Cooperative Boolean systems with generically long attractors I

TL;DR

This paper investigates cooperative Boolean networks with simple regulatory functions, demonstrating that large networks can possess extremely long attractors occupying a significant portion of the state space, contrasting with other dynamical systems.

Contribution

It proves the existence of large cooperative Boolean networks with exponentially long attractors that dominate the state space, highlighting their generic dynamical complexity.

Findings

Existence of Boolean networks with attractors longer than c^N for c<2

Large basins of attraction for these long attractors

Contrast with non-cooperative systems showing fewer long attractors

Abstract

We study the class of cooperative Boolean networks whose only regulatory functions are COPY, binary AND, and binary OR. We prove that for all sufficiently large N and c < 2 there exist Boolean networks in this class that have an attractor of length > c^N whose basin of attraction comprises an arbitrarily large fraction of the state space. The existence of such networks contrasts with results on various other types of dynamical systems that show nongenericity or absence of non-steady state attractors under the assumption of cooperativity.