Homeostasis in Input-Output Networks: Structure, Classification and Applications

The paper’s Theorem 4.12 classifies 3‑node input = output core networks into Family 1 (appendage ρ ⇌ σ) with det(H) = f_{σ,x_σ} f_{ρ,x_ρ} − f_{σ,x_ρ} f_{ρ,x_σ}, and Family 2 (two appendage path components) with det(H) = f_{σ,x_σ} f_{ρ,x_ρ}, concluding that homeostasis arises by null‑degradation in Family 2 and either by null‑degradation or 2‑node appendage cancellation in Family 1. This matches the candidate’s derivation exactly, including the use of the derivative criterion x′ ∝ det(H) under f_{ι,ℑ} ≠ 0 and det(J) ≠ 0. See the input‑output derivative formula and homeostasis criterion (2.21) and the input=output specialization of the homeostasis matrix, together with Theorem 4.12 and Example 4.4.