The ubiquity of massive graph data sets in numerous applications requires fast algorithms for extracting knowledge from these data. We are motivated here by three electrical measures for the analysis of large small-world graphs $G = (V, E)$ -- i.e., graphs with diameter in $O(\log |V|)$, which are abundant in complex network analysis. From a computational point of view, the three measures have in common that their crucial component is the diagonal of the graph Laplacian's pseudoinverse, $L^\dagger$. Computing diag$(L^\dagger)$ exactly by pseudoinversion, however, is as expensive as dense matrix multiplication -- and the standard tools in practice even require cubic time. Moreover, the pseudoinverse requires quadratic space -- hardly feasible for large graphs. Resorting to approximation by, e.g., using the Johnson-Lindenstrauss transform, requires the solution of $O(\log |V| / \epsilon^2)$ Laplacian linear systems to guarantee a relative error, which is still very expensive for large inputs. In this paper, we present a novel approximation algorithm that requires the solution of only one Laplacian linear system. The remaining parts are purely combinatorial -- mainly sampling uniform spanning trees, which we relate to diag$(L^\dagger)$ via effective resistances. For small-world networks, our algorithm obtains a $\pm \epsilon$-approximation with high probability, in a time that is nearly-linear in $|E|$ and quadratic in $1 / \epsilon$. Another positive aspect of our algorithm is its parallel nature due to independent sampling. We thus provide two parallel implementations of our algorithm: one using OpenMP, one MPI + OpenMP. In our experiments against the state of the art, our algorithm (i) yields more accurate results, (ii) is much faster and more memory-efficient, and (iii) obtains good parallel speedups, in particular in the distributed setting.

中文翻译：

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用于复杂网络分析的拉普拉斯伪逆对角线的近似

海量图数据集在众多应用中无处不在，需要快速算法来从这些数据中提取知识。我们在这里受到三个用于分析大型小世界图 $G = (V, E)$ 的电学测量的启发——即直径在 $O(\log |V|)$ 中的图，它们在复数中很丰富网络分析。从计算的角度来看，这三个度量的共同点是它们的关键组成部分是图拉普拉斯伪逆的对角线 $L^\dagger$。然而，完全通过伪反演计算 diag$(L^\dagger)$ 与密集矩阵乘法一样昂贵——而且实践中的标准工具甚至需要三次时间。此外，伪逆需要二次空间——对于大图来说几乎不可行。诉诸近似，例如，使用 Johnson-Lindenstrauss 变换，需要 $O(\log |V| / \epsilon^2)$ 拉普拉斯线性系统的解决方案来保证相对误差，这对于大输入仍然非常昂贵。在本文中，我们提出了一种新的近似算法，它只需要一个拉普拉斯线性系统的解。其余部分纯粹是组合性的——主要是对均匀生成树进行采样，我们通过有效电阻将其与 diag$(L^\dagger)$ 相关联。对于小世界网络，我们的算法以高概率获得了 $\pm\epsilon$-近似值，其时间在 $|E|$ 中几乎是线性的，在 $1 / \epsilon$ 中是二次的。我们算法的另一个积极方面是由于独立采样而具有并行性。因此，我们提供了我们算法的两种并行实现：一种使用 OpenMP，一种使用 MPI + OpenMP。在我们反对最先进技术的实验中，