Periodic Coupling inhibits Second-order Consensus on Networks

TL;DR

This paper reveals how small periodic changes in coupling strength can prevent consensus in multi-agent networks by inducing parametric resonance at specific frequencies linked to the network's spectral properties.

Contribution

It analytically demonstrates the inhibitory effect of periodic coupling modulations on second-order consensus and extends parametric resonance theory to networked systems.

Findings

Periodic coupling inhibits consensus at certain frequencies.

Resonance frequencies are linked to the Laplacian spectrum.

Analytical predictions match numerical simulations.

Abstract

Consensus algorithms on networks have received increasing attention in recent years for various applications ranging from animal flocking to multi-vehicle co-ordination. Building on the established model for second-order consensus of multi-agent networks, 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.