## 2 Methods
### Kolmogorov Entropy Method  
As mentioned above, in this project, we want to formulate a complete description of the traffic situation in real time. To do this, while keeping the basics, it is necessary to revolutionize the elaboration mechanism to implement the system on the embedded microcontroller.  
This is achieved by introducing Kolmogorov entropy. The analytical solution of the Euler-Lagrange system for a traffic node could be described (7)  
\[\{Y_{n}\},\,n=1,\,\ldots,N \tag{7}\]  
and can be unfolded in a multi-dimensional effective phase space using time delay coordinates (\(t\) is a delay time (embedding)) (8).  
\[Y_{n}=(Y_{n-(m-1)t},\,\ldots,Y_{n}) \tag{8}\]  
An important observation is that the Euler-Lagrange equation always has a \(\{Y_{n}\}\) solution, but a distinction must be made between the solutions that belong to the analytical solutions and those that do not, i.e., in which case we can make statements for the completeness of the node.  
If \(\{Y_{n}\}\) is an analytical solution of a Euler-Lagrange dynamical system for a given traffic scenario, it can be shown under certain genericity conditions that the reconstructed point set is a one-to-one image of the original trajectory of the traffic node dynamical system, where completeness and provability are to be defined.  
In our article, to test the comparability of the traffic node, we use Kolmogorov entropy [36], which is an important characteristic describing the degree of solvability of the Euler-Lagrange system [37]. The entropy gives the average rate of information loss about the position of the phase point on the attractor.  
In our article, we are looking for the traffic node, which is completely provable and consistent if (9)  
\[0<K<\infty \tag{9}\]  
Incidentally, our entropy method can also be used to characterize the existing traffic node state because of the following:  
* \(K\) is infinite in a random traffic node dynamics system -\(>\) the system is not complete.
* \(K=0\) in a chaotic traffic node dynamics system -\(>\) the system is not complete.  
The Kolmogorov entropy \(K_{q}\) can be defined as follows; -\(>\) Let \(Y(t)\) and \(t>0\) be the trajectory of the Euler-Lagrange equation for a traffic node in an \(n\)-dimensional phase space sampled at discrete time intervals \(\Delta t\). Let us divide the phase space into the \(n\)-dimensional hypercubes of side \(r\). Let \(P_{i1,i2,\,\cdots,\,iN}\) be the joint probability that the trajectory \(Y(t)\) on the Euler-Lagrange equation subsequently visits cubes \(i_{1}\), \(i_{2}\), \(\cdots\), and \(i_{N}\) at times \(t=\Delta t\), \(t=2\Delta t\), \(\cdots\), and \(N\Delta t\).  
The Kolmogorov entropy Is then (\(1^{\prime\prime}\))  
\[K_{q}=-\underset{r\to 0\Delta t\to 0N\to\infty}{\lim}\underset{N\to\infty}{\lim} \frac{1}{N\Delta t}\frac{1}{q-1}ln\underset{i_{1}i_{2}\ldots i_{N}}{\sum}P_{ i_{1}i_{2}\ldots i_{N}} \tag{10}\]In our article, we use the \(K_{2}\) form (11)  
\[K_{2}\sim\lim_{m\rightarrow\infty r\to 0}K_{2}^{m}(r), \tag{11}\]  
where (12)  
\[K_{2}^{m}=\frac{1}{k\Delta t}ln\frac{C^{m}(r)}{C^{m+k}(r)} \tag{12}\]  
and \(k\) is a sufficiently small integer number and \(m\) is the embedding dimension. The saturation value of \(K_{2}^{m}\) as m increases is regarded as \(K_{2}\) and \(C^{m}(r)\), and the correlations integral defined by (13)  
\[C^{m}(i,r)=\frac{1}{M-1}\sum_{\begin{subarray}{c}j=1\\ i\
eq j\end{subarray}}^{M}\theta\big{(}r-\left\|Y_{i}-Y_{j}\right\|\big{)} \tag{13}\]  
where \(M=N-(m-1)/\Delta t\) and \(N\) is the number of data points in the Euler-Lagrange trajectory. \(\theta(x)\) is a Heaviside step function defined by (14)  
\[\theta(x)=\left\{\begin{matrix}1\;for\;x\geq 0\\ 0\;for\;x<0\end{matrix}\right\} \tag{14}\]  
In the following, we show how we can use entropy to prove the completeness of a transport node.