# DELER: Distributed Clustering with Local Direction Centrality and Density Measure
## 3 The DEALER Algorithm
### Data Partitioning
#### 3.3.2 Data Partition Adjustment
Since the parameters used in this study are similar, the CDC algorithm is selected for comparison with DEALER. However, due to the weak connectivity of the DS2, Flame, and

Figure 16: Clustering visualization: DS2. Figure 17: Clustering visualization: Flame. Figure 18: Clustering visualization: Spiral. Spiral datasets, the CDC algorithm performs poorly on these datasets. Therefore, the more stable DS1 dataset is used for evaluation, comparing external clustering metrics such as ACC, AMI, ARI, and F-score. Tables 5 and 6 present the variations in these metrics under different parameter settings. Specifically, Table 5 reports the external metrics when the independent variable is the number of k-nearest neighbors (\(k\)), while Table 6 shows the results when the independent variable is the \(DCM\) threshold (\(T_{DCM}\)). From Tables 5 and 6, it is evident that DEALER maintains superior clustering performance across a broader range of parameter variations compared to CDC, demonstrating greater stability. The number of k-nearest neighbors (\(k\)) has a relatively smaller impact on clustering results over a wide range of values compared to \(T_{DCM}\). ### Case Analysis

This section provides a detailed analysis of how the DEALER algorithm ensures the validity of clustering results at each key step. To facilitate a better understanding and visualization of the analysis, we use two datasets, DS1 and Flame, which are easier to visualize. These datasets allow for a more intuitive perception of the data distribution and clustering results. Below, we analyze the parts of the clustering process that directly influence the clustering outcomes. Based on the analysis in Section 3, it is clear that the resolution of weak connectivity issues depends on the clustering method, especially for datasets with complex structures. Therefore, the choice of clustering method is critical to the clustering results. For better visualization and analysis, we again use the two datasets mentioned above: the synthetic dataset DS1 and the real-world dataset Flame. There are two clustering approaches to compare: one is the method used in the CDC algorithm, which involves surrounding core points with boundary points to form clusters, and the other is the local expansion approach used in DEALER, where clusters are first formed by expanding from center points. Figures 19-22 show the scatter plots of the clustering results using the CDC algorithm and the DEALER algorithm, respectively, on the DS1 and Flame datasets. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline  & \multicolumn{4}{c}{**CDC**} & \multicolumn{4}{c}{**DEALER**} \\ \cline{2-9} \(T_{DCM}\) & **ACC** & **NMI** & **ARI** & **F-Score** & **ACC** & **NMI** & **ARI** & **F-Score** \\ \hline
0.15 & 0.9009 & 0.9689 & 0.8969 & 0.8676 & 0.9980 & 0.9957 & 0.9956 & 0.9980 \\
0.20 & 0.9009 & 0.9689 & 0.8969 & 0.8676 & 0.9980 &



Figure 19 and 20 show the scatter plots of the clustering results for the DS1 dataset using the CDC and DEALER clustering methods, respectively. It is evident that the clustering results obtained with the CDC algorithm are inaccurate. This is because the CDC algorithm uses a boundary-point-enclosing approach for clustering the center points. When the distance between two adjacent boundary points is large, it becomes easy for the algorithm to merge with other clusters that are closer in proximity, leading to the occurrence of weak connectivity issues. Figure 21: Scatter plot of clustering results of dataset Flame in CDC clustering mode. Figure 19: Scatter plot of clustering results of data at DS1 in the CDC clustering mode. Figures 21 and 22 present the scatter plots of clustering results for the Flame dataset using the CDC and DEALER clustering methods, respectively. It is clear that the DEALER algorithm performs well, producing high-quality clustering results. The local expansion approach used in DEALER allows for clustering by expanding from center points, with expansion halting when a boundary point is encountered.