Fulton proves that the matrix Schubert variety overline{X_{pi }} cong Y_{pi } times mathbb {C}^q can be defined via certain rank conditions encoded in the Rothe diagram of pi in S_N. In the case where Y_{pi }:={{,textrm{TV},}}(sigma _{pi }) is toric (with respect to a (mathbb {C}^*)^{2N-1} action), we show that it can be described as a toric (edge) ideal of a bipartite graph G^{pi }. We characterize the lower dimensional faces of the associated so-called edge cone sigma _{pi } explicitly in terms of subgraphs of G^{pi } and present a combinatorial study for the first-order deformations of Y_{pi }. We prove that Y_{pi } is rigid if and only if the three-dimensional faces of sigma _{pi } are all simplicial. Moreover, we reformulate this result in terms of the Rothe diagram of pi .