Pattern reconstruction through generalized eigenvectors on defective networks

Abstract

Self-organization in natural and engineered systems causes the emergence of ordered spatio-temporal motifs. In presence of diffusive species, Turing theory has been widely used to understand the formation of such patterns on continuous domains obtained from a diffusion-driven instability mechanism. The theory was later extended to networked systems, where the reaction processes occur locally (in the nodes), while diffusion takes place through the networks links. The condition for the instability onset relies on the spectral property of the Laplace matrix, i.e., the diffusive operator, and in particular on the existence of an eigenbasis. In this work we make one step forward and we prove the validity of Turing idea also in the case of a network with defective Laplace matrix. Moreover, by using both eigenvectors and generalized eigenvectors we show that we can reconstruct the asymptotic pattern with a relatively small discrepancy. Because a large majority of empirical networks are non-normal and often defective, our results pave the way for a thorough understanding of self-organization in real-world systems.

References in corpus (8)

• Turing patterns in network-organized activator-inhibitor systems
• Synchronization is optimal in non-diagonalizable networks
• Topology-driven instabilities: the theory of pattern formation on directed networks
• How limit cycles and quasi-cycles are related in systems with intrinsic noise
• Turing patterns in systems with high-order interactions
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Synchronization dynamics in non-normal networks: the trade-off for optimality
• Non-reciprocal interactions enhance heterogeneity

Cited by in corpus (2)

• Turing patterns on discrete topologies: from networks to higher-order structures
• Impact of directionality on the emergence of Turing patterns on m-directed higher-order structures