Pseudo-phase difference guides additional connection between oscillators for synchrony

In complex systems, synchronization plays a pivotal role underlying the coherent operation of various systems (networks) ranging from biology to technology. In a dynamic network, a link between nodes can be newly created implementing a new interaction in the network. Therefore, it is of great importance to understand how to enhance the synchronized state of a system especially when adding a new connection. This study investigates ways to enhance synchronization through optimal link addition, employing the Synchrony Alignment Function (SAF) and Adjusted Lyapunov Function (ALF) that assess the effects of new connections. By applying the ALF method to compare potential link additions, we identify two key factors that contribute to the effectiveness of link addition: the steady-state phase in the linearized dynamics, which we named the pseudo-steady-state phase, and the structural attributes of the network. By applying these methods across diverse network topologies, including Barabási–Albert, Erdős–Rényi, and Cayley tree models, we uncover the dominant role of the phase difference in promoting synchronization. This exploration offers new insights into the dynamics of network synchronization, highlighting the critical impact of specific factors on the efficacy of enhancing network coherence. Our findings also lay a foundation for further research into targeted strategies for network optimization.