  We propose a novel extremal dependence measure called the partial
tail-correlation coefficient (PTCC), in analogy to the partial correlation
coefficient in classical multivariate analysis. The construction of our new
coefficient is based on the framework of multivariate regular variation and
transformed-linear algebra operations. We show how this coefficient allows
identifying pairs of variables that have partially uncorrelated tails given the
other variables in a random vector. Unlike other recently introduced
conditional independence frameworks for extremes, our approach requires minimal
modeling assumptions and can thus be used in exploratory analyses to learn the
structure of extremal graphical models. Similarly to traditional Gaussian
graphical models where edges correspond to the non-zero entries of the
precision matrix, we can exploit classical inference methods for
high-dimensional data, such as the graphical LASSO with Laplacian spectral
constraints, to efficiently learn the extremal network structure via the PTCC.
We apply our new method to study extreme risk networks in two different
datasets (extreme river discharges and historical global currency exchange
data) and show that we can extract meaningful extremal structures with
meaningful domain-specific interpretations.
