On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917039779610624 |
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| author | Pluska, Alexander Malhotra, Sagar |
| author_facet | Pluska, Alexander Malhotra, Sagar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21392 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model Pluska, Alexander Malhotra, Sagar Discrete Mathematics Machine Learning 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. |
| title | On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model |
| topic | Discrete Mathematics Machine Learning |
| url | https://arxiv.org/abs/2510.21392 |