New high-resolution maps of Hawaiian lava flows highlight complex topographically controlled channel networks. Network geometries range from distributary systems dominated by branching around local obstacles, to tributary systems constricted by topography. We combine 2-D network analysis tools developed for river systems and neural networks with 3-D lidar morphologic analysis and historical records of flow emplacement to investigate both the origins of channel networks and their influence on flow morphology and behavior. We find that network complexity is a function of underlying slope and that the degree of flow branching, network connectivity, and longevity of individual channels all influence the final flow morphology (flow and channel widths and levee heights). Furthermore, because channel networks govern the distribution of lava supply within a flow, changes in the channel topology can dramatically alter the effective volumetric flux in any one branch, which affects both flow length and advance rate. Specifically, branching will slow and shorten flows, while merging can accelerate and lengthen them. Consideration of channel networks is thus important for predicting lava flow behavior and mitigating flow hazards with diversion barriers. Observed relationships between network geometry, flow parameters, and morphology also offer insight into the interpretation of these features elsewhere on Earth and other terrestrial planets.