Unlocking heterogeneity of node activation in Boolean networks through dynamical programming

We analyse the dynamics of node activation in networks of Boolean linear threshold units with fully asymmetric connectivity in the presence of noise, for which the dynamical cavity method provides the most efficient way to evaluate probabilities of dynamic trajectories. However, the complexity of the cavity approach grows exponentially with the in-degrees of nodes, which creates a de-facto barrier for the successful analysis of systems with fat-tailed in-degree distributions. In this letter, we present a dynamic programming algorithm that overcomes this barrier by reducing the computational complexity in the in-degrees from exponential to quadratic, whenever couplings are chosen randomly from (or can be approximated in terms of) a discrete set of equidistant values. As a case study, we analyse the dynamics of a random Boolean network with a fat-tailed degree distribution and fully asymmetric binary $\pm J$ couplings, and we use the power of the algorithm to unlock the noise dependent heterogeneity of stationary node activation patterns in such a system.