  We continue our investigation of the modular graph functions and string
invariants that arise at genus-two as coefficients of low energy effective
interactions in Type II superstring theory. In previous work, the
non-separating degeneration of a genus-two modular graph function of weight $w$
was shown to be given by a Laurent polynomial in the degeneration parameter $t$
of degree $(w,w)$. The coefficients of this polynomial generalize genus-one
modular graph functions, up to terms which are exponentially suppressed in $t$
as $t \to \infty$. In this paper, we evaluate this expansion explicitly for the
modular graph functions associated with the $D^8 {\cal R}^4$ effective
interaction for which the Laurent polynomial has degree $(2,2)$. We also prove
that the separating degeneration is given by a polynomial in the degeneration
parameter $\ln (|v|)$ up to contributions which are power-behaved in $v$ as $v
\to 0$. We further extract the complete, or tropical, degeneration and compare
it with the independent calculation of the integrand of the sum of Feynman
diagrams that contributes to two-loop type II supergravity expanded to the same
order in the low energy expansion. We find that the tropical limit of the
string theory integrand reproduces the supergravity integrand as its leading
term, but also includes sub-leading terms proportional to odd zeta values that
are absent in supergravity and can be ascribed to higher-derivative stringy
interactions.
