We associate a weight function to pairs (X, ω) consisting of a smooth and proper variety
X over a complete discretely valued field and a pluricanonical form ω on X.
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 (X, ω), and
we prove that this skeleton is connected when X has geometric genus one and ω is
a canonical form on X. This result can be viewed as an analog of the Shokurov–Koll´ar
connectedness theorem in birational geometry.