We propose a stylized model of a problem-solving organization whose internal communication structure is given by a fixed network. Problems arrive randomly anywhere in this network and must find their way to their respective "specialized solvers" by relying on local information alone. The organization handles multiple problems simultaneously. For this reason, the process may be subject to congestion. We provide a characterization of the threshold of collapse of the network and of the stock of floating problems (or average delay) that prevails below that threshold. We build upon this characterization to address a design problem: the determination of what kind of network architecture optimizes performance for any given problem arrival rate. We conclude that, for low arrival rates, the optimal network is very polarized (i.e. star-like or "centralized"), whereas it is largely homogenous (or "decentralized") for high arrival rates. We also show that, if an auxiliary assumption holds, the transition between these two opposite structures is sharp and they are the only ones to ever qualify as optimal.
