Tree Codes Improve Convergence Rate of Consensus Over Erasure Channels

Abstract

We study the problem of achieving average consensus between a group of agents
over a network with erasure links. In the context of consensus problems, the
unreliability of communication links between nodes has been traditionally
modeled by allowing the underlying graph to vary with time. In other words,
depending on the realization of the link erasures, the underlying graph at each
time instant is assumed to be a subgraph of the original graph. Implicit in
this model is the assumption that the erasures are symmetric: if at time t the
packet from node i to node j is dropped, the same is true for the packet
transmitted from node j to node i. However, in practical wireless communication
systems this assumption is unreasonable and, due to the lack of symmetry,
standard averaging protocols cannot guarantee that the network will reach
consensus to the true average. In this paper we explore the use of channel
coding to improve the performance of consensus algorithms. For symmetric
erasures, we show that, for certain ranges of the system parameters, repetition
codes can speed up the convergence rate. For asymmetric erasures we show that
tree codes (which have recently been designed for erasure channels) can be used
to simulate the performance of the original "unerased" graph. Thus, unlike
conventional consensus methods, we can guarantee convergence to the average in
the asymmetric case. The price is a slowdown in the convergence rate, relative
to the unerased network, which is still often faster than the convergence rate
of conventional consensus algorithms over noisy links.