Stability of linear Boolean networks

Stability is an important characteristic of network models that has implications for other desirable aspects such as controllability. The stability of a Boolean network (BN) depends on various factors, such as the topology of its wiring diagram and the type of the functions describing its dynamics. Linear Boolean networks can be completely described by their wiring diagram, and therefore the structure of linear networks plays a prominent role in determining their stability. In this paper, we study the stability of linear Boolean networks by computing Derrida curves and quantifying the number of attractors and cycle lengths imposed by their network topologies. Derrida curves are commonly used to measure the stability of Boolean networks and several parameters such as the average indegree K and the output bias p can indicate if a network is stable, critical, or unstable. For random unbiased Boolean networks there is a critical connectivity value K-c = 2 such that if K < K-c networks operate in the ordered regime, and if K > K-c networks operate in the chaotic regime. Here, we show that for linear networks, which are the least canalizing and most unstable, the phase transition from order to chaos already happens at an average in-degree of K-c = 1. Consistently, we also show that unstable networks exhibit a large number of attractors with very long limit cycles while stable and critical networks exhibit fewer attractors with shorter limit cycles. Additionally, we present theoretical results to quantify important dynamical properties of linear networks. First, we present a formula for the proportion of attractor states in linear systems. Second, we show that the expected number of fixed points in linear systems is 2, while general Boolean networks possess on average one fixed point. Third, we present a formula to quantify the number of bijective linear Boolean networks and provide a lower bound for the percentage of this type of network.