Failure in most material systems is characterized by strain localization, where deformation concentrates within a narrow region. Recently, a class of truss-based metamaterials has been shown to undergo severe deformation without exhibiting localization1. The mechanisms underlying this unusual delocalized response remain unknown. Here, we employ graph theory to elucidate the origins of this behavior. Each lattice is represented as a pair of graphs—the tension and compression networks—and their topological properties are quantified using graph-theoretic metrics. We find that the onset of localization correlates strongly with connectivity measures of these graphs. Specifically, deformation delocalization arises from an asymmetry in connectivity between these networks: when the tension graph remains more connected than the compression graph, deformation spreads throughout the structure instead of localizing. Connectivity measures such as average global efficiency2 capture this transition quantitatively. This framework provides design principles for creating materials and metamaterials that intrinsically resist failure localization.
