SpaPool: Adaptive Graph Pooling

• SpaPool is a graph neural network pooling method characterized by adaptive clustering that preserves both local and global connectivity.
• It employs a TopKPool-based selection and cosine similarity assignment to efficiently reduce graph size while maintaining essential structure.
• Empirical evaluations demonstrate SpaPool’s superior performance on small-scale graphs, making it ideal for applications in bioinformatics, social networks, and related fields.

SpaPool is a graph neural network (GNN) pooling method characterized by a “dense yet adaptive” assignment approach that combines the structural retention of dense pooling techniques with the flexibility of sparse strategies. Specifically, SpaPool groups nodes into an adaptive number of clusters, reducing graph size efficiently while conserving both local and global connectivity. The method is formulated to address the limitations of fixed-cluster pooling (as in DiffPool) and the node-dropping behavior of sparse methods (e.g., TopKPool). Empirical studies highlight SpaPool’s performance advantages, especially on small-scale graphs, and its suitability for a wide array of graph-centric applications.

1. Motivation and Design Principles

SpaPool was developed in response to bifurcated trends in GNN pooling: dense methods forcibly aggregate nodes into a preset number of clusters, while sparse approaches remove subsets of nodes based on learned scores. Dense methods guarantee preservation of graph structure, yet require manual preselection of cluster count and lack adaptivity. Sparse methods are data-driven, more flexible with graph size, but are susceptible to loss of important connectivity through node elimination.

SpaPool addresses these trade-offs by:

• Facilitating node grouping according to learned importance, thus preserving essential structural elements.
• Enabling adaptive determination of cluster number via integration of TopKPool-based scoring.
• Retaining both local and global graph features through cluster assignment based on cosine similarity. This synthesis provides efficient dimensionality reduction without sacrificing structural information, particularly pertinent for graphs where node-level detail is critical.

2. Methodological Framework

The architecture of SpaPool consists of a repeated three-step pooling block at each layer $\ell$:

• Select: Given an attributed graph $(A^{(\ell)}, H^{(\ell)})$, a GCN layer produces $H_{emb}^{(\ell)}$. TopKPool selects a representative subset:

$$(H_{rep}^{(\ell)}, H_{scores}^{(\ell)}) = \text{TopKPool}(H_{emb}^{(\ell)}, k)$$

Centroids are computed:

$$H_{centroids}^{(\ell)} = H_{rep}^{(\ell)} \cdot H_{scores}^{(\ell)}$$

Assignment matrix $S$ is derived by softmax over cosine similarities:

$$S = \text{Softmax}\left(\left(\frac{H_{emb}^{(\ell)}}{\|H_{emb}^{(\ell)}\|_2}\right)\left(\frac{H_{centroids}^{(\ell)}}{\|H_{centroids}^{(\ell)}\|_2}\right)^T\right)$$

• Reduce: Node features are aggregated:

$H^{(\ell+1)} = S^T H^{(\ell)}$

• Connect: New adjacency matrix is constructed:

$A^{(\ell+1)} = S^T A^{(\ell)} S$

Auxiliary losses are employed to promote meaningful clustering:

• Link prediction loss: $\mathcal{L}_{LP} = ||A^{(\ell)} − S S^T||_F$
• Entropy regularization: $\mathcal{L}_E = \frac{1}{N} \sum H(S_i)$ Both facilitate structure-aware, well-conditioned assignment matrices.

3. Adaptive Clustering Mechanism

Unlike traditional dense clustering, SpaPool’s use of TopKPool in the selection phase imparts adaptivity to the number of clusters. Representative nodes identified via learned scores serve as centroids for subsequent assignment, dynamically reflecting both graph size and feature distribution. The assignment matrix, computed via cosine similarity and normalized softmax, ensures that supernodes exhibit meaningful correspondence to input graph structure, preserving neighborhood and global topology.

A plausible implication is that SpaPool’s assignment matrix $S$ can reflect transient structural features, making it resilient to distributional shifts in graph statistics across datasets.

4. Empirical Evaluation

Experiments were conducted on diverse graph classification benchmarks, including PROTEINS, ENZYMES, DD, Mutagenicity, github_stargazers, reddit_threads, OHSU, twitch_egos, COLLAB, and IMDB-BINARY. The experimental protocol maintained architectural consistency—two GCN and two MLP blocks, only the pooling layers were varied—with hyperparameters set at $k = 0.5$ and a learning rate of $5 \times 10^{-4}$. Dataset splits followed an 80/10/10 regime for training, validation, and testing.

Key findings include:

• SpaPool attained competitive or superior results compared to TopKPool, ASAPool, and SAGPool, particularly on datasets with small graphs (average $\sim$30 nodes).
• Its grouping mechanism preserved key structural elements, whereas node-dropping methods could lose information through hard pruning.
• On larger and heterogeneous graphs, performance variability was minimal. These results suggest SpaPool’s utility is maximized in domains where node retention and structure are paramount.

5. Comparative Analysis

SpaPool’s clustering procedure was contrasted with alternative node selection (e.g., SAGPool-based) and aggregation strategies (scalar product, attention mechanisms). TopKPool-based selection consistently outperformed alternatives within SpaPool’s framework, and cosine similarity aggregation yielded improved result stability.

Table: Comparative features of pooling methods | Method | Cluster Adaptivity | Structure Retention | |-------------|-------------------|--------------------| | DiffPool | No | High | | TopKPool | Yes | Moderate | | SpaPool | Yes | High |

The findings indicate that SpaPool’s strategy for combining representative node selection with soft assignment is effective for both graph reduction and structure conservation.

6. Domains of Application

SpaPool is positioned for deployment in tasks requiring retention of graph structure across variable graph sizes:

• Bioinformatics and cheminformatics: Grouping nodes into clusters can preserve functional or chemical moieties within graphs representing molecules.
• Social network analysis: Community detection and analysis benefit from adaptive supernode formation, reflecting organic group boundaries.
• Recommender systems, computer vision (scene graphs), epidemiology: Heterogeneous and dynamic graphs are maintained via SpaPool’s adaptive pooling.

This suggests SpaPool’s utility on datasets typified by small graphs or heterogeneity is especially pronounced, affording improved information preservation.

7. Concluding Remarks and Future Directions

SpaPool represents an overview of dense and sparse pooling paradigms, enabling adaptive clustering with robust structural conservation. Its advantageous performance on small-scale graphs underscores its value where information density is high. Future research avenues include:

• Augmenting scalability for extremely large graphs and multi-attribute graphs.
• Integrating explainability to discern salient features and structural influences in pooling decisions.
• Further refinement of auxiliary loss terms to enhance assignment fidelity.

In summary, SpaPool provides a principled and empirically validated contribution to GNN pooling methodology, extending dense clustering approaches with adaptivity and achieving favorable results in graph-centric domains requiring efficient and effective structural processing.