Lipschitz continuity under toric equivalence for asynchronous Boolean networks

Mathematical models rooted in network representations are becoming increasingly more common for capturing a broad range of phenomena. Boolean networks (BNs) represent a mathematical abstraction suited for establishing general theory applicable to such systems. A key thread in BN research is developing theory that connects the structure of the network and the local rules to phase space properties or so-called structure-to-function theory. While most theory for BNs has been developed for the synchronous case, the focus of this work is on asynchronously updated BNs (ABNs) which are natural to consider from the point of view of applications to real systems where perfect synchrony is uncommon. A central question in this regard is sensitivity of dynamics of ABNs with respect to perturbations to the asynchronous update scheme. Macauley & Mortveit [Nonlinearity 22, 421-436 (2009)] showed that the periodic orbits are structurally invariant under toric equivalence of the update sequences. In this paper and under the same equivalence of the update scheme, the authors (i) extend that result to the entire phase space, (ii) establish a Lipschitz continuity result for sequences of maximal transient paths, and (iii) establish that within a toric equivalence class the maximal transient length may at most take on two distinct values. In addition, the proofs offer insight into the general asynchronous phase space of Boolean networks.