  We consider the problem of unveiling the implicit network structure of node
interactions (such as user interactions in a social network), based only on
high-frequency timestamps. Our inference is based on the minimization of the
least-squares loss associated with a multivariate Hawkes model, penalized by
$\ell_1$ and trace norm of the interaction tensor. We provide a first
theoretical analysis for this problem, that includes sparsity and low-rank
inducing penalizations. This result involves a new data-driven concentration
inequality for matrix martingales in continuous time with observable variance,
which is a result of independent interest and a broad range of possible
applications since it extends to matrix martingales former results restricted
to the scalar case. A consequence of our analysis is the construction of
sharply tuned $\ell_1$ and trace-norm penalizations, that leads to a
data-driven scaling of the variability of information available for each users.
Numerical experiments illustrate the significant improvements achieved by the
use of such data-driven penalizations.
