  This paper introduces the Hawkes skeleton and the Hawkes graph. These objects
summarize the branching structure of a multivariate Hawkes point process in a
compact, yet meaningful way. We demonstrate how graph-theoretic vocabulary
(`ancestor sets', `parent sets', `connectivity', `walks', `walk weights', ...)
is very convenient for the discussion of multivariate Hawkes processes. For
example, we reformulate the classic eigenvalue-based subcriticality criterion
of multitype branching processes in graph terms. Next to these more
terminological contributions, we show how the graph view may be used for the
specification and estimation of Hawkes models from large, multitype event
streams. Based on earlier work, we give a nonparametric statistical procedure
to estimate the Hawkes skeleton and the Hawkes graph from data. We show how the
graph estimation may then be used for specifying and fitting parametric Hawkes
models. Our estimation method avoids the a priori assumptions on the model from
a straighforward MLE-approach and is numerically more flexible than the latter.
Our method has two tuning parameters: one controlling numerical complexity, the
other one controlling the sparseness of the estimated graph. A simulation study
confirms that the presented procedure works as desired. We pay special
attention to computational issues in the implementation. This makes our results
applicable to high-dimensional event-stream data, such as dozens of event
streams and thousands of events per component.
