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

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.

Authors

• Alexander Pluska
• Sagar Malhotra

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