Turing patterns on adaptive networks

We are surrounded by spatio-temporal patterns resulting from the interaction of the numerous basic units constituting natural or human-made systems. In the presence of diffusive-like coupling, Turing theory has been largely applied to explain the formation of such self-organized motifs on networked systems, where reactions occur in the nodes and the available links are used for species to diffuse. In many relevant applications, those links do not have static weights, as very often assumed, but evolve in time, and more importantly, they adapt their weights as functions of the states of the nodes. In this work, we make one step forward, and we provide a general theory to prove the validity of the Turing idea in the case of adaptive symmetric networks with positive weights. The conditions for the emergence of Turing instability rely on the spectral property of the Laplacian matrix and the model parameters, thus strengthening the interplay between dynamics and network topology. A rich variety of patterns are presented by using two prototype models of nonlinear dynamical systems, the Brusselator and the FitzHugh-Nagumo model. Because many empirical networks adapt to changes in the system states, our results pave the way for a thorough understanding of self-organization in real-world systems.