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

TL;DR

This paper introduces color convergence based on the Weisfeiler-Leman algorithm to analyze local limits of sparse random graphs, and proposes the Refined Configuration Model (RCM) as a universal framework for local convergence.

Contribution

It presents a new notion of local convergence called color convergence and introduces the RCM, a generalized model that captures local limits across various sparse random graph models.

Findings

Color convergence characterizes well-behaved graph classes for message-passing neural networks.

RCM is universal among locally tree-like random graph models.

Complete characterization of local limits of such graphs.

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 Erd\H{o}s-R\'enyi, stochastic block and configuration models. Finally, this framework enables a complete characterization of the random trees that arise as local limits of such graphs.