Graph-dependent pooling: adaptive granularity in hierarchical graph neural networks

Graph pooling aims to compress the graph, including both node embeddings and their underlying topological patterns, into a more compact representation. Previous works focus primarily on the overly fine-grained representation of nodes, progressively coarsening the graph by removing nodes or merging them into clusters, thus neglecting the global-to-local patterns and adaptive granularity of the graph's topological structure. In the real scenario, graphs as a whole can be considered the coarsest level of granularity, encapsulating the global topological structure, with progressively finer-grained local topological structures represented from top to bottom. This process continues until the adaptive granularity for each subdomain is reached. To this end, we propose a novel Topology-Preserving Adaptive Graph Pooling (TPAGP) method that dynamically partitions graphs into granular balls by integrating node features and topological information, enabling the generation of multi-granularity representations that effectively capture both local and global structural patterns. Additionally, we design a multi-granularity graph network model that facilitates feature interaction and optimization across different granularities, significantly enhancing performance in graph classification tasks. Experimental results demonstrate that TPAGP outperforms existing pooling methods across various benchmark datasets, effectively mitigating information loss caused by fixed-granularity strategies.

Graph pooling is commonly applied in graph classification, yet its empirical gains over standard WL-1 expressive GNNs are often marginal or inconsistent. We study this gap by analysing the interaction between node features and graph topology and their effect on pooling objectives. Our analysis reveals that pooling operators require node features that are well-aligned with the graph's topology -- a condition often overlooked and not guaranteed in empirical networks. We formalise fundamental requirements for node features to enable effective pooling, and introduce a quantitative measure of feature quality. Our empirical evaluation shows that, when these requirements are satisfied, pooling can be beneficial and improve performance on appropriate datasets.

Sheaf Neural Networks (SNNs) generalize Graph Neural Networks (GNNs) by replacing scalar node signals with stalk-valued signals and by using restriction maps to measure compatibility across edges. Unlike standard graph diffusion, which encourages neighboring node features to become similar, sheaf diffusion promotes consistency through the restriction maps and can therefore model more general relationships between neighboring nodes. However, existing sheaf neural architectures mainly operate at a fixed graph resolution and do not provide a principled pooling mechanism for building hierarchical representations. In this paper, we introduce Hierarchical Sheaf Pool (HiSP), a sheaf-aware pooling framework based on local spectral coarsening. Given a partition of the graph, HiSP constructs each coarse stalk by projecting fine stalk-valued features onto the low-frequency eigenmodes of the cluster-internal sheaf Laplacian. These local modes define a cochain-level prolongation map, which allows the fine sheaf energy to be represented on the coarse space through a Galerkin operator. We further analyze the approximation induced by coarsening by separating truncation loss, due to discarded local modes, from realization loss, due to representing the projected operator as a coarse sheaf. Finally, we implement HiSP as a GNN pooling layer compatible with SNNs and provide a PyG implementation supporting batching, lifted sheaf Laplacians, and hierarchical architectures.