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→Boolean Network Example
However, for more general Markov chains and more general state groupings, this simple operation of averaging probabilities is not always feasible. This occurs when the merged transition matrix does not meet Markov chain conditions, such as row normalization or element values within [0, 1]. Refer to the “Reduction of Markov Chains” section or the “[[Coarse-graining of Markov Chains]]” entry for details on feasible macroscopic Markov chains.
However, for more general Markov chains and more general state groupings, this simple operation of averaging probabilities is not always feasible. This occurs when the merged transition matrix does not meet Markov chain conditions, such as row normalization or element values within [0, 1]. Refer to the “Reduction of Markov Chains” section or the “[[Coarse-graining of Markov Chains]]” entry for details on feasible macroscopic Markov chains.
=====Boolean Network Example=====
=====Boolean network example=====
Another instance of causal emergence in the literature <ref name=":0"/> involves a [[Boolean network]]. The figure illustrates a Boolean network with 4 nodes. Each node can exist in one of two states, 0 or 1. Each node is connected to two other nodes and operates according to identical microscopic dynamics (figure a). Therefore, this system contains a total of sixteen microscopic states, and its dynamics can be represented by a state transition matrix (figure c).
Another instance of causal emergence in the literature <ref name=":0"/> involves a [[Boolean network]]. The figure illustrates a Boolean network with 4 nodes. Each node can exist in one of two states, 0 or 1. Each node is connected to two other nodes and operates according to identical microscopic dynamics (figure a). Therefore, this system contains a total of sixteen microscopic states, and its dynamics can be represented by a state transition matrix (figure c).
[[文件:含有4个节点的布尔网络.png|居左|700x700像素|离散布尔网络上的因果涌现|替代=含有4个节点布尔网络的因果涌现]]
[[文件:含有4个节点的布尔网络.png|居左|700x700像素|离散布尔网络上的因果涌现|替代=含有4个节点布尔网络的因果涌现]]
=====Causal Emergence in Continuous Variables=====
=====Causal Emergence in Continuous Variables=====