Emergence of geometric Turing patterns in complex networks

Abstract

Turing patterns, arising from the interplay between competing species of diffusive particles, has long been an important concept for describing non-equilibrium self-organization in nature, and has 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 remains unresolved. In this work, we study the Turing instability in the framework of geometric random graph models, where the network topology is explained by 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.

16 pages, 8 figures, 1 table

References in corpus (22)

• Fast unfolding of communities in large networks
• Hyperbolic Geometry of Complex Networks
• What's in a crowd? Analysis of face-to-face behavioral networks
• Sustaining the Internet with Hyperbolic Mapping
• Turing patterns in network-organized activator-inhibitor systems
• Self-similarity of complex networks and hidden metric spaces
• Topology-driven instabilities: the theory of pattern formation on directed networks
• Pattern formation in multiplex networks
• Patterns of dominant flows in the world trade web
• Turing patterns in multiplex networks
• Langevin approach for the dynamics of the contact process on annealed scale-free networks
• Emergence of Soft Communities from Geometric Preferential Attachment
• The theory of Turing patterns on time varying networks
• Annealed and Mean-Field formulations of Disease Dynamics on Static and Adaptive Networks
• Turing patterns in systems with high-order interactions
• Quasi-stationary analysis of the contact process on annealed scale-free networks
• Percolation in self-similar networks
• From Caenorhabditis elegans to the Human Connectome: A Specific Modular Organisation Increases Metabolic, Functional, and Developmental Efficiency
• Detecting the ultra low dimensionality of real networks
• Sharp thresholds limit the benefit of defector avoidance in cooperation on networks
• A geometry-induced topological phase transition in random graphs
• Multiscale Voter Model on Real Networks

Cited by in corpus (9)

• Turing pattern theory on homogeneous and heterogeneous higher-order temporal network system
• Does the brain behave like a (complex) network? I. Dynamics
• Turing patterns on discrete topologies: from networks to higher-order structures
• Effect of clustering on Turing instability in complex networks
• Local Order Controls the Onset of Oscillations in the Nonreciprocal Ising Model
• Weakly nonlinear analysis of Turing pattern dynamics on curved surfaces
• Oscillatory and chaotic pattern dynamics driven by surface curvature
• Multiplexity amplifies geometry in networks
• Hidden multiscale organization and robustness of real multiplex networks