Figure 2. Topology, persistent homology, and barcodes.

(a) A simplicial complex is a simplified representation of the original space with the same topological features. It is a generalization of a network which, apart from nodes and edges, contains higher dimensional polytopes such as triangles and tetrahedrons. (b) An empty torus consists of one connected component, two independent loops (marked in red), and a two-dimensional void. The dimensions of its 0^th, 1^st, and 2^nd homology groups are respectively 1, 2, and 1. (c) In a Vietoris-Rips filtration a simplicial complex is built from the data at each scale ε by considering the intersection of balls of radii ε centered at the points. Points whose balls intersect are connected in the simplicial complex. Persistent homology groups track how the topological features associated to the simplicial complexes change with the scale ε. (d) Barcodes are a suitable representation of persistent homology groups, where each interval indicates the range of ε for which a given topological feature is associated to the data. In this figure, the 0^th and 1^st persistent homology groups are represented in the barcode.