Automated Classification of Homeostasis Structure in Input-Output Networks

Homeostasis is widely observed in biological systems and refers to their ability to maintain an output quantity approximately constant despite variations in external disturbances. Mathematically, homeostasis can be formulated through an input-output function mapping an external parameter to an output variable. Infinitesimal homeostasis occurs at isolated points where the derivative of this input-output function vanishes, allowing tools from singularity theory and combinatorial matrix theory to characterize and classify homeostatic mechanisms in terms of network topology. Although the theoretical framework allows homeostasis subnetworks to be identified directly from combinatorial structures of the input-output network without numerical simulation, the required combinatorial enumeration becomes increasingly intractable as network size grows. Moreover, the reliance on advanced graph-theoretic concepts limits its broader accessibility and practical use across disciplines, particularly in biological applications. To overcome these limitations, we develop a Python-based algorithm that automates the identification of homeostasis subnetworks and their associated homeostasis conditions directly from network topology. Given an input-output network specified solely by its connectivity structure and the designation of input and output nodes, the algorithm automatically identifies the relevant graph-theoretical structures and enumerates all homeostatic mechanisms. We demonstrate the applicability of the algorithm across a range of biological examples, including small and large networks, networks with a single input parameter (with single or multiple input nodes), multiple input parameters, and cases where input and output coincide. This wide applicability stems from our extension of the theoretical framework from single-input-single-output networks to networks with multiple input nodes through an augmented single-input-node representation. The resulting computational framework provides a scalable and systematic approach to classifying homeostatic mechanisms in complex biological networks, facilitating the application of advanced mathematical theory to a broad range of biological systems.