A Phase Transition Between Random (Fragile) and Correlated (Robust) Phases of Input-Output Maps

Systems which accept a sequence-based input and produce a nontrivial output appear widely across scientific disciplines. Examples include protein/RNA primary sequences mapping to their folded structures, gene regulatory network interactions mapping to expression cycles, or the set of interactions in a spin glass mapping to the ground state(s), among others. In uncorrelated systems, the robustness to perturbations of the inputs scales as the frequency of obtaining the output. Since there are typically many outputs, this implies that input-output maps are fragile. It has been observed, however, that many input-output maps exhibit enhanced robustness, which scales as the log of the output frequencies. We present a generalized statistical physics model of discrete input-output maps arising from entropy maximization with a single constraint on the global robustness. By mapping to a Potts model on a Hamming graph with fixed state frequencies, we analytically derive the naturally observed scaling laws for robustness and numerically reproduce observed topological properties of subnetworks which map to a common output. We suggest that there is a universal transition between uncorrelated “fragile” and correlated “robust” phases for input-output maps.