The finest decompositions' architecture of a reaction network

Abstract

Biochemical and environmental modeling typically relies on reaction networks to represent complex transformations. While the Linkage Class Decomposition (LCD) partitions networks based on visual standard connectivity, it often misaligns with the algebraic properties governing long-term dynamics. This work establishes the Finest Decompositions' Architecture (FDA) framework by analyzing hierarchical relationships between the LCD and two algebraic structures: the Finest Independent Decomposition (FID) and the Finest Incidence-Independent Decomposition (FIID). These algebraic decompositions serve as the respective building blocks for characterizing general equilibria and complex-balanced equilibria of a reaction network. Under the partial order of "coarsens to," we categorize reaction networks into six architectures, distinguishing three subclasses of Independent Linkage Classes (ILC) from three subclasses of Dependent Linkage Classes (DLC). To facilitate the classification, we introduce the Deficiency Difference (Delta), measuring the discrepancy between total and subnetwork deficiencies, and the Common Complexes Cardinality CC of the FID. Results show that Delta uniquely identifies all the ILC classes and one DLC subclass, while CC distinguishes the remaining DLC subclasses. A number of results on mass action systems such as the Deficiency One Theorem as well as on power law systems essentially rely on the ILC property of the underlying networks. These suggest that the FDA classification of ILC and DLC networks signify a certain alignment of both structural and kinetic attributes. This work opens up direction for the study of the structure and equilibria analysis of reaction networks across diverse decomposition architectures.