Permutation order governs basin entropy in critical one-input Boolean networks

Abstract

Basin entropy measures uncertainty about a network's long-term behavior. For every critical one-input synchronous Boolean network with $m$ relevant nodes, we show that the permutation structure induced by its feedback loops determines basin entropy, up to a universal error. Although the loops oscillate independently, their relative phases can be offset, producing a deviation from additivity that can grow linearly in $m$. Yet at the ensemble level, a previously derived additive prediction is preserved through the first subleading term, with the discrepancy appearing at order $\ln m\ln\ln m$.