Filtrations of moduli spaces of tropical weighted stable curves

We consider tropical versions of Hassett’s moduli spaces of weighted stable curves M_{g,\mathcal{A}}^{\operatorname{trop}} , {\overline{M}_{g,\mathcal{A}}^{\operatorname{trop}}} and \Delta_{g,\mathcal{A}} associated to a weight datum \mathcal{A}=(a_1,\ldots,a_n)\in(\mathbb{Q}\cap(0,1])^n , their associated graph complexes {G^{(g,\mathcal{A})}} , and study the topology of these spaces as \mathcal{A} changes. We show that for fixed g and n , there are particular filtrations of these topological spaces and their graph complexes which may be used 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.