The qualitative and quantitative relationships between pattern formation and average degree in networked reaction-diffusion systems

The Turing pattern is an important dynamic behavior characteristic of activator-inhibitor systems. Differentiating from traditional assumption of activator-inhibitor interactions in a spatially continuous domain, a Turing pattern in networked reaction-diffusion systems has received much attention during the past few decades. In spite of its great progress, it still fails to evaluate the precise influences of network topology on pattern formation. To this end, we try to promote the research on this important and interesting issue from the point of view of average degree-a critical topological feature of networks. We first qualitatively analyze the influence of average degree on pattern formation. Then, a quantitative relationship between pattern formation and average degree, the exponential decay of pattern formation, is proposed via nonlinear regression. The finding holds true for several activator-inhibitor systems including biology model, ecology model, and chemistry model. The significance of this study lies that the exponential decay not only quantitatively depicts the influence of average degree on pattern formation, but also provides the possibility for predicting and controlling pattern formation.