The ability to undergo bond exchange in a dynamic covalent polymer network has brought many benefits not offered by classical thermoplastic and thermoset polymers. Despite the bond exchangeability, the overall network topologies for existing dynamic networks typically cannot be altered, limiting their potential expansion into unexplored territories. By harnessing topological defects inherent in any real polymer network, we show herein a general design that allows a dynamic network to undergo rearrangement to distinctive topologies. The use of a light triggered catalyst further allows spatio-temporal regulation of the network topology, leading to an unusual opportunity to program polymer properties. Applying this strategy to functional shape memory networks yields custom designable multi-shape and reversible shape memory characteristics. This molecular principle expands the design versatility for network polymers, with broad implications in many other areas including soft robotics, flexible electronics, and medical devices.