Moduli spaces of curves (with nonempty boundary or at least one marked point) admit cell decompositions in which the cells are labelled by ribbon graphs. In fact, the moduli space of normalised metric ribbon graphs with $n$ boundary components is homeomorphic to the product of the moduli space of curves and an $(n-1)$-simplex.
From this geometric statement one can show that the ribbon graph complex (a chain complex spanned by ribbon graphs, with the differential given by summing over all edge contractions) computes the cohomology of moduli spaces of curves.
My question is: does anyone know how to describe the cochain level cup product structure on the ribbon graph complex? By general machinery, if a complex computes the cohomology of a space then it carries an $A_\infty$ structure for which it is equivalent to the cochains on the space with their usual $A_\infty$ structure.
I would like to know if there is a way to write down a combinatorial formula for this $A_\infty$ structure on the ribbon graph complex.
Similarly, the Lie graph complex (in which vertices are labelled by words in a generic Lie algebra) computes the cohomology of the spaces $BOut(F_n)$ (the classifying spaces of outer automorphism groups of free groups). Is there a way to describe the resulting $A_\infty$ structure on the Lie graph complex?