Capturing dynamics of time-varying data via topology

Figure 1.  A filled-in disk (left) has Betti numbers $ (\beta_0, \beta_1, \beta_2, \beta_3, \ldots) = (1, 0, 0, 0, \ldots) $. A hollow square (center) has Betti numbers $ (1, 1, 0, 0, \ldots) $. A hollow two-torus (right) has Betti numbers $ (1, 2, 1, 0, \ldots) $. If we consider the union of these three shapes, the Betti numbers are $ (\beta_0, \beta_1, \beta_2, \beta_3, \ldots) = (3, 3, 1, 0, \ldots) $. Image of the torus taken from Wikimedia Commons https://commons.wikimedia.org/wiki/File:Torus.svg, available for reuse under CCA BY-SA 3.0

Figure 2.  An illustrative example of a crocker stack for $ H_0 $ computed from a simulation from the Viscek model with noise parameter $ \eta = 0.02 $; see Section 5. The variable $ \alpha $ is a smoothing parameter which captures the persistence of topological features

Figure 3.  Persistence diagram (Left) and corresponding persistence barcode (Right). Let the persistence barcode $ V $ consist of the intervals $ [1, 7] $, $ [2, 9] $, $ [3, 11] $, $ [5, 10] $, $ [5, 9] $, and let $ 4 = i \le j = 8 $. Then $ \mathrm{rank}(V(4) \to V(8)) $ is two since there are two intervals in the persistence diagram that contain the interval $ [i, j] = [4, 8] $

Figure 4.  Round points (in red) represent points in persistence diagram $ A $. Triangular points (in blue) represent points in persistence diagram $ B $. The bottleneck distance between persistence diagrams $ A $ and $ B $ is computed by taking the largest $ L_\infty $ distance between matched round and triangular points of a bijection between $ A $ and $ B $, and then taking the infimum over all bijections. The three boxes (in green) show the optimal matching of three pairs of round and triangular points. The unboxed points match to the closest points on the diagonal

Figure 5.  Vines and vineyard. Each point (in red) represents a point on a persistence diagram. Each dashed curve (in blue) is a vine traced out by a persistent point on time-varying persistence diagrams. The horizontal direction denotes time

Figure 6.  The effect of $ \alpha $-smoothing. (Top) A persistence diagram, the corresponding persistence intervals (drawn vertically), and one column of a crocker plot matrix. If we had points moving in time, then we would get a time-varying persistence diagram, a time-varying persistence barcode, and a complete crocker plot matrix (swept out from left to right as time increases). (Bottom) A persistence diagram with the thick line (in red) reflecting the choice of $ \alpha $-smoothing, along with the corresponding $ \alpha $-smoothed persistence intervals, and one column of an $ \alpha $-smoothed crocker plot matrix. The $ y $-intercept of the diagonal thick red line is $ 2\alpha $. All persistence diagram points under the thick line are ignored under $ \alpha $-smoothing

Figure 7.  To update the heading of an agent (centered, round blue point) according to the Vicsek model, we first find the nearby neighbors within a radius $ R $ (denoted by the dashed circle in red) and then take the average of its neighbors' headings, plus some noise

Figure 8.  A plot of order parameters for three simulations of the Vicsek model with different noise parameters $ \eta $. For smaller values of $ \eta $, particles become more aligned, i.e. move in the same direction, over time

Figure 9.  An example $ H_0 $ crocker plot of a simulation from the Viscek model with noise parameter $ \eta = 0.02 $. This is the same as an $ \alpha $-cross section of a crocker stack when $ \alpha $ = 0. In the region below the lowest curve (purple) where $ \beta_0 \geq 5 $, there can be many connected components, which we interpret as noise and is thus not displayed

Figure 10.  An example $ H_0 $ and $ H_1 $ crocker stack for a simulation from the Viscek model with noise parameter $ \eta = 0.02 $. This figure shows the shifts of Betti curves in $ H_0 $ and $ H_1 $ as smoothing parameter $ \alpha $ increases from $ 0 $ to $ 0.01 $ and $ 0.03 $

Figure 11.  The color scale corresponds to values in the distance matrix; denser color (red) means larger distances, and lighter color (yellow) means smaller distances. The 100 simulations of each noise parameter $ \eta = 0.01, 0.5, 1, 1.5, 2 $ are listed in order and annotated in the left matrix. The $ H_{0, 1} $ crocker distance matrix is more structured (Left) than the order parameter distance matrix (Right)

Figure 12.  The $ H_{0, 1} $ crocker distance matrix is more structured (Left) than the order parameter distance matrix (Right)

Figure 13.  The $ H_{0, 1} $ crocker distance matrix is more structured (Left) than the order parameter distance matrix (Right)

Figure 14.  The $ H_{0, 1} $ crocker distance matrix is more structured (Left) than the order parameter distance matrix (Right)

Figure 15.  Let $ Z = \mathbb{R}^2 $ with the Euclidean metric. The thicker solid curve (in red) represents subset $ X $ of $ Z $, and the thinner solid curve (in blue) represents subset $ Y $ of $ Z $. To compute the Hausdorff distance between $ X $ and $ Y $, we first take the supremum over all points in $ Y $ of the distance to the closest point in $ X $. In this figure, the distance is $ a = \sup_{y \in Y} \inf_{x \in X} d(x, y) $. Then, we do the same for the supremum over all points in $ X $ of the distance to the closest point in $ Y $, as shown by $ b = \sup_{x\in X} \inf_{y \in Y} d(x, y) $. Finally, we take the maximum of the two suprema, $ d_H^Z(X, Y) = \max\{a, b\} = a $

Figure 16.  Consider the persistence diagrams $ \mathrm{Dgm}_k(V) = \{ (3, 6), (2, 8) \} $, denoted with red circles, and $ \mathrm{Dgm}_k(W) = \{ (1, 7), (3, 7.5) \} $, denoted with blue triangles. The bottleneck distance between the two persistence diagrams is 1.5, while the erosion distance is 1

Figure 17.  Suppose $ A $ and $ B $ are dynamic metric spaces. The distance between any two of the four round points (in red) in $ A $ is 1 for all times $ t $. The distance between any two of the four triangular points (in blue) in $ B $ is $ 1+\varepsilon $ for all times $ t $. At an arbitrary time step $ t $ and scale parameter $ 1+\frac{\varepsilon}{2} $, we have $ \beta_0^A = 1 $ and $ \beta_0^B = 4 $