Traditional clustering algorithms, such as k-means, output a clustering that is disjoint and exhaustive, that is, every single data point is assigned to exactly one cluster. However, in real datasets, clusters can overlap and there are often outliers that do not belong to any cluster. This is a well recognized problem that has received much attention in the past, and several algorithms, such as fuzzy kmeans have been proposed for overlapping clustering. However, most existing algorithms address

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... either overlap or outlier detection and do not tackle the problem in a unified way. In this paper, we propose a simple and intuitive objective function that captures the issues of overlap and nonexhaustiveness in a unified manner. Our objective function can be viewed as a reformulation of the traditional k-means objective, with easy-to-understand parameters that capture the degrees of overlap and non-exhaustiveness. By studying the objective, we are able to obtain a simple iterative algorithm which we call NEO-K-Means (Non-Exhaustive Overlapping K-Means). Furthermore, by considering an extension to weighted kernel k-means, we can tackle the case of non-exhaustive and overlapping graph clustering. This extension allows us to apply our NEO-K-Means algorithm to the community detection problem, which is an important task in network analysis. Our experimental results show that the new objective and algorithm are effective in finding ground-truth clusterings that have varied overlap and non-exhaustiveness; for the case of graphs, we show that our algorithm outperforms state-of-the-art overlapping community detection methods.