更改

Since a Boolean network has only 2^N possible states, a trajectory will sooner or later  reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

Since a Boolean network has only 2^N possible states, a trajectory will sooner or later  reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2^N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为 一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。 如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。 导致吸引子的状态集称为吸引子的盆地。 仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。 到达吸引子所需的时间称为瞬态时间。

由于布尔网络只有 2^N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''<font color="#FF8000">伊甸园状态

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

{| class="wikitable sortable"

{| class="wikitable sortable"

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。 布尔网络的稳定性取决于其节点的连接。 布尔网络可以表现出稳定，关键或混乱的行为。 该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''<font color="#FF8000">海明距离 Hamming Distance </font>'''作为距离度量来表征。 在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。 在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''<font color="#FF8000">N-K模型 N-K-Model </font>'''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

 

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。 根据Bernard Derrida和Yves Pomeau的说法

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点

, the critical value of the average  number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average  number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average  number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average  number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

， 平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

 

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if  <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if  <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}/rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}/rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}/rangle >K_{c}</math> ，则为不稳定。

 

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>