The Hawkes process has garnered attention in recent years for its suitability to describe the behavior of online information cascades. Here we present a fully tractable approach to analytically describe the distribution of the number of events in a Hawkes process, which, in contrast to purely empirical studies or simulation-based models, enables the effect of process parameters on cascade dynamics to be analyzed. We show that the presented theory also allows predictions regarding the future distribution of events after a given number of events have been observed during a time window. Our results are derived through a differential-equation approach to attain the governing equations of a general branching process. We confirm our theoretical findings through extensive simulations of such processes. This work provides the basis for more complete analyses of the self-exciting processes that govern the spreading of information through many communication platforms, including the potential to predict cascade dynamics within confidence limits.

中文翻译：

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量化流行度预测模型中的不确定性。

霍克斯过程近年来因其适合描述在线信息级联的行为而受到关注。在这里，我们提出了一种完全易于处理的方法，以分析方式描述霍克斯过程中事件数量的分布，与纯粹的经验研究或基于仿真的模型相比，该方法可以分析过程参数对级联动力学的影响。我们表明，在一定时间范围内观察到给定数量的事件之后，提出的理论还允许对事件的未来分布进行预测。我们的结果是通过微分方程方法获得的，从而获得了一般分支过程的控制方程。我们通过对此类过程的广泛模拟来确认我们的理论发现。