Hyperedge-Triggered Hawkes processes for diagnosing higher-order temporal dependence in event streams

Monte Carlo simulation is used to assess a likelihood-based EM procedure for diagnosing higher-order temporal dependence in multivariate event streams. Multivariate Hawkes processes provide a likelihood-based framework for modelling self- and cross-excitation in continuous-time event streams, but their standard pairwise formulation can confound dense dyadic excitation with genuinely group-triggered temporal dependence. We introduce a hyperedge-triggered Hawkes model in which a candidate group contributes additional intensity only after all of its member nodes fire within a prescribed trigger window. A most-recent-anchor convention makes this higher-order component non-accumulating and leads to a corresponding piecewise compensator in the likelihood. The model is estimated by a latent-branching EM algorithm that assigns event-level responsibilities to background, pairwise, and hyperedge-triggered sources. Controlled simulations evaluate recovery, regularisation, convergence, separation from pairwise baselines, trigger-window and kernel-timescale sensitivity, computational scaling, and comparison with a parameter-matched third-order surrogate. The results show that higher-order excitation can be recovered in informative regimes, while pairwise--hyperedge confounding remains the main statistical limitation. Applications to retinal and visual-cortex spike trains are presented as exploratory predictive analyses: the hyperedge-triggered component can improve candidate-penalised held-out fit, but the fitted effects should be interpreted as diagnostics of higher-order temporal dependence rather than calibrated discoveries of individual biological hyperedges.