Randomized Gossiping with Effective Resistance Weights: Performance Guarantees and Applications

The effective resistance between a pair of nodes in a weighted undirected graph is defined as the potential difference induced when a unit current is injected at one node and extracted from the other, treating edge weights as the conductance values of edges. The effective resistance is a key quantity of interest in many applications, e.g., solving linear systems, Markov Chains, and continuous-time averaging networks. We consider effective resistances (ER) in the context of designing randomized gossiping methods for the consensus problem, where the aim is to compute the average of node values in a distributed manner through iteratively computing weighted averages among randomly chosen neighbors. We show that employing ER weights improves the averaging time corresponding to the traditional choice of uniform weights -the amount of improvement depends on the network structure. We illustrate these results through numerical experiments. We also present an application of the ER gossiping to distributed optimization: we numerically verified that using ER gossiping within EXTRA and DPGA-W methods improves their practical performance in terms of communication efficiency.