  Modeling the popularity dynamics of an online item is an important open
problem in computational social science. This paper presents an in-depth study
of popularity dynamics under external promotions, especially in predicting
popularity jumps of online videos, and determining effective and efficient
schedules to promote online content. The recently proposed Hawkes Intensity
Process (HIP) models popularity as a non-linear interplay between exogenous
stimuli and the endogenous reactions. Here, we propose two novel metrics based
on HIP: to describe popularity gain per unit of promotion, and to quantify the
time it takes for such effects to unfold. We make increasingly accurate
forecasts of future popularity by including information about the intrinsic
properties of the video, promotions it receives, and the non-linear effects of
popularity ranking. We illustrate by simulation the interplay between the
unfolding of popularity over time, and the time-sensitive value of resources.
Lastly, our model lends a novel explanation of the commonly adopted periodic
and constant promotion strategy in advertising, as increasing the perceived
viral potential. This study provides quantitative guidelines about setting
promotion schedules considering content virality, timing, and economics.
