Effect of clustering on Turing instability in complex networks

Abstract

Turing instability in complex networks have been shown in the literature to be dominated by the distribution of the nodal degrees. The conditions for Turing instability have been derived with an explicit dependence on the eigenvalues of the Laplacian, which in turn depends on the network topology. This study reveals that apart from average degree of the network, another global network measure - the nodal clustering - also plays a crucial role. Analytical and numerical results are presented to show the importance of clustering for several network topologies ranging from the $\mathbb{S}^1$ / $\mathbb{H}^2$ hyperbolic geometric networks that enable modelling 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 $\mathbb{S}^1$ / $\mathbb{H}^2$ model. Analysis of eigenvector localization properties in these networks are shown to reveal distinct signatures that enable identifying the so called Turing patterns even in complex networks.