Turing patterns in network-organized activator-inhibitor systems

TL;DR

This paper explores how Turing patterns, known from biological systems, manifest in large, complex networks, revealing unique behaviors such as node differentiation based on degree and the absence of periodic structures.

Contribution

First systematic study of Turing patterns in large random networks, uncovering novel differentiation phenomena and nonlinear dynamics distinct from classical lattice systems.

Findings

Nodes differentiate into activator-rich and low groups based on degree

No ordered periodic Turing patterns form in large networks

Multiple coexisting states and hysteresis effects observed

Abstract

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 activator-rich 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.