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On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model

Alexander Pluska, Sagar Malhotra

TL;DR

The paper introduces color convergence, a Weisfeiler-Leman-inspired local limit for sparse graphs, and proves it exactly characterizes probabilistic consistency of empirical risk for $k$-layer MPNNs. It then develops the Refined Configuration Model (RCM), a universal sparse graph model that captures color-convergent limits and local convergence to Galton–Watson trees, subsuming common models like Erdős–Rényi, SBM, and CM in the limit. The work formalizes a bridge between color refinement and learnability, providing a tractable generative framework with linear-time sampling under finite expected degree. Collectively, this framework yields a complete description of local limits in sparse graph regimes relevant to GNN learning and offers a principled basis for future extensions to higher-order WL methods and quantitative generalization bounds.

Abstract

Local convergence has emerged as a fundamental tool for analyzing sparse random graph models. We introduce a new notion of local convergence, color convergence, based on the Weisfeiler-Leman algorithm. Color convergence fully characterizes the class of random graphs that are well-behaved in the limit for message-passing graph neural networks. Building on this, we propose the Refined Configuration Model (RCM), a random graph model that generalizes the configuration model. The RCM is universal with respect to local convergence among locally tree-like random graph models, including Erdős-Rényi, stochastic block and configuration models. Finally, this framework enables a complete characterization of the random trees that arise as local limits of such graphs.

On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model

TL;DR

The paper introduces color convergence, a Weisfeiler-Leman-inspired local limit for sparse graphs, and proves it exactly characterizes probabilistic consistency of empirical risk for -layer MPNNs. It then develops the Refined Configuration Model (RCM), a universal sparse graph model that captures color-convergent limits and local convergence to Galton–Watson trees, subsuming common models like Erdős–Rényi, SBM, and CM in the limit. The work formalizes a bridge between color refinement and learnability, providing a tractable generative framework with linear-time sampling under finite expected degree. Collectively, this framework yields a complete description of local limits in sparse graph regimes relevant to GNN learning and offers a principled basis for future extensions to higher-order WL methods and quantitative generalization bounds.

Abstract

Local convergence has emerged as a fundamental tool for analyzing sparse random graph models. We introduce a new notion of local convergence, color convergence, based on the Weisfeiler-Leman algorithm. Color convergence fully characterizes the class of random graphs that are well-behaved in the limit for message-passing graph neural networks. Building on this, we propose the Refined Configuration Model (RCM), a random graph model that generalizes the configuration model. The RCM is universal with respect to local convergence among locally tree-like random graph models, including Erdős-Rényi, stochastic block and configuration models. Finally, this framework enables a complete characterization of the random trees that arise as local limits of such graphs.
Paper Structure (18 sections, 31 theorems, 65 equations, 3 figures)

This paper contains 18 sections, 31 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Random Graphs
  4. Galton-Watson Trees
  5. Local Convergence
  6. Message Passing Graph Neural Networks
  7. Color Refinement
  8. Learning and Data Generation Model
  9. Color Convergence
  10. Color Convergence and Generalization Gap in MPNNs
  11. Properties of Color Convergent Random Graphs
  12. Refined Configuration Model
  13. Discussion, Limitations and Broader Impact
  14. Acknowledgements
  15. Color Convergence Proofs
  16. ...and 3 more sections

Key Result

Theorem 2.7

Let $G_t = \mathsf{CM}_t(\mu)$. Then $G_t$ converges locally to a GWT. That is, there exists a GWT $W_t$ such that for all $k\in\mathbb N$ and $B\in\mathcal{B}_k$$b_{k, \infty}(B) = \mathbb{P}(W_k \simeq B).$

Figures (3)

  • Figure 1: Support and distribution of $W_1$.
  • Figure 2: Example demonstrating \ref{['prop:mp']}.
  • Figure 3: From left to right, we have: a table containing the assigned types, stubs occurring in steps 1, 2, and 3, and possible outputs $G_0$ and $G_1$ of the $\mathsf{RCM}$ algorithm.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Theorem 2.7: van2024random
  • Remark 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 64 more