  Positive dependence is present in many real world data sets and has appealing
stochastic properties that can be exploited in statistical modeling and in
estimation. In particular, the notion of multivariate total positivity of order
2 ($ \mathrm{MTP}_{2} $) is a convex constraint and acts as an implicit
regularizer in the Gaussian case. We study positive dependence in multivariate
extremes and introduce $ \mathrm{EMTP}_{2} $, an extremal version of $
\mathrm{MTP}_{2} $. This notion turns out to appear prominently in extremes,
and in fact, it is satisfied by many classical models. For a H\"usler--Reiss
distribution, the analogue of a Gaussian distribution in extremes, we show that
it is $ \mathrm{EMTP}_{2} $ if and only if its precision matrix is a Laplacian
of a connected graph. We propose an estimator for the parameters of the
H\"usler--Reiss distribution under $ \mathrm{EMTP}_{2} $ as the solution of a
convex optimization problem with Laplacian constraint. We prove that this
estimator is consistent and typically yields a sparse model with possibly
nondecomposable extremal graphical structure. Applying our methods to a data
set of Danube River flows, we illustrate this regularization and the superior
performance compared to existing methods.
