Turing patterns in network-organized activator-inhibitor systems

Turing instability in activator-inhibitor systems provides a paradigm of nonequilibrium pattern formation; it has been extensively investigated for biological and chemical processes. Turing pattern formation should furthermore be possible in network-organized systems, such as cellular networks in morphogenesis and ecological metapopulations with dispersal connections between habitats, but investigations have so far been restricted to regular lattices and small networks. Here we report the first systematic investigation of Turing patterns in large random networks, which reveals their striking difference from the known classical behavior. In such networks, Turing instability leads to spontaneous differentiation of the network nodes into activatorrich and activator-low groups, but ordered periodic structures never develop. Only a subset of nodes having close degrees (numbers of links) undergoes differentiation, with its characteristic degree obeying a simple general law. Strong nonlinear restructuring process leads to multiple coexisting states and hysteresis effects. The final stationary patterns can be well understood in the framework of the mean-field approximation for network dynamics.