One of the holy grails of materials science, unlocking structure-property relationships, has largely been pursued via bottom-up investigations of how the arrangement of atoms and interatomic bonding in a material determine its macroscopic behavior. Here, we consider a complementary approach, a top-down study of the organizational structure of networks of materials, based on the interaction between materials themselves. We unravel the complete “phase stability network of all inorganic materials” as a densely connected complex network of 21,000 thermodynamically stable compounds (nodes) interlinked by 41 million tie line (edges) defining their two-phase equilibria, as computed by high-throughput density functional theory. Analyzing the topology of this network of materials has the potential to uncover previously unidentified characteristics inaccessible from traditional atoms-to-materials paradigms. Using the connectivity of nodes in the phase stability network, we derive a rational, data-driven metric for material reactivity, the “nobility index,” and quantitatively identify the noblest materials in nature.