The paper considers a model of competition among firms that produce a homogeneous good in a networked environment. A bipartite graph determines which subset of markets a firm can supply to. Firms compete a la Cournot and decide how to allocate their production output to the markets they are directly connected to. We assume that markets have inverse linear demand and firms have quadratic production costs. First, we show that the resulting Cournot game has a unique equilibrium for any given network and provide a characterization of the production quantities at equilibrium. Our results identify a close connection between the equilibrium outcome and supply paths in the underlying network structure. In particular, we show that whether two firms see their output in different markets as strategic substitutes or complements depends critically on the paths between those markets in the line graph induced by the original bipartite network.