principle for higher-order statistics.

Model extensions

The tutorial surveys four principal extensions beyond fixed linear effects. Time-varying and non-linear effects are represented via spline bases (thin plate or B-splines), avoiding ad hoc decay functions such as exponential weighting with a chosen half-life; the alien-species example shows trade's effect on invasions is positive but decreasing over 125 years. Mixed REMs introduce Gaussian random effects capturing latent actor heterogeneity, e.g., species-specific invasiveness. Dynamic latent space models embed actors in a state-space process whose pairwise distances modulate interaction propensity. Relational hyper event models (RHEMs) generalize dyads to sets of senders and receivers, with subset-repetition statistics averaging prior co-occurrence over sub-subsets; citation networks serve as the canonical example, where the risk set is strictly constrained by publication timing. Stratification of the baseline by event type, sender, or receiver provides a further mechanism for handling heterogeneity.

Inference

Four nested maximum-likelihood strategies are presented, each with a direct R implementation:

Strategy	Equivalent regression	Key property
Full likelihood	Poisson GLM with offset	Requires piecewise-constant hazard
Partial likelihood	Cox regression (coxph)	Eliminates baseline; needs only event order
Sampled partial likelihood	Conditional logistic regression (clogit)	Nested case-control sampling of $s$0 controls
Case-control partial likelihood	Binary GAM without intercept ($s$1)	Cost scales as $s$2, independent of $s$3

The case-control formulation is central: it reduces computational cost from $s$4 to $s$5 while enabling penalized splines and random effects (the latter handled as smooth terms with ridge penalties, smoothing parameters inversely proportional to variance components). Two further advances address remaining bottlenecks. First, because global covariates cancel in standard partial likelihoods, the authors present the shifted counting process of Lembo et al., in which dyad-specific random time shifts ensure that global covariate differences do not vanish between case and control. Second, for very large event streams, stochastic gradient ascent (ADAM) and neural additive models fitted on GPUs provide scalable alternatives. Simulation results confirm unbiasedness of fixed-effect estimates and correct recovery of smooth trends up to the expected constant offset induced by centring constraints.

Model selection and goodness of fit

Model comparison uses a corrected conditional AIC appropriate for models with smoothing terms, which the authors note balances the conservative bias of marginal AIC against the anti-conservative bias of classical conditional estimators. Importantly, they stress that AIC selection does not guarantee adequacy. For absolute goodness-of-fit, the tutorial adopts a martingale-residual-based approach: cumulative sums of differences between observed and model-expected model-matrix entries converge, after appropriate rescaling, to (multivariate) Brownian bridges, yielding Kolmogorov–Smirnov-type tests with exact $s$6-values for linear terms and simulation-based $s$7-values for penalized terms. This is computationally cheaper than simulation-based GOF procedures and supports flexible REM specifications.

Empirical applications

Three applications illustrate increasing complexity. The World Trade Center radio communications data (481 events among 37 actors, full risk set of 1,332 dyads) fit via Cox regression show positive effects of sender activity, receiver popularity, reciprocity, closure, and coordinator roles — coordinators are more than three times as likely to receive communications — and a negative repetition effect. Supplementary results across sampled risk sets ($s$8 controls) preserve coefficient signs but inflate standard errors, quantifying the sampling–precision trade-off. The bipartite mammalian invasion application (799 first records, 186 species, 153 regions, 1880–2005) combines a linear temperature effect, a time-varying trade effect, a non-linear distance effect, and species random effects; the most invasive species identified (raccoon, black rat, coypu) match ecological consensus. Finally, 57,791 emails among 159 employees are modelled with non-linear effects of interarrival times for reciprocity, repetition, and closure, plus global calendar covariates via the shifted likelihood; the reciprocity curve's peaks align with working-day boundaries, suggesting delayed replies.

Limitations and open questions

The authors are explicit about several constraints. Computation remains the chief obstacle, particularly for random effects and for history-dependent covariate construction at scale. REMs as presented are descriptive: they capture conditional associations between rates and drivers without causal identification, though separate work on local independence and causal drivers is cited as complementary. Non-linear effect estimates are interpretable only up to shape and relative change due to centring constraints, and the piecewise-constant assumption underlying Poisson-based full-likelihood inference may be restrictive. Open questions include efficient inference for mixed REMs at scale, integration of relational event and relational state processes in one framework, and extension of flexible (time-varying, non-linear) formulations within hyper event models.

Conclusion

This tutorial consolidates two decades of REM development into a coherent, implementable workflow spanning simulation, estimation, validation, and interpretation. Its main pedagogical contribution is demonstrating that modern flexible REMs — with splines, random effects, global covariates, and hyperevents — remain within reach of standard GLM/GAM software, lowering the barrier to rigorous analysis of time-stamped interaction data across social, ecological, and financial domains.