Designing strong and robust bioinspired structures requires an understanding of how function arises from the architecture and geometry of materials found in nature. We draw from trabecular bone, a lightweight bone tissue that exhibits a complex, anisotropic microarchitecture, to generate networked structures using multiobjective topology optimization. Starting from an identical volume, we generate multiple different models by varying the objective weights for compliance, surface area, and stability. We examine the relative effects of these objectives on how resultant models respond to simulated mechanical loading and element failure. We adapt a network-based method developed initially in the context of modeling trabecular bone to describe the topology-optimized structures with a graph-theoretical framework, and we use community detection to characterize locations of fracture. This complementary combination of computational methods can provide valuable insights into the strength of bioinspired structures and mechanisms of fracture.