Computing the Pseudo-Inverse of a Graph´s Laplacian Using GPUs

Many applications in network analysis require the computation of the network´s Laplacian pseudo-inverse - e.g., Topological centrality in social networks or estimating commute times in electrical networks. As large graphs become ubiquitous, the traditional approaches - with quadratic or cubic complexity in the number of vertices - do not scale. To alleviate this performance issue, a divide-and-conquer approach has been recently developed. In this work, we take one step further in improving the performance of computing the pseudo-inverse of Laplacian by parallelization. Specifically, we propose a parallel, GPU-based version of this new divide-and-conquer method. Furthermore, we implement this solution in Mat lab, a native environment for such computations, recently enhanced with the ability to harness the computational capabilites of GPUs. We find that using GPUs through Mat lab, we achieve speed-ups of up to 320x compared with the sequential divide-and-conquer solution. We further compare this GPU-enabled version with three other parallel solutions: a parallel CPU implementation and CUDA-based implementation of the divide-and-conquer algorithm, as well as a GPU-based implementation that uses cuBLAS to compute the pseudo-inverse in the traditional way. We find that the GPU-based implementation outperforms the CPU parallel version significantly. Furthermore, our results demonstrate that a best GPU-based implementation does not exist: depending on the size and structure of the graph, the relative performance of the three GPU-based versions can differ significantly. We conclude that GPUs can be successfully used to improve the performance of the pseudo-inverse of a graph´s Laplacian, but choosing the best performing solution remains challenging due to the non-trivial correlation between the achieved performance and the characteristics of the input graph. Our future work attempts to expose and exploit this correlation.