Emergence of Turing patterns in complex networks: A partial link activation approach

Turing patterns, typical examples of self-organization phenomena, are widely observed in the natural world. From Turing's pioneering work [Philos. Trans. R. Soc. Lond. B 237, 37 (1952)10.1098/rstb.1952.0012] to recent advances by Nakao et al. [Nat. Phys. 6, 544 (2010)10.1038/nphys1651], a fundamental framework has been established for analyzing pattern formation in network-organized reaction-diffusion systems. Notably, most existing studies assume identical diffusion coefficients for all network links to find the conditions for Turing patterns emergence. This naturally raises a question: is pattern formation possible if only a subset of network links has diffusion coefficients that fulfill the requirements? Links characterized by diffusion coefficients satisfying these conditions are termed as activated links. This paper investigates the emergence of Turing patterns through a partial link activation approach, encompassing both random and targeted activation schemes. The mean-field theory is employed to derive the conditions for pattern formation when network links are randomly activated. Numerical simulations are conducted to compare the efficiency of random and targeted activation schemes in triggering Turing patterns. The results demonstrate that the partial link activation approach can indeed generate Turing patterns, offering a more efficient mechanism. And the targeted activation scheme exhibits enhanced performance in promoting pattern formation relative to a random activation scheme. This study provides new perspectives on the role of local regulation in governing global network dynamics.