## 2 Methodology
### Algorithms of Deterministic Annealing Clustering  
Under the setting of the deterministic annealing clustering algorithm, the expected distortion is defined as follows:  
\[D=\sum_{i=1}^{N}p(c_{i})\sum_{j=1}^{m}p(y_{j}|c_{i})D(c_{i},y_{j}) \tag{8}\]where \(C=\{c_{i}\}_{i=1}^{N}\) is the geospatial coordinate of each RSU node, \(Y=\left\{y_{j}\right\}_{j=1}^{m}\) is the cluster center point that needs to be calculated, that is, the position coordinates of the RSU controller. \(p(y_{j}|c_{i})\) is the probability of correlation between the point \(c_{i}\), the cluster center \(y_{j}\), \(D(c_{i},y_{j})\) is the deviation measure, and the square Euclidean distance is usually selected. \(p(c_{i})\) will be defined as giving the relative importance of the first node, assuming \(p(x_{i})=1/N\), if there is no other explanation. System entropy can be defined as \(H=-\sum_{i=1}^{N}p(c_{i})\sum_{j=1}^{m}p(y_{j}|c_{i})\log p(y_{j}|c_{i}).\) In addition, the free energy of the system \(F=D-TH\) is defined, where \(T\) is the temperature of the system. Note that \(F\) can be regarded as a Lagrangian function with the main goal of minimizing \(D\), where \(T\) is a Lagrangian multiplier. The cluster center iteration of the DAC algorithm can be summarized as sequentially optimizing the self-parameters corresponding to \(F\), namely the correlation probability and the cluster center position. Firstly, the distortion \(D(c_{i},y_{j})=d(c_{i},y_{j})+\gamma\sum_{h=1}^{m}d(y_{j};y_{j_{h}})\) is defined. Specifically, the distortion between an edge Road-Side Unit (RSU) node and a controller is determined not solely by the communication delay between these two nodes. It also hinges on the distance from other controllers. This means that the distortion between the edge RSU node \(c_{i}\) and the controller \(y_{j}\) depends not only on the communication delay between the two nodes, but also on the distance \(y_{j}\) from other controllers \(y_{j_{1}}\). In order to observe the relationship between (3) and (7), the total distortion of the model can be summarized as follows:  
\[D=\sum_{i=1}^{N}\sum_{j=1}^{m}p(y_{j}|c_{i})d(c_{i},y_{j})+\gamma\sum_{j_{1}=1} ^{m}\sum_{j=1}^{m}\left(d(y_{j},y_{j_{1}})\sum_{i=1}^{N}p(y_{j}|c_{i})\right) \tag{9}\]  
We replace the hard assignment with the soft correlation probability \(q_{ij}\). As mentioned above, we define the free energy of the system \(F=D-TH\), and let the partial derivative of the free energy term with respect to the correlation probability be zero:  
\[p(y_{j}|c_{i})=\frac{\exp(-\frac{D(c_{i},y_{j})}{T})}{Z_{i}},\ \ Z_{i}=\sum_{j=1}^{m}p(y_{j}|c_{i}) \tag{10}\]  
Therefore, the correlation probability obeys the Boltzmann distribution. Similarly, let the derivative with respect to the cluster center \(y_{j}\) be zero, and the following linear equations can be obtained:  
\[\eta y_{j}-\gamma\sum_{j_{1}\
eq j}y_{j_{1}}=B_{j},\ j=1,\ldots,m \tag{11}\]  
where \(\eta=\gamma(m-1)+1\), and \(B_{j}=\sum_{i=1}^{N}p(c_{i}|y_{j})c_{i}\) can be calculated by Bayesian rule \(p(y_{j}|c_{i})\). This gives us a linear system of variables \(md\), where \(m\) and \(d\) are the number of cluster centers and the dimension of the data, respectively. For the convergence of the deterministic annealing clustering algorithm, it is essential that the linear equations always have solutions. **Theorem 1**. : _Given a linear system of Equation (11) as defined above, if \(\gamma\
eq\frac{1}{n-m}\), \(\frac{1}{n-2m}\) then there is always a unique solution \(\left\{y_{j}\right\}_{j=1}^{m}\), where the coefficient matrix associated with the system of equations is non-degenerate, its determinant is \(\left(\frac{(\gamma m+1)^{n}(\gamma(n-m)-1))}{\gamma(n-2m)-1}\right)^{d}\); the detailed DAC algorithm pseudo-code is as follows:_  
As shown in Algorithm 1, a large number of experiments are first needed to determine the tolerance level \(\delta\). For the convergence test, if \(||F_{\tau}-F_{\tau-1}||<\delta\), then for a predetermined level of tolerance \(\delta\), stop the iteration \(\tau\).