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The author gives four specific examples of Markov chains. The state transition matrix of this Markov chain is shown in the figure. We can compare the <math>EI</math> and approximate dynamical reversibility (the <math>\Gamma</math> in the figure, that is, <math>\Gamma_{\alpha = 1}</math>) of this Markov chain. Comparing figures a and b, we find that for different state transition matrices, when <math>EI</math> decreases, <math>\Gamma</math> also decreases simultaneously. Further, figures c and d are comparisons of the effects before and after coarse-graining. Among them, figure d is the coarse-graining of the state transition matrix of figure c (merging the first three states into a macroscopic state). Since the macroscopic state transition matrix in figure d is a deterministic system, the normalized <math>EI</math>, <math>eff\equiv EI/\log N</math> and the normalized [math]\Gamma[/math]: <math>\gamma\equiv \Gamma/N</math> all reach the maximum value of 1.
The author gives four specific examples of Markov chains. The state transition matrix of this Markov chain is shown in the figure. We can compare the <math>EI</math> and approximate dynamical reversibility (the <math>\Gamma</math> in the figure, that is, <math>\Gamma_{\alpha = 1}</math>) of this Markov chain. Comparing figures a and b, we find that for different state transition matrices, when <math>EI</math> decreases, <math>\Gamma</math> also decreases simultaneously. Further, figures c and d are comparisons of the effects before and after coarse-graining. Among them, figure d is the coarse-graining of the state transition matrix of figure c (merging the first three states into a macroscopic state). Since the macroscopic state transition matrix in figure d is a deterministic system, the normalized <math>EI</math>, <math>eff\equiv EI/\log N</math> and the normalized [math]\Gamma[/math]: <math>\gamma\equiv \Gamma/N</math> all reach the maximum value of 1.
====Dynamic independence====
Dynamic independence is a method to characterize the macroscopic dynamical state after coarse-graining being independent of the microscopic dynamical state [40]. The core idea is that although macroscopic variables are composed of microscopic variables, when predicting the future state of macroscopic variables, only the historical information of macroscopic variables is needed, and no additional information from microscopic history is needed. This phenomenon is called dynamic independence by the author. It is another means of quantifying emergence. The macroscopic dynamics at this time is called emergent dynamics. The independence, causal dependence, etc. in the concept of dynamic independence can be quantified by transfer entropy.
=====Quantification of dynamic independence=====
Transfer entropy is a non-parametric statistic that measures the amount of directed (time-asymmetric) information transfer between two stochastic processes. The transfer entropy from process <math>X</math> to another process <math>Y</math> can be defined as the degree to which knowing the past values of <math>X</math> can reduce the uncertainty about the future value of <math>Y</math> given the past values of <math>Y</math>. The formula is as follows:
<math>T_t(X \to Y) = I(Y_t : X^-_t | Y^-_t) = H(Y_t | Y^-_t) - H(Y_t | Y^-_t, X^-_t)</math>
Here, <math>Y_t</math> represents the macroscopic variable at time <math>t</math>, and <math>X^-_t</math> and <math>Y^-_t</math> represent the microscopic and macroscopic variables before time <math>t</math> respectively. [math]I[/math] is mutual information and [math]H[/math] is Shannon entropy. <math>Y</math> is dynamically decoupled with respect to <math>X</math> if and only if the transfer entropy from <math>X</math> to <math>Y</math> at time <math>t</math> is <math>T_t(X \to Y)=0</math>.
The concept of dynamic independence can be widely applied to a variety of complex dynamical systems, including neural systems, economic processes, and evolutionary processes. Through the coarse-graining method, the high-dimensional microscopic system can be simplified into a low-dimensional macroscopic system, thereby revealing the emergent structure in complex systems.
In the paper, the author conducts experimental verification in a linear system. The experimental process is: 1) Use the linear system to generate parameters and laws; 2) Set the coarse-graining function; 3) Obtain the expression of transfer entropy; 4) Optimize and solve the coarse-graining method of maximum decoupling (corresponding to minimum transfer entropy). Here, the optimization algorithm can use transfer entropy as the optimization goal, and then use the gradient descent algorithm to solve the coarse-graining function, or use the genetic algorithm for optimization.
=====Example=====
The paper gives an example of a linear dynamical system. Its dynamics is a vector autoregressive model. By using genetic algorithms to iteratively evolve different initial conditions, the degree of dynamical decoupling of the system can also gradually increase. At the same time, it is found that different coarse-graining scales will affect the degree of optimization to dynamic independence. The experiment finds that dynamic decoupling can only be achieved at certain scales, but not at other scales. Therefore, the choice of scale is also very important.