We analyze how distributed or decentralized estimation can be performed over networks, when there is a price to be paid whenever nodes in the network communicate with each other. The work here has application especially in the network control systems. Assume that different nodes in the network can track perfectly or with imperfectly some stochastic processes, while other nodes in the network need to estimate these stochastic processes. The nodes which can observe the stochastic processes can send information directly to the nodes which need to estimate the processes, or information can be sent to intermediate nodes. When each transmission is performed a cost for communication is paid. The goal of the network is to optimize jointly a cost which consists both of a function of the estimation error and a function of the transmission cost. We show here that for some simple topologies the decision to send information over the network is a threshold policy, while the estimators are linear estimators which resemble with the Kalman-filter. For the result dealing with simple topologies we have proved the results using majorization theory.

It is also shown here both analytically and numerically that things can immediately become quite complicated. If we take into consideration multidimensional problems or problems with multiple agents and/or transmission noise, the optimal strategies can no longer be found analytically and it can be quite difficult to compute numerically the optimal strategies.