Effective resistance, which originates from the field of circuits analysis, is an important graph distance in spectral graph theory. It has found numerous applications in various areas, such as graph data mining, spectral graph sparsification, circuits simulation, etc. However, computing effective resistances accurately can be intractable and we still lack efficient methods for estimating effective resistances on large graphs. In this work, we propose an efficient algorithm to compute effective resistances on general weighted graphs, based on a sparse approximate inverse technique. Compared with a recent competitor, the proposed algorithm shows several hundreds of speedups and also one to two orders of magnitude improvement in the accuracy of results. Incorporating the proposed algorithm with the graph sparsification based power grid (PG) reduction framework, we develop a fast PG reduction method, which achieves an average 6.4X speedup in the reduction time without loss of reduction accuracy. In the applications of power grid transient analysis and DC incremental analysis, the proposed method enables 1.7X and 2.5X speedup of overall time compared to using the PG reduction based on accurate effective resistances, without increase in the error of solution.

翻译：有效电阻起源于电路分析领域，是谱图理论中一种重要的图距离度量。它在图数据挖掘、谱图稀疏化、电路仿真等多个领域得到了广泛应用。然而，精确计算有效电阻往往非常困难，目前仍缺少在大规模图上高效估算有效电阻的方法。为此，本文提出了一种基于稀疏近似逆技术的高效算法，用于计算一般加权图上的有效电阻。与近期同类方法相比，本算法在计算速度上提升了数百倍，同时结果精度提高了1到2个数量级。结合该算法与基于图稀疏化的电网简化框架，我们进一步开发了一种快速电网简化方法，在保持简化精度的前提下，平均简化时间提升了6.4倍。在电网瞬态分析和直流增量分析应用中，与基于精确有效电阻的电网简化方法相比，本方法在未增加求解误差的情况下，分别实现了整体计算时间1.7倍和2.5倍的加速。