Research in complex networks has focused on the use of random graph generators as models for the topology of complex systems. Although a range of generators has been proposed, the standard generators cannot capture the topology of many systems with high fidelity. For example, for the Watts-Strogatz generator we need additional constraints to determine the value of the rewiring probability, p_r, that will generate a network with appropriate characteristics. We show that higher-order functional constraints, in addition to the spatial parameter in the Spatial Preferential Attachment generator, are needed to create high-fidelity topologies for two widely different technological systems, discrete circuits and process-control systems, and for brain networks. By introducing higher-order constraints such as total edge-length, we show that generators such as the generalized random graph (GRG) can create networks with as high a fidelity as the more computationally costly optimization approaches. We argue that higher-order constraints are necessary for all complex system generators as applied to systems with spatial properties.