  We investigate the equidistribution of the eigenfunctions on quantum graphs
in the high-energy limit. Our main result is an estimate of the deviations from
equidistribution for large well-connected graphs. We use an exact
field-theoretic expression in terms of a variant of the supersymmetric
nonlinear sigma-model. Our estimate is based on a saddle-point analysis of this
expression and leads to a criterion for when equidistribution emerges
asymptotically in the limit of large graphs. Our theory predicts a rate of
convergence that is a significant refinement of previous estimates,
long-assumed to be valid for quantum chaotic systems, agreeing with them in
some situations but not all. We discuss specific examples for which the theory
is tested numerically.
