Stability of Partitions Induced by Nearest-Center Assignment Under Perturbations

We study clustering through the partitions it induces on a finite labeled set $[n]=\{1,\dots,n\}$, and analyze how these partitions change under perturbations of a point configuration $X=(x_1,\dots,x_n)\in(\mathbb{R}^d)^n$. We equip the space of partitions $\Pi_n$ with a normalized pairwise disagreement metric $d(\cdot,\cdot)$, and define the stability radius $r(X,A)=\sup\{\varepsilon\ge0: A(X')=A(X)$ whenever $|X-X'\|\le\varepsilon\}$, where $\|X-X'\|=\max_i\|x_i-x'_i\|$. Our main results concern nearest-center assignment with fixed centers $\{c_1,\dots,c_k\}\subset\mathbb{R}^d$. For each point, we define the margin $\gamma_i=\min_{j\ne\ell_i}(\|x_i-c_j\|-\|x_i-c_{\ell_i}\|)$ and $\gamma_{\min}=\min_i\gamma_i$, where $\ell_i$ denotes the assigned center. We show that if $\varepsilon<\gamma_{\min}/2$, then no assignments change under perturbation and hence $A(X')=A(X)$. Conversely, any point that changes its assigned center must satisfy $\gamma_i\le2\varepsilon$, showing that instability is localized near decision boundaries. We construct configurations in which arbitrarily small perturbations $\|X-X'\|\le\varepsilon$ alter the induced partition, demonstrating that the margin condition is sufficient but not necessary for stability. We further extend the framework to a discrete-time setting, showing that if $\sum_t \delta_t<r(X(0),A)$, then $A(X(t))=A(X(0))$, and we give a probabilistic bound on $\mathbb{E}[d(A(X),A(X'))]$ in terms of tail probabilities relative to $\gamma_i$. This framework identifies the margin as the key quantity governing both worst-case and average stability, and provides explicit conditions under which clustering-induced partitions remain invariant in a fixed-center Euclidean model.