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

Alexander Pluska (Technische Universität Wien) · Sagar Malhotra (Technische Universität Wien)

characterizationcolor convergenceconfiguration modelerdős–rényigraph neural networks local convergencelocal limitslocally tree-likemessage-passingrandom treesrefined configuration modelsparse random graph modelsstochastic block modeluniversal propertiesweisfeiler–leman algorithm

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.