Determining the core structure of complex network systems allows us to simplify them. Using h-bridge and h-strength measurements in a weighted network, we extract the h-backbone core structure. We find that focusing on the h-backbone in a network allows greater simplification because it has fewer edges and thus fewer adjacent nodes. We examine three practical applications: the co-citation network in an information system, the open flight network in a social system, and coauthorship in network science publications.