Abstract:Given an undirected graph G, the effective resistance r(s,t) measures the dissimilarity of node pair s,t in G, which finds numerous applications in real-world problems, such as recommender systems, combinatorial optimization, molecular chemistry, and electric power networks. Existing techniques towards pairwise effective resistance estimation either trade approximation guarantees for practical efficiency, or vice versa. In particular, the state-of-the-art solution is based on a multitude of Monte Carlo random … Show more
Effective resistance (ER) is a fundamental metric for measuring node similarities in a graph, and it finds applications in various domains including graph clustering, recommendation systems, link prediction, and graph neural networks. The state-of-the-art algorithm for computing effective resistance relies on a landmark technique, which involves selecting a node that is easy to reach by all the other nodes as a landmark. The performance of this technique heavily depends on the chosen landmark node. However, in many real-life graphs, it is not always possible to find an easily reachable landmark node, which can significantly hinder the algorithm's efficiency. To overcome this problem, we propose a novel multiple landmarks technique which involves selecting a set of landmark nodes V
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such that the other nodes in the graph can easily reach any one of a landmark node in V
l
. Specifically, we first propose several new formulas to compute ER with multiple landmarks, utilizing the concept of Schur complement. These new formulas allow us to pre-compute and maintain several small-sized matrices related to V
l
as a compact index. With this powerful index technique, we demonstrate that both single-pair and single-source ER queries can be efficiently answered using a newly-developed V
l
-absorbed random walk sampling or V
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-absorbed push technique. Comprehensive theoretical analysis shows that all proposed index-based algorithms achieve provable performance guarantees for both single-pair and single-source ER queries. Extensive experiments on 5 real-life datasets demonstrate the high efficiency of our multiple landmarks-based index techniques. For instance, our algorithms, with a 1.5 GB index size, can be up to 4 orders of magnitude faster than the state-of-the-art algorithms while achieving the same accuracy on a large road network.
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