We can conveniently represent post-critically finite topological branched selfcovers of the sphere to itself using maps of graphs. With this representation, there is also a positive characterization of hyperbolic rational maps among these topological branched self-covers, using energies that control elastic 'stretchiness'. In broad terms, a map is rational iff a network of elastic bands gets looser and looser as you pull it back. This complements the older negative characterization of W. Thurston.