  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.
