Degree Fluctuations and the Convergence Time of Consensus Algorithms

Details Degree Fluctuations and the Convergence Time of Consensus Algorithms Degree Fluctuations and the Convergence Time of Consensus Algorithms

2011-04-04

arxiv.org/abs/1104.0454v2

Mathematics

Optimization and Control

Scientific paper

We consider a consensus algorithm in which every node in a time-varying undirected connected graph assigns equal weight to each of its neighbors. Under the assumption that the degree of any given node is constant in time, we show that the algorithm achieves consensus within a given accuracy epsilon on n nodes in time O(n^3 log(n/epsilon)). Because there is a direct relation between consensus algorithms in time-varying environments and inhomogeneous random walks, our result also translates into a general statement on such random walks. Moreover, we give simple proofs that the convergence time becomes exponentially large in the number of nodes n under slight relaxations of the above assumptions. We prove that exponential convergence time is possible for consensus algorithms on fixed directed graphs, and we use an example of Cao, Spielman, and Morse to give a simple argument that the same is possible if the constant degrees assumption is even slightly relaxed.

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