  Online social networking services allow their users to post content in the
form of text, images or videos. The main mechanism driving content diffusion is
the possibility for users to re-share the content posted by their social
connections, which may then cascade across the system. A fundamental problem
when studying information cascades is the possibility to develop sound
mathematical models, whose parameters can be calibrated on empirical data, in
order to predict the future course of a cascade after a window of observation.
In this paper, we focus on Twitter and, in particular, on the temporal patterns
of retweet activity for an original tweet. We model the system by
Time-Dependent Hawkes process (TiDeH), which properly takes into account the
circadian nature of the users and the aging of information. The input of the
prediction model are observed retweet times and structural information about
the underlying social network. We develop a procedure for parameter
optimization and for predicting the future profiles of retweet activity at
different time resolutions. We validate our methodology on a large corpus of
Twitter data and demonstrate its systematic improvement over existing
approaches in all the time regimes.
