  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.
