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This paper investigates the use of a networked system ($e.g.$, swarm of robots, smart grid, sensor network) to monitor a time-varying phenomenon of interest in the presence of communication and computation latency. Recent advances in edge computing have enabled processing to be spread across the network, hence we investigate the fundamental computation-communication trade-off, arising when a sensor has to decide whether to transmit raw data (incurring communication delay) or preprocess them (incurring computational delay) in order to compute an accurate estimate of the state of the phenomenon of interest. We propose two key contributions. First, we formalize the notion of $processing$ $network$. Contrarily to $sensor$ $and$ $communication$ $networks$, where the designer is concerned with the design of a suitable communication policy, in a processing network one can also control when and where the computation occurs in the network. The second contribution is to provide analytical results on the optimal preprocessing delay ($i.e.$, the optimal time spent on computations at each sensor) for the case with a single sensor and multiple homogeneous sensors. Numerical results substantiate our claims that accounting for computation latencies (both at sensor and estimator side) and communication delays can largely impact the estimation accuracy.