  Modular graph functions arise in the calculation of the low-energy expansion
of closed-string scattering amplitudes. For toroidal world-sheets, they are
${\rm SL}(2,\mathbb{Z})$-invariant functions of the torus complex structure
that have to be integrated over the moduli space of inequivalent tori. We use
methods from resurgent analysis to construct the non-perturbative corrections
arising when the argument of the modular graph function approaches the cusp on
this moduli space. ${\rm SL}(2,\mathbb{Z})$-invariance will in turn strongly
constrain the behaviour of the non-perturbative sector when expanded at the
origin of the moduli space.
