Reliable Topology for Dynamic Data: Mathematical Foundations and Applications

TL;DR

This work develops a rigorous stability theory for Crocker diagrams, a coarse, discretized topological summary used on time-evolving point clouds. It reveals a two-tier stability structure: exact invariance under a computable geometric separation (grid-threshold clearance $\Gamma$) when point perturbations satisfy $\delta < \Gamma/2$, and a geometry-driven bounded-change regime otherwise, with bounds depending on local density $\Lambda(\widehat{\varepsilon}_j)$. The authors extend the analysis to probabilistic stability under Gaussian noise, deriving conditions under which the probability of topology change decays exponentially with $\tau^* = \frac{\Gamma_{grid}}{\sqrt{2}\sigma} - \sqrt{d}$, and they address point churn with linear-in-$q$ bounds. Two analytical showcases—the analytically tractable breathing polygon and a feasibility analysis for epithelial-cell imaging—illustrate how geometry and density govern stability, guiding experimental design and interpretation. Together, these results provide practitioners with practical diagnostics and guarantees for Crocker-based analyses in biology, materials science, and beyond, explaining observed empirical robustness and highlighting limitations when handling large, dense point clouds.

Abstract

Across many scientific domains, practitioners rely on coarse, discretized summaries to track the evolving structure of complex systems under noise, measurement error, and changing system size. Understanding when such summaries are reliable -- and when apparent robustness is illusory -- remains a fundamental challenge. Topological data analysis (TDA) provides a case study: Crocker diagrams track the number of topological features across spatial scale and time, and because they are computationally efficient and easy to interpret, they have been widely used for exploratory analysis, bifurcation detection, model selection, and parameter inference. Despite their popularity, Crocker diagrams have lacked rigorous stability guarantees ensuring robustness to small data distortions. We develop a conditional stability theory for Crocker diagrams constructed from evolving point clouds. Our main results include deterministic conditions guaranteeing exact invariance when pairwise distances are well separated from the diagram's discretization thresholds, together with bounds on how much the diagrams can change when these conditions fail. We also establish probabilistic stability guarantees under Gaussian noise and bounds on topological change caused by adding or removing points, scaling linearly with the number of modified points. We illustrate these results using two complementary examples: an analytically tractable breathing polygon model that reveals how stability thresholds depend on geometry, and a feasibility analysis of epithelial cell imaging data showing when bounded-change guarantees provide the appropriate robustness framework. Together, these results reveal a two-tier stability structure for coarse, discretized topological summaries: exact invariance under verifiable geometric separation conditions, and geometry-controlled bounded change otherwise.

Figures (4)

• Figure 1: Geometry of the breathing pentagon and the origin of stability bounds. (A)--(C) A regular pentagon at minimum, mean, and maximum circumradius $a(t) = 1 + \frac{1}{2}\sin t$; the side length $c_1$ (blue) and diagonal $c_2$ (red dashed) scale proportionally with $a(t)$. (D) At the minimum radius $a = 1/2$ (where the gap $\Delta$ is smallest), we compare the chord-family spacing for $m=5$ versus $m=6$. The pentagon has two chord families with gap $\Delta \approx 0.363$; the shaded region shows $\Delta/2 \approx 0.18$ for scale. The hexagon (inset) has three chord families: the diameter chord $c_3$ (green) compresses the smallest gap to $\Delta \approx 0.134$, with $\Delta/2 \approx 0.07$. This explains the alternating pattern in Table \ref{['tab:stability']}: even-sided polygons have smaller values of $\Delta$ because diameter chords create tighter chord-family spacing. Note that the actual grid clearance $\Gamma$ depends on where grid values land relative to critical distances and should be computed directly.
• Figure 1: Filtration of the Vietoris--Rips complex for a regular pentagon with unit circumradius at time $t^\star$ (vertices $v_0, v_1, v_2, v_3, v_4$) after inserting point $v_\ast = (0.096, 0.294)$ at the geometric midpoint between non-adjacent vertices $v_0$ and $v_2$. Each panel shows the complex at threshold $\varepsilon$ using $\varepsilon/2$-balls (gray) around vertices; edges connect vertices at distance $\le \varepsilon$. (A)--(C) At small scales, $v_\ast$ first forms an isolated component, then connects to $v_1$, then to $v_0$ and $v_2$. (D) At $\varepsilon = c_1 \approx 1.176$, the pentagon side edges appear; since $v_\ast v_0$, $v_\ast v_1$, and $v_\ast v_2$ are already present from smaller scales, the flag property implies the triangles $(v_0,v_1,v_\ast)$ and $(v_1,v_2,v_\ast)$ (blue) appear immediately. (E) At $\varepsilon = 1.263$, edges $v_\ast$--$v_3$ and $v_\ast$--$v_4$ appear, making $v_\ast$ adjacent to all pentagon vertices. The resulting fan of five triangles fills the pentagon cycle, so $\beta_1 = 0$. (F) Without the insertion, the pentagon cycle would persist until $\varepsilon = c_2 \approx 1.902$. Thus the insertion causes $\Delta\beta_1 = -1$ on $[1.263, 1.902)$: it destroys the cycle early rather than creating a new one. The observed $|\Delta\beta_1| = 1$ lies within our theoretical bound of 15 based on the local neighbor cap $\Lambda = 6$.
• Figure 2: Crocker diagrams for the breathing pentagon ($m=5$), displaying $\beta_1(\varepsilon, t)$ on a discrete grid of scales and times. (A) The unperturbed Crocker diagram shows a characteristic band where $\beta_1=1$, corresponding to the loop that exists when the scale satisfies $c_1(t) < \varepsilon < c_2(t)$, where $c_1(t)=2a(t)\sin(\pi/5)$ and $c_2(t)=2a(t)\sin(2\pi/5)$. (B) In the exact-stability regime of Theorem \ref{['thm:exact_stability']} (i.e., $\delta < \Gamma/2$ where $\Gamma$ is the grid clearance), the Crocker diagram is unchanged; we display an identical diagram to emphasize the cell-for-cell invariance guaranteed by the theorem. (C) To illustrate how changes become possible once we move beyond the exact-stability regime, we display a perturbed instance in which the critical curves are shifted enough to cross some grid boundaries, producing localized changes in $\beta_1$. (D) The entrywise difference (C)$-$(A) highlights where grid cells change: red cells gained a loop ($0\to1$) and green cells lost one ($1\to0$). These diagrams are illustrative demonstrations of the stability theory in a controlled setting, not empirical validation from experimental data.
• Figure 3: Schematic pipeline for the feasibility calculation in Section \ref{['sec:cellmigration']}. (A) A cartoon epithelial sheet (confluent patch) with a small local sparse region. (B) Corresponding centroid locations (schematic); light circles illustrate an example localization uncertainty of radius $\delta$ around selected centroids. (C) A schematic Vietoris--Rips complex at a chosen scale $\varepsilon$: edges connect centroids whose pairwise distance is at most $\varepsilon$, and filled triangles indicate 2-simplices. A highlighted interior 1-cycle contributes to $\beta_1$.

Theorems & Definitions (27)

• Definition 1: Grid Clearance
• Theorem 2: Exact Stability
• Proof 1
• Remark 1
• Definition 3: Local Density Parameter
• Lemma 4: Simplex Participation Bound
• Proof 2
• Proposition 5: Change in Betti Number per Point
• Proof 3
• Theorem 6: Global Stability Bound