Pattern reconstruction through generalized eigenvectors on defective networks

Self-organization in natural and engineered systems causes the emergence of ordered spatio-temporal motifs. In the 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 a 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 is non-normal and often defective, our results pave the way for a thorough understanding of self-organization in real-world systems.