e maximization of KSE and the minimization of the mixing time**. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics and gives a physical meaning to the KSE. We first show that on average, the greater the KSE, the smaller the mixing time, and we correlated this result to its link with the transition matrix eigenvalues. Then, we show that the dynamics that maximises KSE is close to the one minimizing the mixing time, both in the sense of the optimal diffusion coefficient and the transition matrix.

Consider a network with $m$ nodes, on which a particle jumps randomly. This process can be described by a finite Markov chain defined by its adjacency matrix $A$ and its transition matrix $P$. $A(i,j)=1$ if and only if there is a link between the nodes $i$ and $j$ and 0 otherwise. $P=(p_{ij})$ where $p_{ij}$ is the probability for a particle in $i$ to hop on the $j$ node. Let us introduce the probability density at time $n$ $\mu_n=(\mu_n^i)_{i=1...m}$ where $\mu_n^i$ is the probability that a particle is at node $i$ at time $n$. Starting with a probability density $ \mu_0$, the evolution of the probability density writes: $\mu_{n+1}=P^t\mu_{n}$ where $P^t$ is the transpose matrix of $P$.\
Within this paper, we assume that the Markov chain is irreducible and thus has a unique stationary state.

Let us define: $$\label{eqdn}
 d(n)= max{ || (P^t)^n\mu - \mu_{stat}|| \text{  } \forall \text{ } \mu },$$ where $||.||$ is a norm on $\mathbb{R}^n$. For $ \epsilon >  0$, the mixing time, which corresponds to the time such that the system is within a distance $\epsilon$ from its stationary state is defined as follows:

$$\label{eq:mix1}
t(\epsilon)= \min{  n,  \, d(n) \leq \epsilon}.$$

For a Markov chain the KSE takes the analytical form [@billingsley1965ergodic]:

$$\label{eqhks}
 h_{KS}=-\sum_{ij} \mu_{stat_{i}}p_{ij}\log(p_{ij}).$$

Random $m$ size Markov matrices are ge