Graph spectral clustering is a fundamental technique in data analysis, which utilizes eigenpairs of the Laplacian matrix to partition graph vertices into clusters. However, classical spectral clustering algorithms require eigendecomposition of the Laplacian matrix, which has cubic time complexity. In this work, we describe pass-efficient spectral clustering algorithms that leverage recent advances in randomized eigendecomposition and the structure of the graph vertex-edge matrix. Furthermore, we derive formulas for their efficient implementation. The resulting algorithms have a linear time complexity with respect to the number of vertices and edges and pass over the graph constant times, making them suitable for processing large graphs stored on slow memory. Experiments validate the accuracy and efficiency of the algorithms.

Cancer is a complex cellular ecosystem where malignant cells coexist and interact with immune, stromal and other cells within the tumor microenvironment (TME). Recent technological advancements in spatially resolved multiplexed imaging at single-cell resolution have led to the generation of large-scale and high-dimensional datasets from biological specimens. This underscores the necessity for automated methodologies that can effectively characterize molecular, cellular and spatial properties of TMEs for various malignancies. This study introduces SpatialCells, an open-source software package designed for region-based exploratory analysis and comprehensive characterization of TMEs using multiplexed single-cell data. The source code and tutorials are available at https://semenovlab.github.io/SpatialCells. SpatialCells efficiently streamlines the automated extraction of features from multiplexed single-cell data and can process samples containing millions of cells. Thus, SpatialCells facilitates subsequent association analyses and machine learning predictions, making it an essential tool in advancing our understanding of tumor growth, invasion and metastasis.