Reducing Boolean Networks via Analysis of Dynamic Network Subgraph Behavior

Abstract

Boolean networks provide a compact framework for modeling regulatory systems, yet their rapidly expanding state spaces make systematic dynamical analysis challenging. Here, we systematically enumerate all non-isomorphic two-node signed regulatory subgraphs with their admissible Boolean update rules and exhaustively characterize their state-transition graphs (STGs) under both synchronous and asynchronous updating. By extracting quantitative dynamical descriptors, we move beyond static wiring diagrams toward a behavior-based classification of network structures. The synchronous analysis identifies deterministic fixed-point and cyclic attractor regimes, while the asynchronous analysis captures one-node-at-a-time dynamics through terminal strongly connected components of the asynchronous STGs. Comparing the two update schemes distinguishes dynamical behaviors that are robust across update assumptions from those that depend on synchronous updating. Despite the diversity of graph-rule combinations, many realizations converge to a limited repertoire of dynamical classes, revealing substantial redundancy between structural and dynamical representations. We further propose a testable framework for investigating whether such local dynamical classes can be used to characterize selected subnetworks within larger Boolean models. To facilitate exploration and reproducibility, we developed BORNA, an interactive web application for exploring the complete network catalogue, STGs, and synchronous and asynchronous simulations, available at https://jafarilab.github.io/BORNA/. Together, these results provide a systematic reference for minimal Boolean network dynamics and a foundation for extending behavior-based analysis to larger regulatory systems.