Emergence of Geometric Turing Patterns in Complex Networks

Turing patterns, arising from the interplay between competing species of diffusive particles, have long been an important concept for describing nonequilibrium self-organization in nature and have been extensively investigated in many chemical and biological systems. Historically, these patterns have been studied in extended systems and lattices. Recently, the Turing instability was found to produce topological patterns in networks with scale-free degree distributions and the small-world property, although with an apparent absence of geometric organization. While hints of explicitly geometric patterns in simple network models (e.g., Watts-Strogatz) have been found, the question of the exact nature and morphology of geometric Turing patterns in heterogeneous complex networks remained unresolved. In this work, we study the Turing instability in the framework of geometric random graph models, where the network topology is explained using an underlying geometric space. We demonstrate that not only can geometric patterns be observed, their wavelength can also be estimated by studying the eigenvectors of the annealed graph Laplacian. Finally, we show that Turing patterns can be found in geometric embeddings of real networks. These results indicate that there is a profound connection between the function of a network and its hidden geometry, even when the associated dynamical processes are exclusively determined by the network topology.