# DELER: Distributed Clustering with Local Direction Centrality and Density Measure
## 3 The DEALER Algorithm
### Data Partitioning
#### 3.3.2 Data Partition Adjustment
The Silhouette Coefficient is a widely used metric for evaluating clustering quality, measuring both the cohesion of data points within a cluster and their separation from other clusters. The Davies-Bouldin Index is an internal evaluation metric that assesses clustering performance based on intra-cluster compactness and inter-cluster separation. (1) Comparison Algorithms: The comparison algorithms used in this section include the classical k-means clustering algorithm, DBSCAN, DPC, and CDC. The parameter settings for these algorithms are based on their original research papers. (2) Accuracy Comparison: Table k presents the performance of the five algorithms across four evaluation metrics. Among these metrics, the Davies-Bouldin Index has \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{ACC} & \multicolumn{4}{c}{NMI} \\ \cline{2-9} algorithm & DS1 & DS2 & Flame & Spiral & DS1 & DS2 & Flame & Spiral \\ \hline k-means & 0.9940 & 0.8279 & 0.8417 & 0.3397 & 0.9874 & 0.8878 & 0.4267 & 0.0033 \\ DBSCAN & 0.7998 & 0.7669 & 0.6333 & **0.9968** & 0.9643 & 0.9551 & 0.8687 & **0.9918** \\ DPC & **0.9990** & 0.9216 & **0.9917** & 0.6090 & **0.9976** & 0.9111 & **0.9355** & 0.2845 \\ CDC & 0.9009 & 0.8105 & 0.6375 & 0.3718 & 0.9689 & **1.0000** & 0 & 0.0589 \\ DEALER & 0.9980 & **1.0000** & **0.9917** & **0.9968** & 0.9957 & **1.0000** & 0.9269 & **0.9918** \\ \hline \hline \end{tabular}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{ARI} & \multicolumn{4}{c}{F-score} \\ \cline{2-9} algorithm & DS1 & DS2 & Flame & Spiral & DS1 & DS2 & Flame & Spiral \\ \hline k-means & 0.9867 & 0.7834 & 0.4649 & \(-\)0.0063 & 0.9940 & 0.8623 & 0.8447 & 0.3390 \\ DBSCAN & 0.8926 & 0.9501 & 0.9336 & **0.9951** & 0.7670 & 0.7680 & 0.6333 & **0.9952** \\ DPC & **0.9978** & 0.8520 & **0.9666** & 0.2097 & **0.9990** & 0.8864 & 0.9916 & 0.6056 \\ CDC & 0.8969 & **1.0000** & 0 & 0.0185 & 0.8676 & 0.8105 & 0.4964 & 0.3289 \\ DEALER & 0.9956 & **1.0000** & **0.9666** & **0.9951** & 0.9980 & **1.0000** & **0.9917** & **0.9952** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Clustering metrics of various algorithms on datasets. a lower bound of 0, where a smaller value indicates better clustering performance. The Silhouette Coefficient ranges from \(-1\) to 1, where values closer to 1 indicate better clustering quality, meaning that samples are well clustered within their own groups while being well separated from other clusters. However, both metrics are more suitable for convex-shaped clusters and may not be effective for non-convex structures, such as ring-shaped clusters. Additionally, when the dataset contains only a single cluster, both metrics may produce errors and be invalid. Table 4 presents the clustering quality evaluation metrics for five different algorithms across four datasets. From the data in the table, it is evident that for DS1, where clusters are predominantly convex, most algorithms produce reasonable clustering results. Among them, DEALER demonstrates the best overall performance. However, on the other three datasets, the clustering quality metrics are generally suboptimal. In conjunction with the clustering visualizations shown in Figures 15-18, this issue primarily arises because, in these three datasets (excluding DS1), many clusters exhibit non-convex shapes, making these two parameters less suitable and leading to irregular performance. Specifically, in the Flame dataset, weak connectivity causes the CDC algorithm to misclassify the entire dataset as a single cluster. As a result, the Davies-Bouldin Index and Silhouette Coefficient metrics for CDC fail to compute, leading to errors and missing results. ### Impact of Parameter Variations on Clustering Results

This section presents an experimental analysis of the relationship between clustering results and parameter variations in the DEALER algorithm. Specifically, it examines the impact of the number of k-nearest neighbors (\(k\)) and the _DCM_ threshold (\(T_{DCM}\)) on clustering performance.