Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback

TL;DR

This paper investigates how local feedback affects the coherence of large-scale networks like vehicular formations, revealing that higher spatial dimensions are necessary for maintaining coherence under stochastic disturbances.

Contribution

It demonstrates the dimension-dependent limitations of local feedback in maintaining coherence in large networks, highlighting the necessity of higher dimensions for effective control.

Findings

Coherence scales poorly in low dimensions with only local feedback.

Higher dimensions (3D and above) enable better coherence in large networks.

Local feedback cannot sustain large coherent formations in 1D networks.

Abstract

We consider distributed consensus and vehicular formation control problems. Specifically we address the question of whether local feedback is sufficient to maintain coherence in large-scale networks subject to stochastic disturbances. We define macroscopic performance measures which are global quantities that capture the notion of coherence; a notion of global order that quantifies how closely the formation resembles a solid object. We consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1, 2 and higher dimensions, with vehicular platoons corresponding to the 1 dimensional case. A common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures, with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions. In particular, we show that it is impossible to have large coherent one dimensional vehicular platoons with only local feedback. We analyze these effects in terms of the underlying energetic modes of motion, showing that they take the form of large temporal and spatial scales resulting in an accordion-like motion of formations. A conclusion can be drawn that in low spatial dimensions, local feedback is unable to regulate large-scale disturbances, but it can in higher spatial dimensions. This phenomenon is distinct from, and unrelated to string instability issues which are commonly encountered in control problems for automated highways.