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Graph sampling methods have been used to reduce the size and complexity of big complex networks for graph mining and visualization. However, existing graph sampling methods often fail to preserve the connectivity and important structures of the original graph. This paper introduces a new divide and conquer approach to spectral graph sampling based on graph connectivity, called the BC Tree (i.e., decomposition of a connected graph into biconnected components) and spectral sparsification. Specifically, we present two methods, spectral vertex sampling    $$BC\_SV$$  B C _ S V and spectral edge sampling    $$BC\_SS$$  B C _ S S by computing effective resistance values of vertices and edges for each connected component. Furthermore, we present    $$DBC\_SS$$  D B C _ S S and    $$DBC\_GD$$  D B C _ G D , graph connectivity-based distributed algorithms for spectral sparsification and graph drawing respectively, aiming to further improve the runtime efficiency of spectral sparsification and graph drawing by integrating connectivity-based graph decomposition and distributed computing. Experimental results demonstrate that    $$BC\_SV$$  B C _ S V and    $$BC\_SS$$  B C _ S S are significantly faster than previous spectral graph sampling methods while preserving the same sampling quality.    $$DBC\_SS$$  D B C _ S S and    $$DBC\_GD$$  D B C _ G D obtain further significant runtime improvement over sequential approaches, and    $$DBC\_GD$$  D B C _ G D further achieves significant improvements in quality metrics over sequential graph drawing layouts.

BC tree-based spectral sampling for big complex network visualization