  Conditional independence, graphical models and sparsity are key notions for
parsimonious statistical models and for understanding the structural
relationships in the data. The theory of multivariate and spatial extremes
describes the risk of rare events through asymptotically justified limit models
such as max-stable and multivariate Pareto distributions. Statistical modelling
in this field has been limited to moderate dimensions so far, partly owing to
complicated likelihoods and a lack of understanding of the underlying
probabilistic structures. We introduce a general theory of conditional
independence for multivariate Pareto distributions that allows the definition
of graphical models and sparsity for extremes. A Hammersley-Clifford theorem
links this new notion to the factorization of densities of extreme value models
on graphs. For the popular class of H\"usler-Reiss distributions we show that,
similarly to the Gaussian case, the sparsity pattern of a general extremal
graphical model can be read off from suitable inverse covariance matrices. New
parametric models can be built in a modular way and statistical inference can
be simplified to lower-dimensional marginals. We discuss learning of minimum
spanning trees and model selection for extremal graph structures, and
illustrate their use with an application to flood risk assessment on the Danube
river.
