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

TL;DR

This paper introduces a novel approach using parity and time-reversal to analyze complex behaviors in stochastic models, enabling efficient attractor detection in large Boolean networks and revealing insights into their scaling properties.

Contribution

It presents a new attractor identification algorithm based on parity and time-reversal, significantly improving scalability and uncovering the attractor scaling law in critical Boolean networks.

Findings

Attractor count scales with a low exponent of approximately 0.12.

The new algorithm is 80 times faster, handling networks with over 16,000 nodes.

The scaling law differs from previous analytical bounds, indicating a more constrained repertoire of behaviors.

Abstract

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 longstanding 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 surprisingly low scaling exponent of $0.12\pm 0.05$ -- 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.