Enriched Cell-Leaf Graph (ECLG)

• Enriched Cell-Leaf Graph (ECLG) is a hybrid graph-based method that fuses random forest-derived cell-leaf structures with K-nearest-neighbor graphs to capture both gene regulation and expression similarity.
• It employs the LINE network embedding to preserve first- and second-order proximities, leading to enhanced visualization, clustering, and rare cell detection in single-cell RNA-seq data.
• Empirical results demonstrate improved performance with average NNE around 4%, ARI up to 0.85, and superior detection of rare populations compared to traditional PCA-based embeddings.

The Enriched Cell-Leaf Graph (ECLG) is a hybrid graph-based construction designed to leverage both gene expression similarity and data-driven gene-gene regulatory relationships in the embedding of single-cell RNA sequencing (scRNA-seq) data. By integrating a bipartite cell-leaf structure derived from random forest models with a traditional K-nearest-neighbor (KNN) graph over cells, the ECLG provides a richer and more informative representation of cellular heterogeneity. The ECLG is subsequently embedded using the LINE network embedding algorithm, enabling improved detection of rare cell populations, enhanced clustering, and superior visualization compared to expression-based embeddings alone (Goudarzi et al., 1 Sep 2025).

1. Formal Construction and Mathematical Framework

Let $U = \{c_1, \ldots, c_n\}$ denote the set of $n$ single cells with expression matrix $X \in \mathbb{R}^{n \times p}$ for $p$ highly variable genes. The ECLG is constructed by fusing two constituent graphs:

• Cell-Leaf Graph (CLG):

The CLG is a bipartite graph $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$, where the set of leaves $V = \{\ell_1, \ldots, \ell_m\}$ are extracted from all trees in an ensemble of $T$ random-forest regressors—one per target gene. An edge $(c_i, \ell_j) \in E_{\text{CLG}}$ exists if cell $c_i$ falls into leaf $\ell_j$ in at least one tree. The edge weight $n$0 is typically 1 or counts the number of trees where $n$1 visits $n$2. The block adjacency matrix is:

$n$3

where $n$4.

• K-Nearest-Neighbor Graph (KNNG):

The KNNG is built over $n$5 using PCA-reduced features (typ. $n$6). Pairwise distances $n$7 are computed (Euclidean or $n$8). For each $n$9, neighbors $X \in \mathbb{R}^{n \times p}$0 are found, yielding edge set $X \in \mathbb{R}^{n \times p}$1. Edges are weighted via a Gaussian RBF:

$X \in \mathbb{R}^{n \times p}$2

with $X \in \mathbb{R}^{n \times p}$3 minimizing Nearest-Neighbor Error (NNE). The adjacency $X \in \mathbb{R}^{n \times p}$4 is $X \in \mathbb{R}^{n \times p}$5, nonzero iff $X \in \mathbb{R}^{n \times p}$6.

• Enriched Cell-Leaf Graph (ECLG):

The joint adjacency is defined as:

$X \in \mathbb{R}^{n \times p}$7

with $X \in \mathbb{R}^{n \times p}$8 (frequently $X \in \mathbb{R}^{n \times p}$9) determining the trade-off between gene-interaction and expression similarity. Degree normalization may be applied: $p$0, where $p$1.

2. Graph Construction Algorithms

(a) Random-Forest-Based CLG Extraction

• Input: $p$2 (log-normalized, HVG-filtered).
• For each gene $p$3 ($p$4), train a random-forest regressor ($p$5 trees) predicting $p$6 from $p$7, minimum leaf size $p$8.
• Each tree defines leaf nodes $p$9; for each cell $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$0, record leaf membership across trees.
• Bipartite edges $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$1 constructed with weight 1 or frequency.
• Optionally, regulator importance scores $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$2 impurity decreases where $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$3 splits $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$4 (from GENIE3), but gene nodes are not retained in the CLG.

(b) KNNG Construction

• PCA to $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$5 dimensions on $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$6 (typ. $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$7).
• Distance matrix $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$8 computed.
• For each $G_{\text{CLG}} = (U \cup V, E_{\text{CLG}})$9, select $V = \{\ell_1, \ldots, \ell_m\}$0 nearest neighbors ($V = \{\ell_1, \ldots, \ell_m\}$1 by heuristic or validation).
• Edge weights $V = \{\ell_1, \ldots, \ell_m\}$2; $V = \{\ell_1, \ldots, \ell_m\}$3 from NNE minimization.
• Adjacency assembled and optionally symmetrized, $V = \{\ell_1, \ldots, \ell_m\}$4.

