Emergent Behavior Is Robust to Communication Delays at the Cost of Slower System Evolution

Previous works have shown that strong coupling among heterogeneous agents enforces practical synchronization, leading to collective behavior governed by emergent dynamics. However, it remains an open question whether emergent behavior persists in the presence of communication delays, as high-gain methods are typically sensitive to delays. To address this problem, we instead slow down the agent dynamics, reproducing the synchronization mechanism of strong coupling on a slow time scale. As a result, the emergence of collective behavior is guaranteed despite arbitrary constant communication delays. This, however, comes at a cost: delays induce a derivative-like term, thereby scaling down the vector field of the emergent dynamics. The amount of the scaling is explicitly characterized as a function of the coupling weights and the delay lengths. We further suggest that this delay-induced slowdown, which is of independent interest, may be a general feature of diffusive coupling under suitable assumptions.