  The rich class of multivariate Pareto distributions forms the basis of
recently introduced extremal graphical models. However, most existing
literature on the topic is focused on the popular parametric family of
H\"usler--Reiss distributions. It is shown that the H\"usler--Reiss family is
in fact the only continuous multivariate Pareto model that exhibits the
structure of a pairwise interaction model, justifying its use in many
high-dimensional problems. Along the way, useful insight is obtained concerning
a certain class of distributions considered in arXiv:2303.04258, a result of
independent interest.
