线性布尔网络的稳定性
Stability of Linear Boolean Networks
AI总结：
本文研究线性布尔网络的稳定性，通过 Derrida 曲线和吸引子分析证明其有序到混沌的相变发生在平均入度 Kc=1，并给出吸引子比例、不动点期望数量及双射网络数量的理论结果。
摘要：
稳定性是网络模型的一个重要特性，它对可控性等其他理想性质具有影响。布尔网络的稳定性取决于多种因素，例如其线路图的拓扑结构以及描述其动态的函数类型。本文通过计算 Derrida 曲线并量化由网络拓扑所决定的吸引子数量和周期长度，研究线性布尔网络的稳定性。Derrida 曲线通常用于衡量布尔网络的稳定性，平均入度 K 和输出偏置 p 等若干参数可以指示网络是稳定的、临界的还是不稳定的。对于随机无偏布尔网络，存在一个临界连接值 Kc=2，使得当 K<Kc 时网络运行在有序区域，而当 K>Kc 时网络运行在混沌区域。在此，我们证明对于线性网络——它们是最少渠化且最不稳定的——从有序到混沌的相变已经在平均入度 Kc=1 时发生。与此一致，我们还证明不稳定网络表现出大量具有非常长极限环的吸引子，而稳定和临界网络则表现出更少的吸引子和更短的极限环。此外，我们给出了量化线性网络重要动力学性质的理论结果。首先，我们给出了线性系统中吸引子状态比例的公式。其次，我们证明线性系统中不动点的期望数量为 2，而一般布尔网络平均拥有一个不动点。第三，我们给出了量化双射线性布尔网络数量的公式，并为这类网络的百分比提供了一个下界。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
