Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations
Mark M. Bailey
Abstract
Swarm and constellation reconfiguration can be viewed as motion of an unordered point configuration in an ambient space. Here, we provide persistence-stable, symmetry-invariant geometric representations for comparing and monitoring multi-agent configuration data. We introduce a quotient formation space $\mathcal{S}_n(M,G)=M^n/(G\times S_n)$ and a formation matching metric $d_{M,G}$ obtained by optimizing a worst-case assignment error over ambient symmetries $g\in G$ and relabelings $σ\in S_n$. This metric is a structured, physically interpretable relaxation of Gromov--Hausdorff distance: the induced inter-agent metric spaces satisfy $d_{\mathrm{GH}}(X_x,X_y)\le d_{M,G}([x],[y])$. Composing this bound with stability of Vietoris--Rips persistence yields $d_B(Φ_k([x]),Φ_k([y]))\le d_{M,G}([x],[y])$, providing persistence-stable signatures for reconfiguration monitoring. We analyze the metric geometry of $(\mathcal{S}_n(M,G),d_{M,G})$: under compactness/completeness assumptions on $M$ and compact $G$ it is compact/complete and the metric induces the quotient topology; if $M$ is geodesic then the quotient is geodesic and exhibits stratified singularities along collision and symmetry strata, relating it to classical configuration spaces. We study expressivity of the signatures, identifying symmetry-mismatch and persistence-compression mechanisms for non-injectivity. Finally, in a phase-circle model we prove a conditional inverse theorem: under semicircle support and a gap-labeling margin, the $H_0$ signature is locally bi-Lipschitz to $d_{M,G}$ up to an explicit factor, yielding two-sided control. Examples on $\mathbb{S}^2$ and $\mathbb{T}^m$ illustrate satellite-constellation and formation settings.