This PDF file includes:

• Section S1. Calculation of the T = 0 K universal phase diagram
• Section S2. Degree distribution of the network of all materials
• Section S3. New information encoded in the nobility index
• Table S1. Sample compute times for calculating the existence of a tie-line between two phases.
• Fig. S1. Fitting node connectivity data to candidate distributions.
• Fig. S2. Comparison of nobility index versus common elemental properties.
• Fig. S3. Comparison of number of compounds formed by an element versus its node degree.

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