  We extend the notion of canonical measures to all (possibly non-compact)
metric graphs. This will allow us to introduce a notion of "hyperbolic
measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann
surfaces describes the limiting behavior of canonical (Arakelov) measures on
finite covers in relation to the hyperbolic measure. We will prove a
generalized version of this theorem for metric graphs, allowing any infinite
Galois cover to replace the universal cover. We will show all such limiting
measures satisfy a version of Gauss-Bonnet formula which, using the theory of
von Neumann dimensions, can be interpreted as a "trace formula". In the special
case where the infinite cover is the universal cover, we will provide explicit
methods to compute the corresponding limiting (hyperbolic) measure. Our ideas
are motivated by non-Archimedean analytic and tropical geometry.
