Decentralized Computation of Effective Resistances and Acceleration of Consensus Algorithms

TL;DR

This paper introduces a distributed algorithm for efficiently computing effective resistances in graphs and demonstrates how it can significantly accelerate consensus algorithms in networked systems.

Contribution

The paper presents a novel linearly convergent distributed method for calculating effective resistances and applies it to enhance the speed of consensus algorithms.

Findings

The algorithm computes effective resistances efficiently in a distributed manner.

Applying the algorithm accelerates consensus convergence depending on network structure.

Numerical studies validate the effectiveness of the proposed methods.

Abstract

The effective resistance between a pair of nodes in a weighted undirected graph is defined as the potential difference induced between them when a unit current is injected at the first node and extracted at the second node, treating edge weights as the conductance values of edges. The effective resistance is a key quantity of interest in many applications and fields including solving linear systems, Markov Chains and continuous-time averaging networks. We develop an efficient linearly convergent distributed algorithm for computing effective resistances and demonstrate its performance through numerical studies. We also apply our algorithm to the consensus problem where the aim is to compute the average of node values in a distributed manner. We show that the distributed algorithm we developed for effective resistances can be used to accelerate the convergence of the classical consensus iterations considerably by a factor depending on the network structure.