Samuelsson Björn, Troein Carl
Complex Systems Division, Department of Theoretical Physics, Lund University, Sölvegatan 14A, S-223 62 Lund, Sweden.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Oct;72(4 Pt 2):046112. doi: 10.1103/PhysRevE.72.046112. Epub 2005 Oct 13.
Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per node can be seen as graphs of random maps. We introduce an approach to investigating random maps and finding analytical results for attractors in random Boolean networks with the corresponding topology. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors of various lengths. We also investigate observables related to the average number of attractors in relation to the typical number of attractors. Here, we find strong differences that highlight the difficulties in making direct comparisons between random Boolean networks and real systems. Furthermore, we demonstrate the power of our approach by deriving some results for random maps. These results include the distribution of the number of components in random maps, along with asymptotic expansions for cumulants up to the fourth order.
尽管随机布尔网络表面看似简单，却展现出丰富多样的动力学行为。许多工作都聚焦于吸引子的性质和数量。每个节点有一个输入的随机布尔网络的拓扑结构可被视为随机映射图。我们引入一种方法来研究随机映射，并为具有相应拓扑结构的随机布尔网络中的吸引子找到解析结果。将其他一些非混沌网络近似为此类网络后，我们将解析结果应用于它们。对于这种近似，我们观察到在不同长度吸引子的数量上有惊人的良好一致性。我们还研究了与吸引子平均数量相关的可观测量以及典型吸引子数量。在此，我们发现了显著差异，突出了在随机布尔网络与实际系统之间进行直接比较的困难。此外，我们通过推导随机映射的一些结果来展示我们方法的威力。这些结果包括随机映射中组件数量的分布，以及直至四阶累积量的渐近展开。
