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Across many scientific and engineering disciplines, it is important toconsider how much the output of a given system changes due to perturbations ofthe input. Here, we investigate the glassy phase of $\pm J$ spin glasses atzero temperature by calculating the robustness of the ground states to flips inthe sign of single interactions. For random graphs and theSherrington-Kirkpatrick model, we find relatively large sets of bondconfigurations that generate the same ground state. These sets can themselvesbe analyzed as subgraphs of the interaction domain, and we compute many oftheir topological properties. In particular, we find that the robustness,equivalent to the average degree, of these subgraphs is much higher than onewould expect from a random model. Most notably, it scales in the samelogarithmic way with the size of the subgraph as has been found ingenotype-phenotype maps for RNA secondary structure folding, protein quaternarystructure, gene regulatory networks, as well as for models for geneticprogramming. The similarity between these disparate systems suggests that thisscaling may have a more universal origin.

Robustness and Stability of Spin Glass Ground States to Perturbed Interactions