Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper

To demonstrate the adaptability of our Chemical Mapper framework in exploring diverse chemical latent spaces, we employ three chemical datasets in this study: (1) PubChem, a large-scale molecular repository; (2) a toxicity dataset containing rat oral acute toxicity data; and (3) a water solubility dataset with $logS$ values that quantify how readily compounds dissolve in water.

First, to illustrate the representational capability of mapper in capturing molecular distributions, we focus on a randomly selected subset of two million chemical structures from the PubChem database [21], a comprehensive repository of small molecules. Second, to highlight the ability of Chemical Mapper to capture both molecular structures and associated physicochemical properties, we incorporate two additional datasets from Deep-PK [31], which focus on toxicity and water solubility. The toxicity dataset contains 10,130 molecules and measures acute oral toxicity in rats, typically expressed as the lethal dose for 50% of test animals (LD₅₀). The water solubility dataset comprises 9,964 molecules, each labeled with the logarithm of water solubility (logS) at 20-25^∘C, expressed in log mol/L.

For structural analyses using Chemical Mapper (Section 3.2), we employ GEM [13] as the primary model for generating molecular structural representations. GEM utilizes a geometry-based graph neural network (GNN) architecture enhanced with self-supervised learning strategies designed to capture fine-grained molecular geometry. Molecular representations are produced using a pre-trained GEM model for two million molecules, each represented as a 32-dimensional real-valued vector.

To further examine the training behavior of other geometric deep learning models, we also evaluate GraphMAE [17] and Mole-BERT [48] under pre-trained and fine-tuned conditions (Section 3.3). GraphMAE, inspired by masked autoencoding techniques in natural language processing, learns effective graph representations through self-supervised learning by masking portions of node features and training the network to reconstruct them. In contrast, Mole-BERT adopts a transformer-based architecture to capture rich structural and chemical information. Drawing inspiration from BERT, it employs masking strategies to model contextual relationships within molecular graphs, enabling more flexible and comprehensive molecular feature learning.

The mapper construction is a powerful topological data analysis technique that captures the intrinsic shape of high-dimensional datasets [45]. Its 1D form, the mapper graph, provides a simplified graphical summary of complex point clouds, where nodes represent clusters and edges capture overlapping relationships among them. When applied to chemical space, the mapper graph enables hierarchical exploration of molecular scaffolds, functional groups, and properties, revealing meaningful topological patterns and chemical insights.

The mapper algorithm takes as input a high-dimensional point cloud $𝕏$ equipped with a real-valued function $f:𝕏\to ℝ$, referred to as a filter function. It produces a graphical summary of $𝕏$ in the form of a mapper graph, where each node represents a cluster interpreted as a topological neighborhood, and an edge connects two nodes if their corresponding neighborhoods overlap.

We illustrate the construction of a mapper graph using a 1D point cloud $𝕏$ sampled from the letter $𝒴$, rendered in a mathematical script font. As shown in Figure 1 (left), $𝕏$ is equipped with a height function $f$, which serves as the filter function. To obtain a topological summary of $𝕏$, we begin with a finite cover $𝒱=\{V_1,V_2,\mathrm{\dots }\}$ of $f(𝕏)$, formed by a set of partially overlapping intervals such that $f(𝕏)\subseteq {\bigcup }_j{V}_j$ (see Figure 1, middle). For each interval $V_j$, we consider the clusters induced by points in $f^{-1}(V_j)$; these clusters form a finite cover $𝒰=\{U_1,U_2,\mathrm{\dots }\}$ of $𝕏$ such that $𝕏\subseteq {\bigcup }_j{U}_j$, highlighted by rectangles. For example, points in $f^{-1}(V_1)$ form two clusters, $U_1$ and $U_2$, while those in $f^{-1}(V_2)$ yield four. The 1D nerve of $𝒰$ defines the mapper graph in Figure 1 (right), where each cluster $U_i$ corresponds to a node $i$, and an edge connects nodes $i$ and $j$ whenever clusters $U_i$ and $U_j$ have a nonempty intersection. For example, an edge connects the nodes representing clusters $U_2$ and $U_3$. The resulting mapper graph captures key topological features of the letter $𝒴$, including its branching and looping structures.

In our study, we use the $l^2$-norm as the filter function, which quantifies how strongly a model is activated by an input molecule and has been shown to yield meaningful insights when analyzing latent representations in deep learning [38, 39]. Given a fixed filter function, the mapper graph construction is determined by: (1) the number of intervals and their overlap, which control the resolution of the cover; and (2) the parameters of the clustering algorithm. The overlap among cover elements is crucial for maintaining connectivity among topological neighborhoods at a given resolution, thereby revealing both local and global topological structure. In practice, we employ DBSCAN [12] for clustering, with parameters $ϵ$ and minPts, where $ϵ$ defines the neighborhood radius determining how close points must be to belong to the same cluster, and minPts specifies the minimum number of points required to form a dense cluster. See Section 4.1 for a discussion on parameter turning.

