Fertility Heterogeneity as a Mechanism for Power Law Distributions of Recurrence Times

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent \alpha, where the fertility of an event is the number of aftershocks of first generation that it triggers, on the probability distribution function (pdf) f(\tau) of the recurrence times \tau between successive events. The other input of the model is an exponential Omori law quantifying the pdf of waiting times between an event and its first generation aftershocks, whose characteristic time scale is taken as our time unit. At short time scales, we discover two intermediate power law asymptotics, f(\tau) ~ \tau^{-(2-\alpha)} for \tau « \tau_c and f(\tau) ~ \tau^{-\alpha} for \tau_c « \tau « 1, where \tau_c is associated with the self-excited cascades of aftershocks. For 1 « \tau « 1/\nu, we find a constant plateau f(\tau) ~ const, while at long times, 1/\nu < \tau, f(\tau) ~ e^{-\nu \tau} has an exponential tail controlled by the arrival rate \nu of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.

💡 Research Summary

The paper investigates how the distribution of inter‑event times (recurrence times) emerges in a self‑exciting Hawkes conditional Poisson process, which is the simplest point‑process model that incorporates feedback from past events to future event rates. The authors focus on two essential ingredients of the model. First, the waiting time between a parent event and each of its first‑generation aftershocks follows an exponential Omori law; the characteristic time of this exponential is taken as the unit of time, so that the “bare” triggering kernel is memoryless. Second, the number of first‑generation aftershocks generated by a single event – the event’s fertility – is assumed to follow a heavy‑tailed power‑law distribution (p(k)\propto k^{-\alpha}) with exponent (1<\alpha<2). This fertility distribution captures the empirical observation that some events (large earthquakes, major market moves, viral posts) produce disproportionately many offspring, while most events generate few or none.

Using Laplace transforms and a careful analysis of the singularities of the generating functions, the authors derive the full probability density function (f(\tau)) of the recurrence time (\tau) for all time scales. They identify four distinct regimes:

1. Very short times (\tau\ll\tau_c). In this regime the recurrence time is dominated by direct first‑generation aftershocks. The heavy‑tailed fertility distribution imprints a power‑law tail (f(\tau)\sim \tau^{-(2-\alpha)}). The exponent (2-\alpha) is less than one, reflecting the strong clustering caused by highly fertile events. The crossover time (\tau_c) depends on the branching ratio (n=\langle k\rangle\mu) (where (\mu) is the mean of the exponential Omori kernel) and on (\alpha); as the system approaches criticality ((n\to1)), (\tau_c) grows, extending the influence of this regime.

2. Intermediate short times (\tau_c\ll\tau\ll 1). Here the first‑generation aftershocks have already saturated, and multi‑generation cascades dominate. The recurrence‑time density inherits directly the fertility exponent: (f(\tau)\sim \tau^{-\alpha}). This regime is a clear signature that the underlying power‑law distribution of fertilities survives the branching process and shapes the observable inter‑event statistics.

3. Intermediate times (1\ll\tau\ll 1/\nu). In this window the external (exogenous) Poisson background with rate (\nu) is still negligible, so the process is effectively a pure branching system. The density flattens to a constant plateau, (f(\tau)\approx \text{const}). This plateau reflects the balance between the decay of the exponential Omori kernel and the continual replenishment of activity by the branching cascade.

4. Long times (\tau\gg 1/\nu). At sufficiently large (\tau) the external Poisson arrivals dominate. The recurrence‑time density reverts to the exponential tail of a standard Poisson process, (f(\tau)\sim e^{-\nu\tau}). The rate (\nu) therefore controls the ultimate cutoff of the power‑law regimes.

The analytical predictions are validated by extensive Monte‑Carlo simulations of the Hawkes process with the prescribed fertility distribution. The simulations confirm the predicted exponents, the location of the crossover (\tau_c), the existence of the plateau, and the exponential cutoff. Moreover, the authors show that as (\alpha) approaches 1 the short‑time exponent (2-\alpha) approaches 1, producing an especially heavy tail that matches empirical observations in seismology (aftershock sequences), high‑frequency finance (trade clustering), and online social dynamics (viral cascades).

The paper’s contributions are threefold. First, it demonstrates a novel mechanism by which power‑law inter‑event time distributions can arise: the combination of a power‑law fertility distribution with self‑excitation and cascade dynamics. Second, it introduces a characteristic crossover time (\tau_c) that quantifies the transition from direct aftershock dominance to cascade dominance, linking it to the branching ratio and the fertility exponent. Third, it provides a unified analytical framework that simultaneously captures short‑time clustering, intermediate‑time plateaus, and long‑time exponential decay, thereby reconciling disparate empirical findings within a single stochastic model.

These results have broad implications for any field where events trigger further events in a heavy‑tailed fashion. In seismology, the model offers a mechanistic explanation for the observed power‑law waiting‑time statistics of earthquakes beyond the traditional Omori‑Utsu law. In finance, it suggests that the bursty nature of trades and price changes can be traced back to a few highly “fertile” market moves that spawn cascades of subsequent activity. In social media, the framework explains how a small fraction of highly influential posts generate long tails of reposting intervals. Overall, the study enriches our understanding of how microscopic heterogeneity (fertility) and macroscopic feedback (self‑excitation) together shape the statistical architecture of temporal event sequences.