Expand description
Topological features via persistent homology.
Tracks topological changes in point clouds over time using Betti numbers from Vietoris-Rips persistent homology. Detects structural shifts like “the topic space fragmented” or “two clusters merged”.
§Betti Numbers
- β₀ = number of connected components (clusters)
- β₁ = number of loops (cyclic patterns)
Changes in Betti numbers over time reveal structural evolution:
- Increasing β₀ → fragmentation
- Decreasing β₀ → convergence
- β₁ appearing → cyclic behavior emerges
§Implementation
Uses Vietoris-Rips filtration on pairwise distances between points. Applied on region centroids (K~80 at L3), NOT raw points, for tractability.
§References
- Edelsbrunner, H. & Harer, J. (2010). Computational Topology. AMS.
- Zigzag persistence for temporal networks. EPJ Data Science, 2023.
Structs§
- Persistence
Diagram - Persistence diagram: collection of birth-death intervals.
- Persistence
Interval - A persistence interval: a topological feature that is “born” at one filtration radius and “dies” at another.
- Topological
Features - Topological summary features extracted from a persistence diagram.
Functions§
- l2_dist 🔒
- L2 distance between two vectors.
- topological_
summary - Extract topological summary features from a point cloud.
- vietoris_
rips_ h0 - Compute Vietoris-Rips persistent homology (dimension 0 only).