Persistent Homology (Part 5)

So if we take the maximum scale, $\epsilon = 4$, our simplicial complex is: $S = \text{ { {0}, {1}, {2}, {0,1}, {2,0}, {1,2}, {0,1,2} } }$

But if we keep track of the pair-wise distances between points (i.e. the length/weight of all the edges), then we already have the information necessary for a filtration.

Here are the weights (lengths) of each edge (1-simplex) in this simplicial complex (the vertical bars indicate weight/length): $|{0,1}| = 1.4 \\ |{2,0}| = 2.2 \\ |{1,2}| = 3 And this is how we would use that information to build a filtration:$ S_0 \subseteq S_1 \subseteq S_2
S_0 = \text{ { {0}, {1}, {2} } }
S_1 = \text{ { {0}, {1}, {2}, {0,1} } }
S_2 = \text{ { {0}, {1}, {2}, {0,1}, {2,0}, {1,2}, {0,1,2} } }
$$

Basically each simplex in a subcomplex of the filtration will appear when its longest edge appears. So the 2-simplex {0,1,2} appears only once the edge {1,2} appears since that edge is the longest and doesn’t show up until $\epsilon \geq 2.2$