The mapper construction offers a key advantage in its ability to integrate topological structure into clustering analysis. As an initial study, we compare t-SNE [47] and mapper [45] for visualizing the chemical space of molecular representations.

Our study uses two million chemical structures from the PubChem database [22], comprising ten distinct scaffolds (i.e., core molecular frameworks), each represented by $200,000$ molecules. Each chemical graph is converted into a $32$-dimensional feature vector using the pre-trained GEM model. Detailed experimental settings are provided in Section 2.

In the t-SNE visualization shown in Figure 2, each data point is colored by its chemical scaffold. In contrast, the mapper graphs used throughout this work represent each node as a group of molecular representations, visualized as a pie chart whose color proportions reflect the distribution of scaffolds within the node (see Figures 10 and 11), except in Section 3.2.3. Nodes rendered in a single color indicate high purity, meaning all molecules share the same scaffold (e.g., the mapper graph in Figure 5 contains only one impure node out of 5,104 total nodes). Although nodes are displayed with a uniform size by default, they may contain varying numbers of molecules, and their sizes can be adjusted if needed. The corresponding scaffold colormap, along with representative molecular structures, is shown in Figure 3.

While t-SNE organizes molecular representations into ten well-separated clusters, it does not capture relationships among them. In contrast, the mapper graph in Figure 5 reveals both inter- and intra-cluster connectivity and exposes finer-grained topological features – such as branches, chains, and loops – that capture meaningful structural distinctions within the chemical space. By capturing both local and global organization, the mapper graph offers a more comprehensive view of molecular structure, surpassing the representational capacity of conventional clustering methods. In subsequent sections, our results demonstrate that the mapper graph effectively uncovers topological organization within the chemical space, revealing clusters and their relationships and providing deeper insights into molecular structure and function (see Sections 3.2 and 3.3 for details).

We integrate molecular representations learned from geometric deep learning models into the mapper algorithm to create an exploratory framework, Chemical Mapper. As illustrated in Figure 4, the workflow begins with molecular datasets from which latent representations are extracted using deep learning models. These representations form a high-dimensional point cloud that is analyzed using the mapper algorithm. The resulting mapper graph captures relationships among scaffolds, functional groups, and molecular properties. Downstream analyses leverage this graph to reveal structural patterns and property distributions, highlighting the framework’s capacity to uncover organizational principles within chemical latent space.

We analyze a large-scale dataset of two million molecules spanning ten scaffolds from the PubChem database [22]. Each molecule is embedded into a 32-dimensional latent space using a pre-trained GEM model. Figure 5 shows the mapper graph constructed from these representations using 100 intervals with 50% overlap, $ϵ=1$, and $\text{minPts}=10$.

In Figure 5(left), the mapper graph is drawn using an anchored layout, where each node is initialized at the mean of the 2D t-SNE coordinates of its assigned data points, effectively grounding the graph in the underlying embedding. Figure 5(right) shows the corresponding force-directed layout, which refines this initialization by iteratively adjusting node positions through attractive forces along edges and repulsive forces between nodes to improve separation and reduce visual clutter. For the remainder of the paper, we use the force-directed layout, as it yields clearer and more interpretable visualizations.

We analyze the topological structure of this graph and its relationship to variations in the chemical latent space, focusing on connected components (Section 3.2.1) and branches (Section 3.2.2). We further examine how the graph organization correlates with chemical functions and properties such as toxicity and solubility (Section 3.2.3).

In this section, we demonstrate how mapper graphs can serve as a powerful tool for characterizing and evaluating molecular representations learned by geometric deep learning models such as GraphMAE and Mole-BERT. We further show that mapper graphs enable qualitative comparisons of the learning capabilities between pre-trained and fine-tuned models.

Following the procedure for the GEM model (Section 3.2), we apply Chemical Mapper to the molecular representations of the same two million molecules from the PubChem database [22], learned by the pre-trained GraphMAE model. A subgraph of the resulting mapper graph is shown in Figure 10 (left), where each node is represented as a pie chart indicating the distribution of molecular scaffolds within that node. Each node clearly contains a mixture of scaffolds, suggesting that the pre-trained GraphMAE model does not separate the chemical space according to scaffold structure (cf. Section 3.2.1). Moreover, the mapper graph exhibits no clear branching patterns, in contrast to the structured branches observed for the GEM model (Section 3.2.2).

Subsequently, the pre-trained GraphMAE model is fine-tuned on the task of predicting molecular scaffolds, and representations for the same two million molecules are extracted from the fine-tuned model. Using identical parameters, a new mapper graph is constructed. Figure 10 (right) shows a portion of this graph, where the nodes are noticeably more homogeneous, with molecules sharing the same scaffold clustering together. Fine-tuning the GraphMAE model, therefore, induces a clearer partitioning of the chemical space according to molecular scaffolds. The resulting mapper graph contains 1,606 nodes – a substantial increase from the 420 nodes in the mapper graph of the pre-trained model – indicating that fine-tuning expands the representational space and more effectively separates molecules with different scaffolds.

