# Boundary Matching and Interior Connectivity-Based Cluster Validity Analysis
## 1 Introduction  
Clustering analysis is an unsupervised technique that can be used for finding the structure in a dataset [1; 2; 3]. The evaluation of clustering results plays a vital role in clustering analysis and usually is performed by a clustering validity index or several [4; 5]. In the past decades, a large number of validity indices have been proposed to evaluate the clustering results and to determine the optimal number of clusters, which is an essential character of a dataset. These frequently used validity indices contain Davies-Bouldin measure [6], Tibshirani Gap statistics [7], [PERSON]'s separation measure [8], and etc. Moreover, the commonly used Bayesian Information Criterion (BIC) has been applied to estimate the number of clusters [9; 10]. For example, [11] provided a closed-form BIC expression by imposing the multivariate Gaussian assumption on the distribution of the datasets. Then, a novel two-step cluster enumeration algorithm has been proposed by combining the cluster analysis problem. Thus, this new BIC method contains information about the dataset in both data-fidelity and penalty terms. Compared with the existing BIC-based cluster enumeration algorithms, the penalty term of the proposed criterion involves information about the actual number of clusters. [PERSON] et al. [12] made comparisons among the existing validity indices. Recently, an unsupervised validity index [13] independent of any clustering algorithm was proposed for any dataset of spherical clusters. In addition, the idea has been proposed to deal with the clustering evaluation under the condition of big data [14; 15], while very little work is available in the literature that discusses validity indices for big data. One of the few papers onthis topic that we are aware of is [16], but the method does not consider the complex data structures in today's computing environment. However, the existing validity indices are greatly restrained by the following three disadvantages at least. Specific algorithm. Usually, the existing validity indices can only evaluate the clustering results obtained by a specified algorithm (e.g., C-means [17; 18] or Fuzzy C-means [19; 20]) rather than an arbitrary clustering algorithm. If the clustering algorithm chosen for the dataset is not suitable, the evaluating results will not be guaranteed. Specific parameter. Different parameters in a clustering algorithm lead to different clustering results. Most cluster validity indices aim to select the best one among all clustering results, and thus they regard the clustering parameter as their variable. These existing indices can only regard the number of clusters as a variable rather than other clustering parameters, such as the density threshold in Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm [21] and the grid size in the CLIQUE algorithm [22]. Recently, a density peak-point-based clustering (DPC) algorithm [23] and its variants [24; 25; 26] have attracted considerable attention; but the number of peak points therein remains so uncertain that the correctness of clustering results is difficult to guarantee. Untapped result. A cluster consists of a high-density center and a group of relatively low-density neighbors around the center in DPC, a group of core points and corresponding boundary points around these core points in DBSCAN, a center and a group of points that are assigned to the cluster by the nearest neighbor principle in C-means. Consequently, all points in a cluster are partitioned into two types. An accurate clustering partition must result from the correct identification of the two types of points. Although the existing algorithms may partition all points into the two types of points, and the partitioning results fail to be taken into a validity index to evaluate the clustering results. Especially when several clustering algorithms are performed in the same dataset, it is impossible to choose the best clustering result.