Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering

We present and analyze a new framework for graph clustering based on a specially weighted version of correlation clustering, that unifies several existing objectives and satisfies a number of attractive theoretical properties. Our framework, which we call LambdaCC, relies on a single resolution parameter λ, which implicitly controls both the edge density and sparsest cut of all output clusters. We prove that our new clustering objective interpolates between the cluster deletion problem and the minimum sparsest cut problem as we vary λ, and is also closely related to the well-studied maximum modularity objective. We provide several algorithms for optimizing our new objective, including a 5-approximation for the case where λ ≥ 1/2, and also the first constant factor approximation algorithm for the NP-hard cluster deletion problem. We demonstrate the effectiveness of our framework and algorithms in finding communities in several real-world networks.