## 7 Discussion  
The theoretical motivation for a generalized NBR clustering framework is based on the relationship between resilience and clustering. This relationship is explicit in the sparsest cuts clustering framework, and has been extended in a novel manner to NBR measures, which presents unique challenges as well as advantages for noise removal and overlapping community detection. This paper's theoretical motivation is strengthened by defining an infinite family of graphs, as illustrated in Figure 2, such that using any node-based resilience measure for clustering immediately (in one iteration) yields the correct partitions. Furthermore, a generalized node-based resilience clustering framework (NBR-Clust) is presented that is naturally applicable for variations of the traditional clustering problem in which a complete partitioning is not appropriate, such as clustering in the presence of noise as well as in the presence of overlapping clusters. The primary goal of this work has been to demonstrate the effectiveness of NBR-Clustering methodology via extensive experimental results across variations of the clustering problem and a diverse set of datasets, including real and synthetic, as well as native graph data and point set data. There are computational hardness results pertaining to all resilience measures considered in this research's NBR-Clustering framework. The heuristic, \(O(|V|^{2}|E|)\) time Greedy-BC method is utilized to _simultaneously_ compute all relevant measures given the acceptable empirical approximation factors attained for that approach in [10] as well as the super-linear improvement in the time complexity of Greedy-BC over Greedy-BC with hill-climbing. Additionally, use of a parallelized algorithm as described in [55] allowed the clustering of larger graphs of up to 100,000 nodes in reasonable times. In the context of this work, the main concern is the quality of the clusterings obtained via NBR-Clustering as well as examination of NBR measures and data scenarios for which NBR-Clustering \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**VAT**} & \multicolumn{3}{c}{**Integrity**} \\ \hline  & \multicolumn{3}{c}{**\# Overlap Nodes**} & \multicolumn{3}{c}{**\# Overlap Nodes**} \\ \hline
**Memberships** & **25** & **50** & **100** & **25** & **50** & **100** \\ \hline
2 & 0.98 & 0.97 & 0.98 & **0.93** & **0.92** & **0.96** \\
4 & 0.97 & 0.96 & 0.95 & **0.97** & **0.97** & **0.95** \\
6 & 0.96 & 0.93 & 0.88 & **0.96** & **0.93** & **0.84** \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Normalized mutual information for scale free LFR benchmark networks with overlap. Bold text indicates one-step clustering. performs (i) accurate one-iteration clustering without the number of clusters known a priori; (ii) noise removal with high precision and recall and (iii) high-quality detection of overlapping communities. NBR-Clust exhibited competitive results compared to the popular Girvan-Newman and Louvain community detection methods on a series of well-known real networks, outperforming those methods on the food web dataset and tying them on the karate club network. Regarding performance on synthetic graph data, this paper has demonstrated that NBR-Clust gives robust results for both regular and scale-free degree distributions, for varying cluster size distributions, across a reasonable range of mixing factors, and even on complex networks with attributes. NBR-Clustering results on real and synthetic datasets generated from point-set data transformed into kNN graphs have been similarly promising, both in noiseless and noisy (10% uniformly random noise) settings. On a series of kNN graphs generated from Gaussian-mixture point-set data, VAT, integrity and tenacity clustered noiseless datasets almost perfectly, with VAT also clustering all noisy point-set data almost perfectly. VAT also resulted in accurate clustering for the BTER complex network dataset.