Communications
Most swarm algorithms assume the network. These treat it as part of the plant: the radio graph has a state, that state degrades as the fleet spreads, and a controller can act on it before it fails.
Why a smooth link model matters
The usual disk model — connected inside a radius, not outside — is convenient and useless to a controller. Its gradient is zero everywhere and undefined at the rim, so the swarm gets no signal that a link is about to break, only that it already has. The models here are smooth and strictly decreasing in range, so the weight carries "this link is getting weak" long before it carries "this link is gone".
Why λ₂ rather than "is it connected"
Connectivity is a boolean, and a boolean tells a controller nothing until it is already too late. Algebraic connectivity — the second smallest eigenvalue of the weighted graph Laplacian — is strictly positive exactly while the network is connected, degrades smoothly as links stretch, and has a closed form gradient with respect to position. That is what turns connectivity from a property you check into one you fly.
Core Questions
- What does the radio actually do at range, and does the model's gradient carry usable information before the link fails?
- Which links are load-bearing? The Fiedler vector answers this, and the answer is rarely "the longest ones".
- What is coverage worth if it cannot be reported?
Algorithms
Prerequisites
- Spectral graph theory: Laplacians, eigenvalues, the Fiedler vector
- Radio path loss and link budgets
- Gradient-based multi-agent control