Soft materials can sustain large, elastic, and reversible deformations, finding widespread use as elastomers and hydrogels. These materials constitute 3-D polymer networks and are typically synthesized by cross-linking polymer chains or copolymerizing monomer and cross-linker. Seminal investigations have enabled control over the network architecture by cross-linking chains of poly(dimethylsiloxane), poly(1,4-butadiene), or tetra-poly(ethylene glycol); however, as soft materials become attractive for robotics, electronics, and prosthetics, codesigning the network architecture, mechanical, and functional properties has become pressing. We investigate the relationship among reaction pathway, network architecture, and mechanical properties in poly(ethyl glycidyl ether) networks synthesized by epoxide ring-opening polymerization with two organoaluminum catalysts. The key result is that uncontrolled polymerizations yield loosely cross-linked, entangled, soft, and extensible networks, whereas more controlled polymerizations, instead, lead to highly cross-linked, stiff, and brittle networks. Such catalytic control over network architecture and mechanical properties could enable design of novel soft, tough, and functional materials.