Fig. 3. Hierarchy in the materials network and underlying energetic considerations.

(A) The mean node degree or average number of tie-lines 〈k〉 (green, open) decreases as a function of number of components 𝒩 (i.e., binary, ternary, and so on), which results from high-𝒩 materials having to compete with low-𝒩 materials for stability. The number of known stable 𝒩-ary materials (red) itself actually peaks at 𝒩 = 3 (ternaries). (B) Gaussian kernel density estimates of compound formation energies for all stable materials separated by number of components in the material. Dashed vertical lines indicate the respective median of each distribution. High-𝒩 materials need notably lower formation energies than low-𝒩 materials to become stable, e.g., −2.08 versus −0.47 eV per atom for quaternary and binary materials, respectively.