  We associate a weight function to pairs consisting of a smooth and proper
variety X over a complete discretely valued field and a differential form on X
of maximal degree. This weight function is a real-valued function on the
non-archimedean analytification of X. It is piecewise affine on the skeleton of
any regular model with strict normal crossings of X, and strictly ascending as
one moves away from the skeleton. We apply these properties to the study of the
Kontsevich-Soibelman skeleton of such a pair, and we prove that this skeleton
is connected when X has geometric genus one. This result can be viewed as an
analog of the Shokurov-Kollar connectedness theorem in birational geometry.
