Spectral analysis of synchronization in mobile networks

We here analyze a system consisting of agents moving in a two-dimensional space that interact with other agents if they are within a finite range. Considering the motion and the interaction of the agents, the system can be understood as a network with a time-dependent topology. Dynamically, the agents are assumed to be identical oscillators, and the system will eventually reach a state of complete synchronization. In a previous work, we have shown that two qualitatively different mechanisms leading to synchronization in such mobile networks exist, namely global synchronization and local synchronization, depending on the parameters that characterize the oscillatory dynamics and the motion of the agents [1]. In this contribution we show that the spectral pattern differs between the two synchronization mechanisms. For global synchronization the spectrum is flat, which means that all eigenmodes contribute identically. For local synchronization, instead, the synchronization dynamics is determined mostly by the eigenmodes whose eigenvalues are close to zero. This result suggests that the global synchronization mechanism achieves fast synchronization by efficiently using the fast decaying eigenmodes (larger eigenvalues).