Furthermore, the mapper graph derived from the fine-tuned model consists predominantly of chains and exhibits few branches. As discussed in Section 3.2.2, branching in the chemical space reflects distinctions among functional groups. Because the fine-tuning task focuses solely on scaffold prediction and does not consider functional groups, the lack of branching suggests that the chemical space is organized primarily around scaffold structure rather than functional group variation.

We conduct a similar experiment with Mole-BERT on the same two million PubChem molecules, obtaining results qualitatively consistent with GraphMAE. Figure 11 compares mapper graphs from the pre-trained (left) and fine-tuned (right) Mole-BERT models for the scaffold classification task.

For completeness, we may also compare the pre-trained and fine-tuned GEM model. Figure 12 shows the complete mapper graph of the chemical latent space derived from the GEM model fine-tuned on the scaffold classification task, with nodes colored by molecular scaffolds. We perform a comparative analysis between the pre-trained and fine-tuned GEM models. As shown in Figure 5 and Figure 12, the mapper graph of the fine-tuned GEM model exhibits the same qualitative trends observed in the pre-trained model (Section 3.2): molecules with different scaffolds remain clearly separated into distinct connected components, and branching structures continue to reflect variations in functional groups.

Because the chemical space produced by the pre-trained GEM model already partitions molecules according to scaffolds (as shown in Figure 5), fine-tuning on this task does not substantially affect the purity of the cluster nodes. However, comparing the mapper graphs in Figure 5 and Figure 12 reveals notable topological changes. Certain scaffold classes become more dispersed, exhibiting a larger number of isolated nodes and fewer interconnections (e.g., c1ccccc1), whereas others contract significantly (e.g., c1ccc(-c2ccccc2)cc1 and c1ccc(COc2ccccc2)cc1). Investigating the causes and implications of these topological changes represents an important direction for future research.

The Chemical Mapper framework involves several key parameters: the number of intervals $m$, the percentage of overlap $p$, and the DBSCAN clustering parameters, including the neighborhood radius $ϵ$ and the minimum number of points minPts required to form a core point. These parameters are collectively denoted as $(m,p,ϵ,\text{minPts})$. In practice, they are often hand-tuned (as in this study), with best practice being to select values within ranges where the mapper graph structures (such as branches, loops, and connected components) remain stable. Prior works have investigated automated parameter selection for mapper graphs [1, 6, 7, 8], and $ϵ$ can be estimated using the elbow method [12, 53].

Theoretical studies have investigated the structural stability of mapper graphs [4, 7] and their multiscale variants [11]. In particular, mapper graphs are stable under perturbations of the data $(𝕏,f)$, with stability depending on how the cover $𝒰$ aligns with the critical values of $f$ [4]. However, these theoretical guarantees rely on statistical assumptions that are often not satisfied in real-world datasets. Motivated by this gap, we provide empirical evidence of stability in Figure 13.

Building on these findings, we examine the stability of mapper graphs under varying parameter configurations using two million molecular representations from PubChem, generated by the pre-trained GEM model (same as the dataset used in Section 3). As shown in Figure 13, local branches in the mapper graphs remain consistent across suitable ranges of $m$, where each branch corresponds to molecules sharing a functional group, indicated alongside the branch. In particular, the first row in Figure 13 highlights connected components containing molecules with –O– and –(O=S=O)– functional groups, exhibiting the same bifurcation pattern across three values of $m$. Similarly, the second row in Figure 13 shows a bifurcation between molecules containing –F and –Cl groups across different interval numbers.

Overall, this experiment provides an initial demonstration of the robustness of the mapper framework in capturing chemically meaningful topological structures across parameter settings. Although parameter variations affect the granularity of branching, the key separations corresponding to distinct functional groups remain stable. Future work should investigate how chemical semantics relate to parameter choices, guiding principled parameter selection and further enhancing the mapper’s interpretability and applicability in cheminformatics.

We plan to extend Chemical Mapper to study structural motifs beyond scaffolds, including functional groups as well as aromatic and reactivity motifs. The recent Explainable Mapper framework [49] provides an effective strategy for analyzing high-dimensional latent spaces by integrating topological representations with agentic large language model (LLM)–based explanations. In the chemical domain – where latent spaces may encode relationships among molecular structure, reactivity, and function – this approach opens up rich opportunities for human–AI collaboration. In such a setting, rather than relying solely on human experts to explain mapper elements (Section 3.2), agents powered by LLMs or VLMs (Vision–Language Models) can propose distinguishing descriptors (e.g., fingerprints, substructures, or property ranges) and generate hypotheses about the chemical attributes that unify a given mapper element – such as the connected components, chains, and branches described in Section 3.2. Human experts can then validate or refine these hypotheses using their domain knowledge. We leave this direction for future work.