Counts of special trajectories of quadratic differentials on Riemann surfaces are an example of Donaldson-Thomas invariants, they exhibit wall-crossing phenomena over the moduli space of differentials. I will describe a combinatorial construction of the Kontsevich-Soibelman invariant for this counting problem based on ribbon graphs of degenerate Jenkins-Strebel differentials. Time permitting, I will also sketch generalizations to motivic wall-crossing invariants, and to collections of k-differentials in the context of Hitchin systems.