我们将复杂网络分析应用于随机布尔网络（RBN）的状态空间。RBN包含$N$布尔元素，每个布尔元素具有$K$输入。一个有向状态空间网络（SSN）是通过连接每个动态状态，表示为一个节点，其时间的继任者。我们研究了SSN在局部和全球尺度上的异质性，以及SSN合奏内的样本到样本的波动。我们使用节点的入度作为局部拓扑度量，并且路径多样性[Phys. Rev. Lett. 98，198701（2007）]的SSN作为一个全球拓扑措施。具有$2 \leq K \leq 5$的RBN在局部和全局尺度上都表现出非平凡的波动，而K=2表现出最大的样本到样本的波动，可能是非自平均的波动。我们解释所观察到的“多尺度”的波动在SSNs的临界性和复杂性的K=2 RBN的指示。“伊甸园”（GoE）状态是SSN上具有零度的节点。虽然用于$K>1$ SSN的非GoE节点的入度可以采用0和$2^N$之间的任何整数值，但是对于K=1，SSN中的所有非GoE节点具有相同的入度，其总是2的幂。
英文摘要：
We apply complex network analysis to the state spaces of random Boolean networks (RBNs). An RBN contains $N$ Boolean elements each with $K$ inputs. A directed state space network (SSN) is constructed by linking each dynamical state, represented as a node, to its temporal successor. We study the heterogeneity of an SSN at both local and global scales, as well as sample-to-sample fluctuations within an ensemble of SSNs. We use in-degrees of nodes as a local topological measure, and the path diversity [Phys. Rev. Lett. 98, 198701 (2007)] of an SSN as a global topological measure. RBNs with $2 \leq K \leq 5$ exhibit non-trivial fluctuations at both local and global scales, while K=2 exhibits the largest sample-to-sample, possibly non-self-averaging, fluctuations. We interpret the observed ``multi scale'' fluctuations in the SSNs as indicative of the criticality and complexity of K=2 RBNs. ``Garden of Eden'' (GoE) states are nodes on an SSN that have in-degree zero. While in-degrees of non-GoE nodes for $K>1$ SSNs can assume any integer value between 0 and $2^N$, for K=1 all the non-GoE nodes in an SSN have the same in-degree which is always a power of two.
A Novel Antifragility Measure Based on Satisfaction and Its Application to Random and Biological Boolean Networks
Universidad Nacional Autónoma de México（墨西哥国立自治大学）；Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México（墨西哥国立自治大学复杂性科学中心）；Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México（墨西哥国立自治大学应用数学与系统研究所）；ITMO University（ITMO大学）
