  We study how changing the weight datum
$\mathcal{A}=(a_1,...,a_n)\in(\mathbb{Q}\cap(0,1])^n$ affects the topology of
tropical moduli spaces $M_{g,\mathcal{A}}^{trop}$,
$\overline{M}_{g,\mathcal{A}}^{trop}$ and $\Delta_{g,\mathcal{A}}$ and the
homology of the latter one. We show that for fixed $g$ and $n$, there are
particular filtrations of these topological spaces which we can use to compute
the reduced rational homology of $\Delta_{g,\mathcal{A}}$ and the top weight
cohomology of the moduli space $\mathcal{M}_{g,\mathcal{A}}$ of smooth
$(g,\mathcal{A})$-stable algebraic curves.
