Limits of multifunctionality in tunable networks

Significance Functionally optimized networks are ubiquitous in nature, e.g., in allosteric proteins that change conformation upon binding to a ligand or vascular networks that distribute oxygen and nutrients in animals or plants. Many of these networks are multifunctional, with proteins that can catalyze more than one substrate or vascular networks that can deliver enhanced flow to more than one localized region of the network. This work investigates the question of how many simultaneous functions a given network can be designed to fulfill, uncovering a phase transition that is related to other constraint–satisfaction transitions such as the jamming transition. Nature is rife with networks that are functionally optimized to propagate inputs to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. By adding and removing links, we are able to optimize both types of networks to perform specific functions. We define a single function as a tuned response of a single “target” link when another, predetermined part of the network is activated. Using network structures generated via such optimization, we investigate how many simultaneous functions such networks can be programed to fulfill. We find that both flow and mechanical networks display qualitatively similar phase transitions in the number of targets that can be tuned, along with the same robust finite-size scaling behavior. We discuss how these properties can be understood in the context of constraint–satisfaction problems.