# An Edge-Based Approach to Partitioning and Overlapping Graph Clustering with User-Specified Density
## 4 Proposed Partitioning and Overlapping Clustering Method
This suggests that nodes within clusters tend to have strong connections to their respective clusters, indicating a high level of interconnectivity. The higher values of average embeddedness can be attributed to the fact that this metric considers both the internal degree of nodes and their overall degree. Referring to Table 5, the cut ratio metric yields the most favorable results among the existing external connectivity metrics. This is attributed to its consideration of external degrees within clusters and factors such as cluster size and total graph nodes. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Network**} & \multicolumn{2}{c}{**Cov**} & \multicolumn{2}{c}{**Avg-Embed**} & \multicolumn{2}{c}{**Avg-FOMD**} & \multicolumn{2}{c}{**Avg-TPR**} & \multicolumn{2}{c}{**Avg-IED**} \\ \cline{2-11}  & **0.4** & **0.6** & **0.4** & **0.6** & **0.4** & **0.6** & **0.4** & **0.6** & **0.4** & **0.6** \\ \hline Zachary’s Karate Club & 0.55 & 0.67 & **0.64** & **0.77** & 0.25 & 0.27 & 0.42 & 0.40 & 0.50 & 0.51 \\ \hline Aves-Weaver Social & 0.54 & 0.74 & 0.55 & 0.71 & 0.10 & 0.20 & 0.58 & **0.80** & **0.85** & 0.75 \\ \hline Dolphins Interaction & 0.51 & **0.62** & 0.50 & 0.57 & 0.21 & 0.29 & 0.33 & 0.35 & **0.67** & 0.60 \\ \hline Les Misérables & 0.57 & **0.70** & **0.61** & 0.66 & 0.19 & 0.30 & 0.57 & 0.45 & 0.55 & 0.44 \\ \hline Political Books & 0.58 & **0.66** & 0.55 & 0.52 & 0.30 & 0.33 & **0.67** & 0.65 & 0.41 & 0.45 \\ \hline American College Football & 0.53 & 0.67 & 0.32 & 0.51 & 0.25 & 0.23 & 0.49 & **0.88** & **0.54** & 0.35 \\ \hline Facebook Pages Food & 0.52 & 0.66 & 0.59 & **0.68** & 0.14 & 0.24 & 0.25 & 0.33 & **0.76** & **0.68** \\ \hline US-Grid Power & 0.50 & 0.68 & 0.61 & **0.73** & 0.20 & 0.24 & 0.23 & 0.08 & **0.64** & 0.51 \\ \hline \end{tabular}
\end{table} 
Table 4: Partitioning resultant clusters’ internal connectivity evaluation under \(U(\delta_{r})\) 0.4 and 0.6. Best results are shown in bold. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Network**} & \multicolumn{2}{c}{**Conductance**} & \multicolumn{2}{c}{**Cut Ratio**} & \multicolumn{2}{c}{**Normalized Cut**} & \multicolumn{2}{c}{**Flake-ODF**} \\ \cline{2-9}  & **0.4** & **0.6** & **0.4** & **0.6** & **0.4** & **0.6** & **0.4** & **0.6** \\ \hline Zachary’s Karate Club & 0.43 & 0.27 & **0.10** & **0.06** & 0.65 & 0.51 & 0.38 & 0.35 \\ \hline Aves-Weaver Social & 0.37 & 0.26 & **0.08** & 0.04 & 0.54 & 0.36 & 0.32 & **0.02** \\ \hline Dolphins Interaction & 0.48 & 0.45 & **0.04** & **0.03** & 0.53 & 0.56 & 0.42 & 0.31 \\ \hline Les Misérables & 0.39 & 0.30 & **0.05** & **0.03** & 0.51 & 0.50 & 0.29 & 0.24 \\ \hline Political Books & 0.42 & 0.36 & **0.04** & **0.02** & 0.50 & 0.49 & 0.32 & 0.38 \\ \hline American College Football & 0.48 & 0.33 & **0.06** & **0.05** & 0.67 & 0.50 & 0.45 & 0.47 \\ \hline Facebook Pages Food & 0.43 & 0.36 & **0.00** & **0.00** & 0.49 & 0.40 & 0.29 & 0.21 \\ \hline US-Grid Power & 0.42 & 0.28 & **0.00** & **0.00** & 0.43 & 0.31 & 0.15 & 0.06 \\ \hline \end{tabular}
\end{table} 
Table 5: Partitioning resultant clusters’ external connectivity evaluation under \(U(\delta_{r})\) 0.4 and 0.6. Best results are shown in bold. The trend of higher metric values with higher-density specifications for internal quality measures suggests a bias towards larger clusters and network size. Interestingly, this trend is reversed when considering external evaluation metrics. This bias highlights the sensitivity of the metric to network structure and edge count rather than solely capturing clustering quality. ### Overlapping Clustering Evaluation through Pre-Existing Quality Metrics

Referring to Table 6, in the context of overlapping clustering analysis using existing metrics, a significant revelation emerges regarding that certain metric values may exceed the maximum value of 1. This phenomenon occurs due to the nature of overlapping clusters, where nodes connect with multiple clusters and edges might potentially be considered more than once.