Z. Huang and M. Golubitsky
Classification of infinitesimal homeostasis in four-node input-output networks
J. Math. Biol.
An input-output network has an input node i, an output node o, and regulatory
nodes j. Such a network is a core network if each j is downstream from I and up-
stream from o. Wang, Huang, Antonelli and Golubitsky  show that infinitesimal homeostasis can be classified in biochemical networks through infinitesimal
homeostasis in core subnetworks. Golubitsky and Wang  show that there are
three types of 3-node core networks and three types of infinitesimal homeostasis
in 3-node core networks. This paper uses the theory developed in Wang et al. 
to show that there are twenty types of 4-node core networks (Theorem 1.3) and
seventeen types of infinitesimal homeostasis in 4-node core networks (Theorem 1.7). Biological contexts illustrate the classification theorems and show that the theory
can help when calculating homeostasis in specific biochemical networks.