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Self-organization in natural and engineered systems causes the emergence ofordered spatio-temporal motifs. In presence of diffusive species, Turing theoryhas been widely used to understand the formation of such patterns on continuousdomains obtained from a diffusion-driven instability mechanism. The theory waslater extended to networked systems, where the reaction processes occur locally(in the nodes), while diffusion takes place through the networks links. Thecondition for the instability onset relies on the spectral property of theLaplace matrix, i.e., the diffusive operator, and in particular on theexistence of an eigenbasis. In this work we make one step forward and we provethe validity of Turing idea also in the case of a network with defectiveLaplace matrix. Moreover, by using both eigenvectors and generalizedeigenvectors we show that we can reconstruct the asymptotic pattern with arelatively small discrepancy. Because a large majority of empirical networksare non-normal and often defective, our results pave the way for a thoroughunderstanding of self-organization in real-world systems.

Pattern reconstruction through generalized eigenvectors on defective networks