How local spectral gaps regulate the multistability of Turing patterns on graphs

Due to their self-organised, collective nature, Turing patterns on graphs are an important source of information about the relationship between graph architecture and dynamics. One defining feature of these dynamics is the coexistence of multiple stable patterns (‘pattern diversity’), the dependency of which on network architecture is still not well understood. Here we create standardised situations near the Turing instability threshold and study the multistability of patterns as a function of structural perturbations of the graph. In particular, we analyse the resulting changes in pattern diversity as a binary classification problem. We find an asymmetry between lower and higher eigenvalues near the Turing instability, which can be understood in terms of the interlacing theorem, known from spectral graph theory. This allows us to derive rules governing the multistability of Turing patterns using local spectral gaps of the graph’s Laplacian as input, but also evaluate the contribution of nonlinear interactions between eigenmodes to pattern diversity, independently of the interaction model considered.Author summary: Pattern formation has for a long time been at the core of the exploration of complex systems. Also, the nonlinear models giving rise to spatiotemporal patterns have informed method development and analysis strategies in nonlinear dynamics. Studying such processes on graphs is a relatively new trend with the goal of understanding structure-function relationships in complex networks. Here we consider a class of spatiotemporal patterns arising in reaction-diffusion systems, Turing patterns, and use a machine-learning approach – binary classification – to understand, which network properties affect the diversity of such patterns.