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arxiv: 2606.07680 · v1 · pith:Z7G6LZGTnew · submitted 2026-06-04 · 📊 stat.ME · cs.SI

A Counting Process View of Relational Event Models: Practical Asymptotics

Cornelius Fritz , Alexander Fuchs-Kreiss This is my paper

Pith reviewed 2026-06-27 23:50 UTC · model grok-4.3

classification 📊 stat.ME cs.SI
keywords relational event modelscounting processesasymptotic normalitycox modelsmaximum likelihood estimationnetwork dynamicsdyadic interactions
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The pith

Relational event models achieve asymptotic normality of maximum likelihood estimators when represented as counting processes under limits on network size or observation time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Relational Event Models analyze continuous-time sequences of interactions between pairs of actors while incorporating history-dependent effects such as reciprocity. The paper establishes that representing these models as counting processes allows standard martingale results to deliver asymptotic normality for the estimators in three regimes: network size growing large, observation period growing long, or both. It focuses on Cox-type multiplicative intensity models and spells out the conditions under which the required regularity assumptions hold. Simulations then show how practical modeling decisions, including the choice of temporal windows and logarithmic transformations, influence coverage and convergence in finite samples. The work supplies concrete guidelines for applying the models to real data.

Core claim

By embedding relational event models in the counting process framework, the maximum likelihood estimator for Cox-type multiplicative intensity models is asymptotically normal as the network size n, the observation time T, or both tend to infinity, provided the usual regularity conditions for counting processes hold, such as predictable variation processes converging appropriately and no explosive behavior.

What carries the argument

The counting process representation of the relational event intensity, which converts the likelihood into a product-integral form and permits application of martingale central limit theorems to the score process.

If this is right

  • Asymptotic normality holds in each of the three separate limits (n to infinity, T to infinity, or joint) when the corresponding conditions on the predictable variation are met.
  • Choices such as temporal windowing and log transformations of covariates directly affect finite-sample coverage and should be tuned accordingly.
  • The same counting-process argument extends to models that include history dependence such as triadic closure or reciprocity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • For networks of moderate size the joint n-and-T limit may offer the most robust finite-sample behavior.
  • The counting-process approach could be used to derive similar asymptotic results for other point-process formulations of social interaction data.

Load-bearing premise

The intensity processes and observation scheme satisfy the regularity conditions required for the martingale central limit theorem to apply in the chosen asymptotic regime.

What would settle it

Simulations in which the empirical coverage of asymptotic confidence intervals fails to approach the nominal level, or the estimator bias does not shrink, as n or T is increased would indicate that the regularity conditions do not hold for the data-generating processes considered.

read the original abstract

Relational Event Models (REMs) provide a rigorous framework for analyzing dyadic interactions observed in continuous time, capturing history-dependent dynamics such as triadic closure and reciprocity. Framing REMs through the lens of counting processes embeds the model in a rich theoretical foundation, facilitating its mathematical analysis. While Maximum Likelihood Estimation (MLE) is standard practice for estimating these models, the underlying statistical guarantees rely on specific asymptotic regimes, namely, whether the network size (n), the observational period (T), or both approach infinity. We review the theoretical foundations of such counting-process-based models, formalizing the core assumptions required to achieve asymptotic normality across these different limits. With a specific focus on Cox-type multiplicative models, we detail the circumstances under which these assumptions hold. Supported by simulation studies, we illustrate how structural modeling choices, including temporal windowing and logarithmic transformations, affect empirical coverage and estimator convergence. We thereby derive several guiding principles for specifying such models in realistic contexts, bridging theory and practice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript reviews the counting-process formulation of Relational Event Models (REMs) for dyadic interaction data. It formalizes the assumptions needed for asymptotic normality of maximum-likelihood estimators in Cox-type multiplicative intensity models under three regimes (n o∞ with fixed T, T o∞ with fixed n, and joint n,T o∞). Simulation experiments examine the finite-sample consequences of modeling decisions such as temporal windowing and logarithmic transformations of covariates, from which the authors extract practical guidelines for specification.

Significance. If the review of the counting-process assumptions is accurate and the simulations reliably illustrate the regimes in which the asymptotics are useful, the paper supplies a convenient reference that links standard martingale theory to applied REM practice. The emphasis on concrete modeling choices that affect coverage is a constructive contribution for users of these models.

minor comments (3)
  1. [Simulation studies] The simulation section would benefit from an explicit statement of the data-generating intensity and the precise definition of the temporal windowing operator; without these, it is difficult to judge how representative the reported coverage probabilities are of the three asymptotic regimes.
  2. Several figures display point estimates of coverage without accompanying Monte Carlo standard errors or interval estimates; adding these would strengthen the visual comparison across regimes.
  3. [Theoretical foundations] The notation for the compensator and the predictable variation process in the counting-process setup could be cross-referenced to a standard reference (e.g., Andersen et al.) to aid readers less familiar with the martingale framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive summary and positive assessment of the manuscript's contribution as a reference linking martingale theory to applied REM practice. The recommendation for minor revision is noted, and we will incorporate any editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; review of standard counting-process theory

full rationale

The paper positions itself as a review that formalizes core assumptions from existing counting-process theory for asymptotic normality of Cox-type REM estimators under n→∞, T→∞, and joint limits. It draws on prior literature for the theoretical foundations rather than claiming new derivations. No load-bearing step reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the simulation studies are presented as illustrative rather than as the source of the asymptotic claims. The work is self-contained against external benchmarks in the counting-process literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No concrete free parameters, axioms, or invented entities can be identified from the abstract alone; the work relies on standard counting-process assumptions whose details are not supplied here.

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discussion (0)

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