  This paper considers a multi-cell AirComp network and investigates the
optimal power control policies over multiple cells to regulate the effect of
inter-cell interference. First, we consider the scenario of centralized
multi-cell power control, where we characterize the Pareto boundary of the
multi-cell MSE region by minimizing the sum MSE subject to a set of constraints
on individual MSEs. Though the sum-MSE minimization problem is non-convex and
its direct solution intractable, we optimally solve this problem via
equivalently solving a sequence of convex second-order cone program feasibility
problems together with a bisection search. Next, we consider distributed power
control in the other scenario without a centralized controller, for which an
alternative IT-based method is proposed to characterize the same MSE Pareto
boundary, and enable a decentralized power control algorithm. Accordingly, each
AP only needs to individually control the power of its associated devices, but
subject to a set of IT constraints on their interference to neighboring cells,
while different APs can cooperate in iteratively updating the IT levels by
pairwise information exchange, to achieve a Pareto-optimal MSE tuple. Last,
simulation results demonstrate that cooperative power control using the
proposed algorithms can substantially reduce the sum MSE of AirComp networks.
