Distributed node counting in wireless sensor networks can be important in various applications, such as network maintenance and information aggregation. In this paper, a distributed consensus algorithm for estimating the number of nodes in a wireless sensor network in the presence of communication noise is introduced. In networks with a fusion center, counting the number of nodes can easily be done by letting each node to transmit a fixed constant value to the fusion center. In a network without a fusion center, where nodes do not know the graph structure, estimating the number of nodes is not straightforward. The proposed algorithm is based on distributed average consensus and norm estimation. Different sources of error are explicitly discussed; the Fisher information and the distribution of the final estimate are derived. Several design parameters and how they affect the performance of the algorithm are studied, which provide guidelines toward making the estimation error smaller. Simulation results corroborating the theory are also provided.
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