  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.
