Dense subregion (subgraph & subtensor) detection is a well-studied area, with a wide range of applications, and numerous efficient approaches and algorithms have been proposed. Approximation approaches are commonly used for detecting dense subregions due to the complexity of the exact methods. Existing algorithms are generally efficient for dense subtensor and subgraph detection, and can perform well in many applications. However, most of the existing works utilize the state-or-the-art greedy 2-approximation algorithm to capably provide solutions with a loose theoretical density guarantee. The main drawback of most of these algorithms is that they can estimate only one subtensor, or subgraph, at a time, with a low guarantee on its density. While some methods can, on the other hand, estimate multiple subtensors, they can give a guarantee on the density with respect to the input tensor for the first estimated subsensor only. We address these drawbacks by providing both theoretical and practical solution for estimating multiple dense subtensors in tensor data and giving a higher lower bound of the density. In particular, we guarantee and prove a higher bound of the lower-bound density of the estimated subgraph and subtensors. We also propose a novel approach to show that there are multiple dense subtensors with a guarantee on its density that is greater than the lower bound used in the state-of-the-art algorithms. We evaluate our approach with extensive experiments on several real-world datasets, which demonstrates its efficiency and feasibility.
Influence of initial states and structure defects on non-equilibrium critical behavior of 3D and 2D Ising models
