Convergence time for unbiased quantized consensus

We revisit the quantized consensus problem on undirected connected graphs, and obtain some strong results on expected time to convergence. This is unbiased consensus, because the edges emanating from a node have equal probability of being selected. The paper first develops an approach that bounds the expected convergence time of the underlying discrete-time dynamics. The bounds are tight for some simple networks when there exists some symmetry in the network. Following this, the paper provides a tight expression for the expected convergence time of unbiased quantized consensus over general but fixed networks. We show that the expected convergence time can be expressed in terms of the effective resistances of the associated Cartesian product graph. The approach adopted in the paper uses the theory of harmonic functions for reversible Markov chains. Finally, we extend our results to bound the expected convergence time of the underlying dynamics in time-varying networks. In spite of lack of reversibility for time-varying networks, we prove a tight upper bound for expected convergence time of the dynamics using the spectral representation of the networks. Index Term