Maximally robust neutral networks in the correlated phase 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, and the set of interactions in a spin glass mapping to the ground state(s), among others. Naturally observed systems exhibit substantially higher robustness than would be expected from a random mapping of input to output. Casting the input-output map as vertex labeling of a Hamming graph, the robustness of an output to input mutations becomes a network-theoretic property. Using maximum entropy and a single constraint on global robustness, we present a general statistical physics model for discrete input-output maps which shows a universal transition between correlated (robust) and uncorrelated (fragile) phases. We analytically derive the scaling laws for robustness observed in natural systems and show that our model numerically reproduces other network topological features of natural input-output maps. We also elucidate the properties of subnetworks in the robust phase—maximally robust neutral networks known as "bricklayer's graphs"— including the robustness, which is related to the sums-of-digits function.