  Modular Graph Functions (MGFs) are SL(2,$\mathbb{Z}$)-invariant functions
that emerge in the study of the low-energy expansion of the one-loop closed
string amplitude. To find the string scattering amplitude, we must integrate
MGFs over the moduli space of the torus. In this paper, we use the iterated
integral representation of MGFs to establish a depth-dependent basis for them,
where "depth" refers to the number of iterations in the integral. This basis
has a suitable Laplace equation. We integrate this basis from depth zero to
depth three over the fundamental domain of SL(2,$\mathbb{Z}$) with a cut-off.
