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arXiv · 2510.21392

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

Alexander Pluska·Sagar Malhotra
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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.

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Alexander Pluska, Sagar Malhotra. 2025-10-24. On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model. https://arxiv.org/abs/2510.21392

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