  For a given permutation $\pi \in S_N$, 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 the case
where $Y_{\pi}:=\text{TV}(\sigma_{\pi})$ is toric (with respect to a
$(\mathbb{C}^*)^{2N-1}$ action), we show that it can be described as an 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 Rothe diagram of $\pi$.
