Abstract
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
making 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 novel differential-equation approach to attain the governing
equations of a general branching process. We confirm our theoretical findings
through extensive simulations of such processes and apply them to empirical
data obtained from threads of an online opinion board. This work provides the
ground to perform 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.
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