Stability is an important characteristic of network models that has implications for other desirable aspects such as controllability. The stability of a Boolean network depends on various factors, such as the topology of its wiring diagram and the type of the functions describing its dynamics. 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 in-degree 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 Kc=2 such that if K<Kc networks operate in the ordered regime, and if K>Kc 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 Kc=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.

翻译：稳定性是网络模型的重要特征，对可控性等其他理想性质具有重要影响。布尔网络的稳定性取决于多种因素，例如其连接图的拓扑结构以及描述其动态特性的函数类型。本文通过计算Derrida曲线并量化网络拓扑所施加的吸引子数量和周期长度，研究了线性布尔网络的稳定性。Derrida曲线常用于衡量布尔网络的稳定性，平均入度K和输出偏差p等参数可指示网络处于稳定、临界或不稳定状态。对于随机无偏布尔网络，存在临界连通性值Kc=2：当K<Kc时，网络运行有序状态；当K>Kc时，网络运行混沌状态。本文证明，对于线性网络（最小化约束性且最不稳定的网络），从有序到混沌的相变已在平均入度Kc=1处发生。一致地，我们还表明不稳定网络表现出大量具有极长极限环的吸引子，而稳定和临界网络中吸引子数量较少且极限环较短。此外，我们提出理论结果以量化线性网络的重要动态特性：首先，给出线性系统中吸引子状态比例的公式；其次，证明线性系统中不动点的期望值为2，而一般布尔网络平均仅拥有一个不动点；第三，提出量化双射线性布尔网络数量的公式，并给出此类网络百分比的下界。