Rewiring complex networks to achieve cluster synchronization using graph convolution networks with reinforcement learning

Synchronization on complex networks depends on a myriad of factors such as embedded dynamics, initial conditions, network structure, etc. Current literature simplifies analysis of cluster synchronization leveraging conditions on network topology such as input-equivalence, network symmetries, etc., of which external equitable partition (EEP) is one of the most relaxed conditions. One practical problem is that for a dynamic system, how to alter a network to reach arbitrary achievable cluster synchronization and remaining faithful to the original structure. To solve this problem, we represent graph dynamics in Graph Convolution Network (GCN) modules that sit within an Actor-Critic Reinforcement Learning (AC-RL) framework under the condition of EEP. This allows the framework to select a good policy to sequentially rewire the network, where the sequence of moves matters. We test our method on two types of high-dimensional networked systems, Rossler dynamic networks and Hindmarsh-Rose neuronal circuits, with different network sizes. Our research opens up a way for the discovery of achievable cluster synchronization configurations by altering the network structure in any given networked dynamics.