3. Embedding via LINE Network Embedding

The ECLG is processed via the LINE algorithm, preserving both first- and second-order proximities for each node $V = \{\ell_1, \ldots, \ell_m\}$5 (cell or leaf).

Model Specification:

• Embedding dimension $V = \{\ell_1, \ldots, \ell_m\}$6 (e.g., $V = \{\ell_1, \ldots, \ell_m\}$7).
• First-order proximity: for edge $V = \{\ell_1, \ldots, \ell_m\}$8 with weight $V = \{\ell_1, \ldots, \ell_m\}$9, directly models link presence.
• Second-order proximity: for node $T$0, preserves neighborhood distribution over $T$1.

Loss Functions:

• First-order: $T$2, where $T$3.
• Second-order: $T$4, $T$5.
• Total: $T$6.

Optimization:

• Negative sampling to approximate the log-sigmoid.
• Stochastic gradient descent (or asynchronous SGD) over edges.
• No node features are required; initial $T$7 are randomly initialized.

4. Evaluation Metrics and Downstream Applications

Performance of ECLG-based embeddings is assessed using local and global metrics:

• Rare-Cell Detection: Nearest-Neighbor Error (NNE),

$T$8

where $T$9 is the ground-truth label and $(c_i, \ell_j) \in E_{\text{CLG}}$0 is the label of cell $(c_i, \ell_j) \in E_{\text{CLG}}$1's nearest neighbor in the embedding.

• Clustering Quality: Adjusted Rand Index (ARI) and Normalized Mutual Information (NMI),

$(c_i, \ell_j) \in E_{\text{CLG}}$2

• Visualization: t-SNE or UMAP applied to the learned $(c_i, \ell_j) \in E_{\text{CLG}}$3-dimensional embedding; NNE is also computed in 2D projections.
• Trajectory Inference: Embeddings serve as input to pseudotime and trajectory tools (e.g., Monocle), although not quantitatively benchmarked in (Goudarzi et al., 1 Sep 2025).

5. Quantitative Results and Empirical Performance

On six public scRNA-seq benchmarks, the ECLG+LINE pipeline (denoted DAE) achieved an average NNE of approximately $(c_i, \ell_j) \in E_{\text{CLG}}$4, compared to $(c_i, \ell_j) \in E_{\text{CLG}}$5–$(c_i, \ell_j) \in E_{\text{CLG}}$6 for methods such as PCA, SVD, and t-SNE. Clustering using Phenograph on ECLG embeddings reached ARI up to $(c_i, \ell_j) \in E_{\text{CLG}}$7 and NMI up to $(c_i, \ell_j) \in E_{\text{CLG}}$8, versus $(c_i, \ell_j) \in E_{\text{CLG}}$9–$c_i$0 for PCA-based methods. Visualization using t-SNE or UMAP showed superior separation of rare populations (e.g., microglia at $c_i$1 frequency) when using embeddings derived from the ECLG (Goudarzi et al., 1 Sep 2025).

Method	NNE (%)	ARI (max)	NMI (max)
DAE (ECLG + LINE)	~4.09	0.85	0.87
PCA/SVD/tSNE	6–27	0.60–0.75	0.60–0.75

6. Context and Significance

By integrating both the regulatory structure captured via random-forest leaf assignments (CLG) and the local transcriptional similarity (KNNG), the ECLG addresses a central limitation of conventional embeddings that focus solely on expression measures, often neglecting gene-gene interactions relevant for cell identity and function. The approach outperforms expression-only and graph-only baselines across several key analytic tasks, notably in rare-cell detection and cluster separability (Goudarzi et al., 1 Sep 2025). A plausible implication is that incorporation of implicit, data-driven gene interaction structure can considerably enhance biological signal recovery in scRNA-seq analysis frameworks.

7. Relationship to Prior Methods and Extensions

ECLG generalizes and extends earlier graph-based embeddings in single-cell analysis by (a) extracting CLG structure through random forests inspired by GENIE3, and (b) integrating this with expression neighborhood graphs in a weighted manner. While the graph neural network framework is referenced, the implementation employs the LINE embedding, with optimization of both first- and second-order objectives. Prospective directions include exploration of alternative graph neural network architectures and incorporation of more granular gene-gene interaction information beyond random-forest-derived splits.

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For detailed methodology, benchmarks, and theoretical underpinnings, see "Enhanced Single-Cell RNA-seq Embedding through Gene Expression and Data-Driven Gene-Gene Interaction Integration" (Goudarzi et al., 1 Sep 2025).