Efficient Estimation of Pairwise Effective Resistance

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 scenarios, 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 walks, rendering it rather inefficient in practice especially on large graphs. Motivated by this, this paper first presents an improved Monte Carlo approach, AMC, which reduces both the length and amount of random walks required without degrading the theoretical accuracy guarantee, through careful theoretical analysis and an adaptive sampling scheme. Further, we develop a greedy approach, GEER, which combines AMC with sparse matrix-vector multiplications in an optimized and non-trivial way. GEER offers significantly improved practical efficiency over AMC without compromising its asymptotic performance and accuracy guarantees. Extensive experiments on multiple benchmark datasets reveal that GEER is orders of magnitude faster than the state of the art in terms of computational time when achieving the same accuracy. CCS Concepts: • Mathematics of computing → Graph algorithms; • Theory of computation → Random walks and Markov chains.