On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model
This work addresses the need for better tools to understand sparse random graphs, particularly for applications in graph neural networks, by providing a foundational framework that generalizes existing models.
The paper tackles the problem of analyzing sparse random graph models by introducing color convergence, a new notion of local convergence based on the Weisfeiler-Leman algorithm, and proposes the Refined Configuration Model (RCM), which is universal for locally tree-like random graphs and enables full characterization of their local limits.
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.