Any Graph is a Mapper Graph

TL;DR

We address the inverse Mapper problem: given a dataset $X$ and a target graph $G$, can Mapper parameters be chosen so that the Mapper graph is isomorphic to $G$? We provide constructive routes—a star-cover/topology-based method and a convex-set representation in $\mathbb{R}^3$—showing that with an appropriate lens $f$ and cover $\mathcal{U}$ (and trivial clustering), MT$(\mathcal{U},f)\cong G$, and we extend these ideas to continuous lenses via extension theorems. The framework is further generalized to arbitrary finite simplicial complexes, with dimensionality bounds and extension results ensuring the Mapper construction can realize a wide class of complexes. Collectively, the work demonstrates the expressive power of Mapper and offers practical guidance for parameter design to realize desired topological structures from data.

Abstract

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper parameters such that the output Mapper graph of $X$ is isomorphic to $G$? We provide constructions that affirmatively answer this question. Our results demonstrate that it is possible to engineer Mapper parameters to generate a desired graph.

Figures (4)

• Figure 1: A collection $\mathcal{U}$ of sets and its nerve $\mathrm{Nrv}(\mathcal{U})$. The sets in $\mathcal{U}$ are colored in purple. Its nerve $\mathrm{Nrv}(\mathcal{U})$ consists of vertices, edges, and a triangle, and its geometric realization is colored in black. The space $\bigcup_{U\in \mathcal{U}}U$ and the geometric simplicial complex $\|\mathrm{Nrv}(\mathcal{U})\|$ are homotopy equivalent, as shown in the nerve theorem.
• Figure 2: Mapper Graph Construction. An example of the four-step procedure to construct a Mapper graph from a finite set of points $X\subseteq \mathbb{R}^2$. In steps 1 and 2, the lens function $f\colon X\to \mathbb{R}$ and cover $\mathcal{U}$ of $f(X)$ are initialized by the user. The lens function is the height function and the cover is denoted by the colored overlapping intervals. In step 3, the preimage of each interval is denoted by the points in the shaded regions with the same color. A clustering algorithm chosen by the user is applied to each colored region. The points in each ellipsoid-shaped region are clustered together. In step 4, we construct the Mapper graph by creating a vertex for each cluster, and adding an edge between two vertices if the two clusters contain at least one point from $X$ in common. Observe the Mapper graph is the 1-skeleton of the nerve of the clustered sets in step 3.
• Figure 3: Mapper graph with star cover reconstruction. Left: Each color represents the star of a vertex. The set of all stars of vertices forms the cover. The lens function $f$ maps at least one element of $X$ to each edge and isolated vertex in $G$. Right: The Mapper graph is constructed using the function $f$ and the star cover. We obtain seven vertices that are colored according to their corresponding cover element. The elements of $X$ listed on each vertex are those mapped to that cover element. The Mapper graph is isomorphic to the original graph $G$.
• Figure 4: Mapper graph with convex sets reconstruction. We consider $X=\{x_1, x_2, \dots, x_8\}$ and the same graph as in Figure \ref{['fig:star-open-cover']}. Left: Each color represents a convex set. The set of all convex sets forms a cover. The lens function $f$ maps at least one element of $X$ to each intersection of convex sets and isolated convex set. Right: The Mapper graph is constructed using the function $f$, the convex cover, and trivial clustering. We obtain seven vertices that are colored according to their corresponding cover element. The elements of $X$ listed on each vertex are those mapped to that cover element. The Mapper graph is isomorphic to the original graph.

Theorems & Definitions (13)

• Definition 2.1: Simplicial Complex
• Definition 2.2: Nerve
• Definition 3.1: Mapper with trivial clustering
• Theorem 3.2
• proof
• Theorem 3.3: wegner1967eigenschaften
• Theorem 3.4
• proof
• Proposition 3.5
• proof