Transient Supercriticality in Hawkes Processes: Detection, Identifiability, and Apparent Criticality

Self-exciting (Hawkes) models of epidemics, earthquakes, financial markets, and social cascades are usually calibrated with a branching ratio $n$ below one. This is a condition for stationarity, not for existence: a process with $n>1$ grows exponentially but remains finite on any finite horizon. Imposing stationarity is also misleading, because transient supercritical episodes then masquerade as near-criticality. Reconstructing the latent genealogy of events, we show that a single finite record can reveal an excursion into the regime $n(t)>1$, and back, and date its onset and end. Detectability is governed by the excess $n-1$ times the square root of the number of exposed parents. Even when the memory time of the triggering is unknown, the data determine the growth rate of activity during the episode nearly as precisely as when it is known, and hence whether $n$ crossed unity, whereas the value of $n$ requires independent knowledge of the memory time. Letting the data choose between endogenous amplification and external forcing prevents confusing the two. In the 2015 Korean MERS outbreak, our reconstruction shows that transmission by hospital cohorts had collapsed by 28 May, before the incidence peak and two weeks before the instantaneous reproduction number fell below one; in real time, it establishes the collapse a few days earlier than this standard indicator. Our framework separates stationarity, non-explosion, and transient supercritical amplification, and provides a practical route for detecting short-lived endogenous instabilities in self-exciting systems.