Hybrid marked point processes: characterisation, existence and uniqueness

We introduce a class of hybrid marked point processes, which encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class amalgamates such existing processes, it also contains novel processes with complex dynamics. These processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process interacting with its state process. We show the existence and uniqueness of hybrid marked point processes under general assumptions, extending the results of Massouli\'e (1998) on interacting point processes.Comment: v6: introduction updated with reference to application of state-dependent Hawkes processe

Hawkes process as a model of social interactions: a view on video dynamics

We study by computer simulation the "Hawkes process" that was proposed in a recent paper by Crane and Sornette (Proc. Nat. Acad. Sci. USA 105, 15649 (2008)) as a plausible model for the dynamics of YouTube video viewing numbers. We test the claims made there that robust identification is possible for classes of dynamic response following activity bursts. Our simulated timeseries for the Hawkes process indeed fall into the different categories predicted by Crane and Sornette. However the Hawkes process gives a much narrower spread of decay exponents than the YouTube data, suggesting limits to the universality of the Hawkes-based analysis.Comment: Added errors to parameter estimates and further description. IOP style, 13 pages, 5 figure