  Let $G = (V,E)$ be a connected graph. A probability measure $\mu$ on $V$ is
called "balanced" if it has the following property: if $T_\mu(v)$ denotes the
"earth mover's" cost of transporting all the mass of $\mu$ from all over the
graph to the vertex $v$, then $T_\mu$ attains its global maximum at each point
in the support of $\mu$. We prove a decomposition result that characterizes
balanced measures as convex combinations of suitable "extremal" balanced
measures that we call "basic." An upper bound on the number of basic balanced
measures on $G$ follows, and an example shows that this estimate is essentially
sharp.
