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1、The matrix is invertible； 2、The matrix satisfies the Markov chain normalization condition, meaning for any [math]i\in[1,N][/math], [math]|P_i|_1=1[/math]

1、The matrix is invertible； 2、The matrix satisfies the Markov chain normalization condition, meaning for any [math]i\in[1,N][/math], [math]|P_i|_1=1[/math]

This property is referred to as [[Dynamical Reversibility]]. Therefore, in a certain sense, EI measures a form of [[Dynamical Reversibility]] in the Markov chain.

This property is referred to as [[Dynamical Reversibility]] in ref<ref name="zhang_reversibility"/>. Therefore, in a certain sense, EI measures a form of [[Dynamical Reversibility]] in the Markov chain.

It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.

It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.

And when [math]\alpha\rightarrow 2[/math], then [math]\Gamma_{\alpha}[/math] degenerates into the square of the [[Frobinius Norm]] of matrix P, that is:

And when [math]\alpha\rightarrow 2[/math], then [math]\Gamma_{\alpha}[/math] degenerates into the square of the [[Frobinius Norm]] of matrix P, that is:

<math>

<math>

||P||_F^2=\sum_{i=1}^N\sigma_i^{2}

||P||_F^2=\sum_{i=1}^N\sigma_i^{2}

</math> This indicator measures the degree of certainty of matrix P, because only when all row vectors in matrix P are [[One-hot Vectors]], [math]||P||_F[/math] will be maximized. Therefore, it has a similar measurement effect to Determinism.

</math>  

So, when [math]\alpha\in(0,2)[/math] changes continuously, [math]\Gamma_{\alpha}[/math] can switch between degeneracy and determinacy. Normally, we take [math]\alpha=1[/math], which allows [math]\Gamma_{\alpha}[/math] to achieve a balance between determinacy and degeneracy.

 

So, when [math]\alpha\in(0,2)[/math] changes continuously, [math]\Gamma_{\alpha}[/math] can switch between degeneracy and determinacy. Normally, we take [math]\alpha=1[/math], which allows [math]\Gamma_{\alpha}[/math] to achieve a balance between determinism and degeneracy.

In reference <ref name="zhang_reversibility" />, the authors demonstrated an approximate relationship between EI and dynamic reversibility [math]\Gamma_{\alpha}[/math]：

In reference <ref name="zhang_reversibility" />, the authors demonstrated an approximate relationship between EI and dynamic reversibility [math]\Gamma_{\alpha}[/math]：

</math>

</math>

For further discussions on the [[Approximate Dynamical Reversibility of Markov chains]], refer to the entry on [[Approximate Dynamical Reversibility]] and the relevant paper：<ref name="zhang_reversibility" />

For further discussions on the approximate Dynamical Reversibility of Markov chains, refer to the entry on [[Approximate Dynamical Reversibility]] and the relevant paper：<ref name="zhang_reversibility" />

 

==EI and JS Divergence==

==EI and JS Divergence==

According to the expression of {{EquationNote|2}}, we know that EI is actually a generalized [[(JS) divergence]], namely [[Jensen-Shannon divergence]].

According to the expression of {{EquationNote|2}}, we know that EI is actually a generalized [[(JS) divergence]], namely [[Jensen-Shannon divergence]].