Examples

Simple cover

Simple cover

Suppose $f:X\to \mathbb{R}$ and suppose $\mathcal{C}$ is just the interval $[-L,L]$ containing the entire $f^{-1}(X)$.

In this case Mapper produces a graph with no edges. Its vertices correspond to clusters of $X=f^{-1}([-L,L])$.

2-Component cover

2-Component cover

Suppose $f:X\to \mathbb{R}$ and suppose $\mathcal{C}$ is the pair $U_{\alpha }=[0,5]$ and $U_{\beta }=[6,10]$. Mind the gap $(5,6)$.

Vertices in the Mapper graph are $\epsilon$-clusters of $f^{-1}(U_{\alpha })$ and of $f^{-1}(U_{\beta })$. These clusters must be all disjoint. So there are no edges between the vertices.

If, however, the cover had been chosen to be $U_{\alpha }=[0,5]$ and $U_{\beta }=[4,10]$, there could be vertices between clusters in $f^{-1}(U_{\alpha })$ and clusters in $f^{-1}(U_{\beta })$.

Consider data points sampled around the unit circle. Let $f:{\mathbb{R}}^2\to \mathbb{R}$ by $f(x,y)=x$. Suppose the image $f(X)$ is covered by small overlapping intervals of equal length. Mapper recovers the topology of the circle:

If we had used a simple cover (single interval), the Mapper graph would have no edges and thus would not express the circle’s topology.