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Unifying approach to uniform expressivity of graph neural networks

Huan Luo, Jonni Virtema

TL;DR

The paper introduces Template GNNs (T-GNNs), a unifying framework that updates node features by aggregating over embeddings of graph templates, thereby incorporating subgraph information beyond immediate neighbours. It develops a graded template modal logic $GML(\mathcal{T})$ and a corresponding $\mathcal{T}$-WL/bisimulation to characterize the expressive power of these networks, proving an equivalence between $T$-GNNs and $GML(\mathcal{T})$ in the bounded counting setting. A central metatheorem shows that, for any finite template set $\mathcal{T}$, the uniform expressivity of a $c$-bounded $\mathcal{T}$-GNN with $l$ layers exactly matches the expressive power of $GML(\mathcal{T})$ (and vice versa), unifying many prior results and subsuming AC-GNNs and $k$-hop subgraph GNNs as instantiations. The work provides a principled, logic-guided framework for designing and analyzing highly expressive GNNs, clarifying the limits imposed by counting bounds and offering a roadmap for future extensions to non-bounded settings and recursive architectures.

Abstract

The expressive power of Graph Neural Networks (GNNs) is often analysed via correspondence to the Weisfeiler-Leman (WL) algorithm and fragments of first-order logic. Standard GNNs are limited to performing aggregation over immediate neighbourhoods or over global read-outs. To increase their expressivity, recent attempts have been made to incorporate substructural information (e.g. cycle counts and subgraph properties). In this paper, we formalize this architectural trend by introducing Template GNNs (T-GNNs), a generalized framework where node features are updated by aggregating over valid template embeddings from a specified set of graph templates. We propose a corresponding logic, Graded template modal logic (GML(T)), and generalized notions of template-based bisimulation and WL algorithm. We establish an equivalence between the expressive power of T-GNNs and GML(T), and provide a unifying approach for analysing GNN expressivity: we show how standard AC-GNNs and its recent variants can be interpreted as instantiations of T-GNNs.

Unifying approach to uniform expressivity of graph neural networks

TL;DR

The paper introduces Template GNNs (T-GNNs), a unifying framework that updates node features by aggregating over embeddings of graph templates, thereby incorporating subgraph information beyond immediate neighbours. It develops a graded template modal logic and a corresponding -WL/bisimulation to characterize the expressive power of these networks, proving an equivalence between -GNNs and in the bounded counting setting. A central metatheorem shows that, for any finite template set , the uniform expressivity of a -bounded -GNN with layers exactly matches the expressive power of (and vice versa), unifying many prior results and subsuming AC-GNNs and -hop subgraph GNNs as instantiations. The work provides a principled, logic-guided framework for designing and analyzing highly expressive GNNs, clarifying the limits imposed by counting bounds and offering a roadmap for future extensions to non-bounded settings and recursive architectures.

Abstract

The expressive power of Graph Neural Networks (GNNs) is often analysed via correspondence to the Weisfeiler-Leman (WL) algorithm and fragments of first-order logic. Standard GNNs are limited to performing aggregation over immediate neighbourhoods or over global read-outs. To increase their expressivity, recent attempts have been made to incorporate substructural information (e.g. cycle counts and subgraph properties). In this paper, we formalize this architectural trend by introducing Template GNNs (T-GNNs), a generalized framework where node features are updated by aggregating over valid template embeddings from a specified set of graph templates. We propose a corresponding logic, Graded template modal logic (GML(T)), and generalized notions of template-based bisimulation and WL algorithm. We establish an equivalence between the expressive power of T-GNNs and GML(T), and provide a unifying approach for analysing GNN expressivity: we show how standard AC-GNNs and its recent variants can be interpreted as instantiations of T-GNNs.
Paper Structure (8 sections, 9 theorems, 23 equations, 2 figures)

This paper contains 8 sections, 9 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Template GNNs
  4. Weisfeiler-Leman Test and Bisimulation
  5. Graded Modal Logic and Template GNNs
  6. Graded Template Modal Logic
  7. From GNNs to logic and back
  8. Conclusions and Future Work

Key Result

Proposition 10

Let $G = (V, E, \lambda)$ and $G' = (V', E', \lambda')$ be labelled graphs, $\mathcal{T}$ a finite set of templates, and $l \in \mathbb{N}$. Then $(G',v') \sim^{l}_{\mathcal{T}} (G,v)$ if and only if $\mathrm{col}^l(v) = \mathrm{col}^l(v')$.

Figures (2)

  • Figure 1: Templates $T_1$ and $T_2$ used to capture AC-GNNs and AC+-GNNs. The solid arrow represents an element of $E^+$, while the dashed arrow represents an element of $E^-$.
  • Figure 2: Top: $2$-hop subgraph WL extracts subgraphs rooted at node $v$. Bottom: $\mathcal{T}$-WL arrives at the same colouring by matching node $v$ against $\mathcal{T} = \{T_{\vartriangle}, T_p\}$.

Theorems & Definitions (27)

  • Definition 1: Template
  • Definition 2: Template embedding
  • Definition 3: Template isomorphism
  • Definition 4: Template aggregation function
  • Definition 5: Unary Template GNN
  • Definition 6: n-ary Template GNN
  • Definition 7: $\mathcal{T}$-WL algorithm
  • Example 8
  • Definition 9: Graded $\mathcal{T}$-bisimulation
  • Proposition 10
  • ...and 17 more