Title: On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model
Venue: NeurIPS 2025
Accepted: yes
Keywords: Random Graphs

Sparse Random Graphs

Graph Neural Networks
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 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.


Peer reviews:
comments: SUMMARY:
The paper proposes color convergence, a new graph-limit notion aligned with 1-WL color refinement, and shows it is equivalent to uniform generalization of every $k$-layer MPNN on node-classification tasks. To generate all such limits, the authors introduce the Refined Configuration Model, a type-augmented configuration model that subsumes E-R, SBM, and classical configuration graphs while remaining locally tree-like.

STRENGTHS AND WEAKNESSES:
Strengths:
1. First limit concept tied to MPNNs with bridge between graph limits and GNN learning theory.
2. Proofs of equivalence, universality, and involution-invariance look reasonable, although I did not check the proofs carefully.

Weaknesses:
1. Applications of this work are unclear to me.
2. The contribution is purely theoretical; no empirical or synthetic experiments are provided.

QUESTIONS:
Apart from the comments in the weakness section, I have the following questions:
1. The framework focuses on node classification. Could the framework be extended to provide insights into other graph learning tasks?
2. How do the authors expect one would fit the parameters (e.g., the PMF $\mu$) from an observed real-world sparse graph? Are there consistent estimators, perhaps leveraging methods of moments on refinement-color counts, that could be used in practice?
3. Why only 2 colors in this work? Could the authors extend their framework to multiple colors?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

comments: SUMMARY:
This paper studies the local limits of sparse random graphs, proposes a notion of local convergence, and also a random graph model. The proposed color convergence generalizes the local convergence and draws a strong relation to the learnability of Message Passing Graph Neural Networks (MPNN). Color convergence is defined based on the classic color refinement algorithm by Weisfeiler and Leman. The proposed Refined Configuration Model (RCM) generalizes the configuration model and it is universal with respect to coloar convergence and local convergence to Galton-Watson trees.

STRENGTHS AND WEAKNESSES:
1. This paper is well-organized and easy to follow. Key concepts are defined and delivered clearly. There are also quite a few figures that enhance the understandability.
2. The proposed color convergence notion exactly capture the learnability of MPNN in the sense that there is a necessary and sufficient condition on whether a random graph satisfies probabilistic consistency of empirical risk with respect to $k$-layer MPNNs. Although this is not a very strong consistency guarantee, it opens up future directions on deriving stronger guarantees such as specific error bounds.
3. RCM generalizes the commonly studied configuration model. It has potential to be studied in many follow-up works that concern with MPNNs or need a more general random graph with a nice local limit structure.

QUESTIONS:
I am curious about is there any arguments about weak local limits we can make using the proposed framework? By weak local limit, I am referring to the Definition 2.1 in [1] for example.


[1] Berger, Noam, et al. "Asymptotic behavior and distributional limits of preferential attachment graphs." The Annals of Probability (2014): 1-40.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nLocal convergence characterizes the neighbourhood distribution around a randomly chosen node in the large‑graph limit, providing several structural details. This is exploited by the Weisfeiler–Leman (WL) color‑refinement algorithm that bounds the expressive power of Message Passing Graph Neural Networks (MPNNs). Using this observation, they propose color convergence, a refinement that facilitates the analysis of MPNNs through the framework of local convergence. \n\nThey prove that a random graph sequence is color-convergent if and only if it satisfies probabilistic consistency of empirical risk with respect to k-layer MPNNs. They then designed the Refined Configuration Model (RCM) that is universal with respect to color convergence. It is a type-labelled extension of the classical configuration model. This framework helps them develop structural results for locally tree-like graphs. Through this framework they show that colour-convergent sequences correspond precisely to local limits given by Galton–Watson trees.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- Bridging WL-based expressivity analysis of GNNs with graph-limit theory is new and interesting. The idea of using Color convergence for this is a good balance. It is weaker than local convergence but captures learnability by MPNNs.\n- The writing is clean. Related work is covered well and defintions are clearly stated with examples. \n\n\nWeakness:\n- They do not give intutions for their proofs.\n- No formal guarantees regarding convergence rates or error bounds. \n- Given that colour convergence is a new framework they are proposing aimed at explaining MPNN behaviour, even a small controlled simulation would greatly strengthen the paper.\n\nQUESTIONS:\nTypo: not is written twice in Remark 2.9\n\nA line before Corollary 3.14 explaining it would be nice. Took me a while to see.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a notion of \"local convergence\" that they call \"color convergence\" and also proposes what the authors call the \"refined configuration model\", which is a variant of the configuration model used in random graphs. This can be motivated as follows to an ML audience: one is often interested in understanding if a local aggregation algorithm, such as a graph neural network based on message passing, ends up \"performing well\" on a graph drawn from a probabilistic model. Here, one can ask if there is a general characterization under which the \"recovery\" up to an \\epsilon error in the loss objective occurs with high probability. For procedures that fall under the umbrella of the Weisfeiler-Leman algorithm (including GNNs), the authors show that \"color convergence\" provides such a characterization.\n\nColor convergence is a local-convergence style property that is based on the distribution of trees obtained via a color refinement algorithm. The main result is then that if a random graph family is color convergent, then with high probability, an MPNN procedure on a graph drawn from the family achieves a small generalization gap (difference between the true risk and the empirical risk).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The problem of achieving small generalization gap via GNN type procedures on classes of random graphs is an important one, as it gives insights into when one can expect these methods to work well on real graphs.\n- The results obtained are quite interesting, assuming they can be motivated appropriately to an ML audience.\n- The variant of the configuration model seems interesting as well.\n\nWeaknesses:\n\n- The biggest concerns about the paper are about the writing and how it does not make the case for \"fit\" for an ML conference like NeurIPS. Right now, the paper reads like one that is of interest to the graph limits community, but perhaps not much beyond.\n\n- Even though the paper gives a few concrete examples of random graph models, color refinement, etc., the writing is far from clear. Suppose we focus on applications of Theorem 3.6, are there specific graph classes on which GNN approaches were not known to be consistent but the new result sheds light? (I imagine there are, but it would be good to see examples.)\n\n- The authors should also say if there are any convergence guarantees. The literature on recovery in SBMs, for example, gives concrete bounds on the recovery error (~ \\sqrt{t} misclassifications, for example). Are such results possible in the general case?\n\nOverall, I feel the paper needs a much more accessible exposition for a NeurIPS audience. But this said, the results do seem interesting if presented in the right way.\n\nQUESTIONS:\nPlease comment on each of the points in the weaknesses section. Also, could you discuss more on the significance of some of the results like 3.13 and the definitions around it?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-08"}