Efficient Index Maintenance for Effective Resistance Computation on Evolving Graphs

In this paper, we study a problem of index maintenance on evolving graphs for effective resistance computation. Unlike an existing matrices-based index, we show that the index can be efficiently maintained by directly preserving samples of random walks and loop-erased walks. This approach not only enables efficient storage and rapid query response but also supports effective maintenance. We propose a novel approach to convert edge updates into landmark node updates. Building upon this, we present two new update algorithms for random walk and loop-erased walk samples respectively. Both algorithms update samples without requiring complete resampling, ensuring accuracy and high efficiency. A particularly challenging and innovative technique involves updating loop-erased walks. Here we develop a novel and powerful cycle decomposition technique for loop-erased walks, enabling us to update samples at the cycle level rather than the node level, significantly enhancing efficiency. Furthermore, we show that both of our methods achieve an Õ (1) time complexity per edge update in real-world graphs under a mild assumption. We conduct extensive experiments using 10 large real-world datasets to evaluate the performance of our approaches. The results show that our best algorithm can be up to two orders of magnitude faster than the baseline methods.