A fully distributed algorithm for estimating the center and the radius of the smallest sphere that contains a wireless sensor network is proposed. The center finding problem is formulated as a convex optimization problem in summation form by using a soft-max approximation to the maximum function. Diffusion adaptation method is used where states of nodes converge to the estimated center distributively. Then, distributed max consensus is used to compute the radius. The proposed algorithm is fully distributed in the sense that each node in the network only needs to know its own location and nodes do not need to be pre-labeled. The algorithm works for any connected graph structure. The performance of the proposed algorithm is analyzed and it is shown that there is a tradeoff: a larger design parameter results in a more accurate center estimation but also makes the convergence speed of the distributed algorithm slower. It is also shown that the proposed algorithm can also be used to estimate the center and radius of the (not necessarily location-based) data at sensor nodes in a distributed way, thereby providing information and insights about the global data at each sensor. Simulation results corroborating the theory are also provided.
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