Effect of clustering on Turing instability in complex networks

Turing instability in complex networks is known to be dependent on the degree distribution, and the necessary conditions for Turing instability have been shown in the literature to have an explicit dependence on the eigenvalues of the Laplacian matrix, which, in turn, depends on the network topology. This study reveals that these conditions are not sufficient, and another global network measure-the nodal clustering-also plays a crucial role. Analytical and numerical results are presented to explain the effects of clustering for several network topologies, ranging from the S1/H2 hyperbolic geometric networks that enable modeling the naturally occurring clustering in real-world networks, as well as the random and scale-free networks, which are obtained as limiting cases of the S1/H2 model. Analysis of the Laplacian eigenvector localization properties in these networks is shown to reveal distinct signatures that enable identifying the so called Turing patterns even in complex networks.