Parity and time reversal elucidate both decision-making in empirical models and attractor scaling in critical Boolean networks

Jordan C. Rozum, Jorge Gómez Tejeda Zañudo, Xiao Gan, Dávid Deritei and Réka Albert
Science Advances  16 Jul 2021:
Vol. 7, no. 29, eabf8124
DOI: 10.1126/sciadv.abf8124

We present new applications of parity inversion and time reversal to the emergence of complex behavior from simple dynamical rules in stochastic discrete models. Our parity-based encoding of causal relationships and time-reversal construction efficiently reveal discrete analogs of stable and unstable manifolds. We demonstrate their predictive power by studying decision-making in systems biology and statistical physics models. These applications underpin a novel attractor identification algorithm implemented for Boolean networks under stochastic dynamics. Its speed enables resolving a long-standing open question of how attractor count in critical random Boolean networks scales with network size and whether the scaling matches biological observations. Via 80-fold improvement in probed network size (N = 16,384), we find the unexpectedly low scaling exponent of 0.12 ± 0.05, approximately one-tenth the analytical upper bound. We demonstrate a general principle: A system’s relationship to its time reversal and state-space inversion constrains its repertoire of emergent behaviors.

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