Before revision:  for each \mathcal{I} near \mathcal{I}_0 and infinitesimal homeostasis occurs at \mathcal{I}_0 when (dx_o/d\mathcal{I})(\mathcal{I}} _0) = 0 by showing that there is a square homeostasis matrix H associated to \mathcal{G) for which dx_o/d\mathcal{I} = 0 if and only if } \det(H) = 0. Applying combinatorial matrix theory and graph theory to H allows us to classify types of homeostasis. We prove that the homeostasis types correspond to a set of irreducible blocks in H each associated with a subnetwork and these subnetworks.
After revision:  for each \mathcal{I} near \mathcal{I}_0 and infinitesimal homeostasis occurs at \mathcal{I}_0 when (dx_o/d\mathcal{I})(\mathcal{I}} _0) = 0 by showing that there is a square homeostasis matrix H associated to \mathcal{G) for which dx_o/d\mathcal{I} = 0 if and only if } \det(H) = 0. Applying combinatorial matrix theory to factor the polynomial \det(H) and thereby determine a menu of different types of possible homeostasis associated with each digraph \mathcal{G.