We propose a multi-hop diffusion strategy for a sensor network to performdistributed least mean-squares (LMS) estimation under local and network-wideenergy constraints. At each iteration of the strategy, each node can combineintermediate parameter estimates from nodes other than its physical neighborsvia a multi-hop relay path. We propose a rule to select combination weights forthe multi-hop neighbors, which can balance between the transient and thesteady-state network mean-square deviations (MSDs). We study two classes ofnetworks: simple networks with a unique transmission path from one node toanother, and arbitrary networks utilizing diffusion consultations over at mosttwo hops. We propose a method to optimize each node's information neighborhoodsubject to local energy budgets and a network-wide energy budget for eachdiffusion iteration. This optimization requires the network topology, and thenoise and data variance profiles of each node, and is performed offline beforethe diffusion process. In addition, we develop a fully distributed and adaptivealgorithm that approximately optimizes the information neighborhood of eachnode with only local energy budget constraints in the case where diffusionconsultations are performed over at most a predefined number of hops. Numericalresults suggest that our proposed multi-hop diffusion strategy achieves thesame steady-state MSD as the existing one-hop adapt-then-combine diffusionalgorithm but with a lower energy budget.
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