Persistent Homology: Barcode Diagrams

Vietoris-Rips filtration: At scale ε, connect all pairs of points within distance ε. As ε grows, simplices are added in order of their "birth" scale.
Persistent homology: Track topological features (H₀ = components, H₁ = loops) as ε increases. Each feature has a birth and death scale — long bars = robust topology.
Stability theorem (Cohen-Steiner 2007): d_B(Dgm(f), Dgm(g)) ≤ ‖f−g‖∞. Barcode is stable under perturbation — noise creates only short bars.
Colors: Green = H₀ (components), Pink = H₁ (loops). Drag ε slider to animate filtration.