I am trying to study Vietoris-Rips complexes that arise from a point data sample, in the context of topological data analysis. Each data point maps to a point in a metric space by some measurement function, which will serve as a vertex of the Vietoris-Rips complex.
Suppose some $k$ data points in the sample coincide, which means that the corresponding vertices in the VR complex are superimposed/repeated. (These $k$ vertices are at distance $0$ from each other and so form a $(k-1)$-face, which is homotopy equivalent to a point). I would like to know if there is any result which guarantees that a VR complex with such clusters of repeats is homotopy equivalent to a VR complex where each cluster of coincident vertices is just considered as a single vertex.
Any insights will be appreciated. Thanks.
Yes, it is known that a VR complex with such clusters of repeats is homotopy equivalent to a VR complex where each cluster of coincident vertices is just considered as a single vertex. Indeed, suppose that $v_1,\ldots,v_k$ are $k$ coincident vertices in your dataset. Then as you have pointed out, each $v_i$ and $v_j$ are connected by an edge for $i \neq j$. Also, it is the case that for any other vertex $u$ in the dataset, $u$ is either connected by an edge to all of the $v_i$ or to none of the $v_i$. Therefore, the vertex $v_1$ dominates the vertex $v_2$, which means that these two vertices are connected by an edge, and furthermore that every neighbor of $v_2$ is also a neighbor of $v_1$. Therefore, since $v_2$ is dominated, you can remove vertex $v_2$ without changing the homotopy type of the clique complex (the Vietoris-Rips complex). The reason why removing the dominated vertex $v_2$ doesn't change the homotopy type is because the link of the vertex $v_2$ is contractible --- indeed the link of the vertex $v_2$ is a cone (with apex $v_1$). Continuing as such, vertex $v_1$ dominates vertex $v_3$, and so you can remove vertex $v_3$ without changing the homotopy type. At the end, you have removed all of the vertices $v_2,v_3,\ldots,v_k$ without changing the homotopy type, leaving only a single vertex $v_1$ left at this position.
For example papers that give more information on dominated vertices, see the papers Complexes of graph homomorphisms, or Strong homotopy types, nerves and collapses, or LC reductions yield isomorphic simplicial complexes, or Nerve complexes of circular arcs, for example.