We consider networks of linear, time-invariant systems defined over matrices of rational functions in a complex variable where each element of the matrix represents a link in the network. When these rational functions are proper, but not necessarily strictly proper, we demonstrate the necessary and sufficient conditions under which such a network configuration is well-posed. We include multiple examples of network configurations and their respective well-posedness conditions, including cases where two or more ill-posed network configurations can be interconnected to form a well-posed network.