Periodic Coupling inhibits Second-order Consensus on Networks

Consensus algorithms on networks have received increasing attention in recent years for various applications, ranging from distributed decision making to multivehicle coordination. In particular, second-order consensus models take into account the Newtonian dynamics of interacting physical agents. For this model class, we uncover a mechanism inhibiting the formation of collective consensus states via rather small time-periodic coupling modulations. We treat the model in its spectral decomposition and find analytically that, for certain intermediate coupling frequencies, parametric resonance is induced on a network level-at odds with the expected emergence of consensus for very short and long coupling time scales. Our formalism precisely predicts those resonance frequencies and links them to the Laplacian spectrum of the static backbone network. The excitation of the system is furthermore quantified within the theory of parametric resonance, which we extend to the domain of networks with time-periodic couplings.