We study a number of problems of practical interest regarding the topology of a sensor network in order to guarantee the convergence, and minimize the communication required, in distributed inference. We cast the topology design in the context of spectral graph theory and show that this combinatorial question translates into optimizing spectral graph properties. With noiseless communication channels among sensors, and under constraints on the number of communication channels and on the network power budget, we show that the optimal topology for the sensor network is a Ramanujan graph. When the communication is noisy, we show that there is an optimal number of iterations for distributed inference. When the communication channels fail with a nonzero probability, we provide necessary and sufficient conditions for mean square convergence of the distributed inference algorithm. Finally, given a matrix of communication costs among sensors, we show that the design of the network topology that minimizes the overall network communication cost is an SDP optimization problem. Simulation studies illustrate our analysis and results.
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