Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks

Abstract

Abstract-In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides a tight upper bound for the expected convergence time to consensus based on the parameters of the network. Moreover, the paper identifies a precise order of the maximum expected convergence time for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives a tight upper bound for expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier results for the quantized consensus problem over time-varying graphs.

Resilient randomized quantized consensus

“…In [20], some common classes of agent networks are considered while further studies on static and time-varying networks can be found in [14], [42]. It is however noted that these works deal with the problem of average consensus where the underlying graph is undirected.…”

Synchronization of Coupled Harmonic Oscillators Using Quantized Sampled Position Data

“…On the other hand, as an important issue, data quantization has received extensive attention in multiagent systems. Theoretical results in the past two years include [10][11][12][13][14][15][16]. The consensus of multiagent systems is studied according to the robust learning control approach in [10].…”

Convergence Time of Quantized Metropolis Consensus Over Time-Varying Networks

“…The paper [6] obtained an upper bound of O(n 2 ), but only on complete graphs. Recently, it was shown in [8] that a certain "unbiased" quantized consensus protocol has maximum expected convergence time of O(nmD log n) on static networks where m and D denote the number of edges and the diameter of the network, and has maximum expected convergence time O(nm max D max log 2 n) on connected time-varying networks, where m max and D max denote, respectively, the maximum number of edges and the maximum diameter in the sequence of time-varying networks. Unbiasedness means that each step of the protocol was based on choosing a random edge uniformly among all possible edges in the network.…”