## 7 Discussion
Although the BTER network's scale-free distribution of both cluster size and degree is known to cause difficulty for many clustering algorithms, NBR-Clustering, using VAT, was able to detect all but 3 of 198 clusters. While VAT has been particularly notable in accurate clustering across various datasets, integrity and tenacity have shown promise in _one step_ clustering across a variety of network and point set data when the number of clusters is unknown a priori. Both integrity and tenacity clustered all attributed network datasets in a single iteration, with integrity clustering at an average of 86% accuracy. For the LFR benchmarks, attributed networks with communities and synthetic point set graphs, integrity clustered 42 out of 46 sets (of 10 graphs) in only one step. Tenacity clustered every regular Girvan-Newman graph in one step with an average accuracy greater than 92%. Tenacity also clustered every kNN graph generated from noiseless and noisy synthetic point set data in one step, and integrity clustered the majority of such data sets in one step with relatively higher accuracies. Upon closer examination of the poor performance of tenacity on noisy synthetic datasets, it was observed that many more nodes were removed by tenacity (including many non-noise nodes), resulting in a higher number of clusters than necessary. Thus, even in cases where tenacity does not achieve an accurate clustering in one step, the number of clusters formed by tenacity serves as an upper bound on the correct number of clusters desired. Finally, in the context of clustering in a single iteration, this research shows that, for _all measures_ except scattering, NBR-Clust clustered almost every scale-free LFR benchmark network in a single iteration. Another important contribution of this work is the application to noise removal. The efficacy of NBR-Clust has been examined with respect to noise removal which occurs as an automatic by-product of node-based resilience measure computation, particularly computation of the critical attack set, which is taken as a candidate set of noise nodes. As stated previously, the noisy datasets of this work are minimum connectivity kNN graphs with Gaussian mixture ground truth clusters and an additional 10% uniformly random noise. VAT showed high precision in detecting noise nodes, with the percentage of noise nodes in the critical attack sets ranging from 63% to 94%. VAT's small attack set size implied that the total number of noise nodes found was small. In contrast, tenacity generally detected over 90% of the noise nodes, but with low precision. Integrity had the best combination of precision and recall with noise detection, although results varied. Noise removal with one-shot integrity-based partial clustering was particularly notable on two classes of noisy point sets: equal density D4K8 and equal density D8K8. For the D4K8 datasets, integrity identified approximately 50% of the noise nodes, with a precision of 96%, while still retaining one-shot clustering accuracies of 97% and 100%. It is interesting to consider the graph corresponding to the noisy, D8K8 unequal density dataset shown in Figure 7. The central location of the noise nodes in this scenario yields a natural stochastic generalization of this paper's illustrative example motivating node-based resilience measures for partial clustering in Figure 2.  
It is not necessary for the noise to form a single centrally located group in order for NBR clustering to cluster effectively. For example, consider the noisy, equal density D2K8 dataset shown in Figure 3.  
Attack-set noise nodes are shown as triangles, and the corresponding min-conn kNN graph is shown in Figure 4. All NBR measures, except scattering, clustered the graph perfectly. The noise removal on that dataset was also at least 49% for all resilience measures considered. It can be observed that the uniformly at random distributed noise still tends to form high-betweenness nodes \"between\" natural clusters.