Local Algorithms for Estimating Effective Resistance
This addresses the need for scalable graph analysis in applications like clustering and network reliability, representing an incremental improvement over existing methods.
The paper tackles the problem of efficiently estimating effective resistances in massive graphs by designing local algorithms that only read a small portion of the input, achieving an additive error ε in time O(poly(log n/ε)) for graphs with bounded mixing time.
Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs. In this work, we design several \emph{local algorithms} for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair $s,t$ with an arbitrarily small additive error $\varepsilon$ in time $O(\mathrm{poly}(\log n/\varepsilon))$, whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms.