Invariant-Stratified Propagation for Expressive Graph Neural Networks

TL;DR

Invariant-Stratified Propagation (ISP) is introduced, a framework comprising both a novel WL variant (ISP-WL) and its efficient neural network implementation (ISPGNN), which stratifies nodes according to graph invariants, processing them in hierarchical strata that reveal structural distinctions invisible to 1-WL.

Abstract

Graph Neural Networks (GNNs) face fundamental limitations in expressivity and capturing structural heterogeneity. Standard message-passing architectures are constrained by the 1-dimensional Weisfeiler-Leman (1-WL) test, unable to distinguish graphs beyond degree sequences, and aggregate information uniformly from neighbors, failing to capture how nodes occupy different structural positions within higher-order patterns. While methods exist to achieve higher expressivity, they incur prohibitive computational costs and lack unified frameworks for flexibly encoding diverse structural properties. To address these limitations, we introduce Invariant-Stratified Propagation (ISP), a framework comprising both a novel WL variant (ISP-WL) and its efficient neural network implementation (ISPGNN). ISP stratifies nodes according to graph invariants, processing them in hierarchical strata that reveal structural distinctions invisible to 1-WL. Through hierarchical structural heterogeneity encoding, ISP quantifies differences in nodes' structural positions within higher-order patterns, distinguishing interactions where participants occupy different roles from those with uniform participation. We provide formal theoretical analysis establishing enhanced expressivity beyond 1-WL, convergence guarantees, and inherent resistance to oversmoothing. Extensive experiments across graph classification, node classification, and influence estimation demonstrate consistent improvements over standard architectures and state-of-the-art expressive baselines.

Figures (19)

• Figure 1: ISP-GNN architecture overview. (a) Input graph. (b) ISP coloring based on invariant values. (c) Layered propagation with WL and ISP aggregation at each layer. Triangle detail shows gap encoding for nodes with higher-layer neighbors. (d) Output embeddings for downstream tasks.
• Figure 2: Mechanisms by which ISP-WL distinguishes graphs that 1-WL cannot: (A) Triangle Aggregation:$G_1$ and $G_2$ have identical degree sequences but different triangle structures. (B) Invariant Stratification: Triangle-free graphs $H_1$ and $H_2$ differ in structural properties captured by invariants (e.g., local girth). (C) Hierarchical Gap Encoding:$I_1$ and $I_2$ have identical triangle patterns but different invariant hierarchies (e.g., betweenness centrality).
• Figure 4: Runtime and performance on ogbg-molpcba dataset. (a) Per epoch time in seconds. (b) Average precision score. Results show ISP-GNN with different structural invariants.
• Figure 5: Gap encoding components capture complementary structural properties using degree as invariant. (a) $\delta_1$ differentiates hub vs. peripheral roles. (b) $\delta_2$ captures structural proximity to the closest neighbour. (c) $\delta_3$ measures neighbor similarity.
• Figure 7: Structural metrics comparison across datasets.

Theorems & Definitions (23)

• Definition 3.1: Hierarchical Structural Heterogeneity Encoding
• Proposition 3.1: Permutation Invariance
• Proposition 3.1: Single Assignment Property
• Definition 3.2: Learnable Invariant
• Proposition 3.2: Structural Consistency
• Theorem 4.1: Enhanced Expressivity Beyond 1-WL
• Theorem 4.2: Node Distinguishability
• Theorem 4.3: Invariant-Dependent Expressivity
• Theorem 4.4: Convergence
• Theorem 4.5: Time Complexity