Title: Explore In-Context Message Passing Operator for Graph Neural Networks in A Mean Field Game
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph neural network

mean-field game

reaction diffusion model

partial differential equation
Abstract: In typical graph neural networks (GNNs), feature representation learning naturally evolves through iteratively updating node features and exchanging information based on graph topology. In this context, we conceptualize that the learning process in GNNs is a mean-field game (MFG), where each graph node is an agent, interacting with its topologically connected neighbors. 
However, current GNNs often employ the identical MFG strategy across different graph datasets, regardless of whether the graph exhibits homophilic or heterophilic characteristics.
To address this challenge, we propose to formulate the learning mechanism into a variational framework of the MFG inverse problem, introducing an in-context selective message passing paradigm for each agent, which promotes the best overall outcome for the graph. Specifically, we seek for the application-adaptive transportation function (controlling information exchange throughout the graph) and reaction function (controlling feature representation learning on each agent), \textit{on the fly}, which allows us to uncover the most suitable selective mechanism of message passing by solving an MFG variational problem through the lens of Hamiltonian flows.
Taken together, our variational framework unifies existing GNN models into various mean-field games with distinct equilibrium states, each characterized by the learned in-context message passing operators. Furthermore, we present an agnostic end-to-end deep model, coined \textit{Game-of-GNN}, to jointly identify the message passing mechanism and fine-tune the GNN hyper-parameters on top of the elucidated message passing operators. \textit{Game-of-GNN} has achieved SOTA performance on diverse graph data, including popular benchmark datasets and human connectomes. More importantly, the mathematical insight of MFG framework provides a new window to understand the foundational principles of graph learning as an interactive dynamical system, which allows us to reshape the idea of designing next-generation GNN models.


Peer reviews:
comments: SUMMARY:
The paper introduced a variational framework of mean field game inverse problem that gives rise to a new learning scheme for GNN, called Game-of-GNN. It learns an in-context message passing operators unique to the mean-field control pattern associated with a given graph. Based on this proposed theoretical framework, the paper proposed an end-to-end algorithm based on Hamiltonian flows in solving the inverse problem.

STRENGTHS AND WEAKNESSES:
Strength

1. The formulation of GNN dynamic learning process as a mean field game is novel and interesting

Weakness

1. The notations can be introduced more clearly to avoid misunderstanding.

2. The paper is math heavy and many concepts are not introduced but are taken for granted. This makes the paper hard to follow.

3. Transitioning from theoretical framework to algorithm, the assumptions underlying are not clearly stated and discussed

4. The implementation of the algorithm is not very clearly presented. In addition the Algorithm 10 (line 287) is missing in the main paper.

QUESTIONS:
1. The acronym for Reaction Diffusion Model (RDM) is used in section 2.1.2 without spelling out the full name.

2. Could the authors provide more explanations on how the Lyapunov functional is defined, specifically the integral over $\mathcal{G}$ with respect to $v\in\mathcal{V}$, which is a finite set of nodes? Or is this a notational error?

3. According to the algorithm in the appendix A.6, the learning of mobility functions and the phase are for each individual nodes. However there should be node-to-node information exchange during the training? Are there any missing logics from theory to algorithm, or is this an incorrect understanding?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 1
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors reformulate the inter - layer information propagation in graph neural networks (GNNs) as a mean - field game (MFG) problem for multi - agents, and transform the question of \"what kind of information exchange/update mechanism should be adopted on a specific graph\" into a variational solution of the MFG inverse problem.\n\nSpecifically, the paper regards node features as \"potential energy\" and defines:\n- The transport function Θ₁ controls the information flow between neighbors.\n- The reaction function Θ₂ controls the self - update of nodes.\n\nThese two functions jointly determine the Hamilton flow in the form of a partial differential equation (PDE).\n\nThe paper further presents the end - to - end model Game - of - GNN: use ICNN to predict (Θ₁, Θ₂), then use a PDE - based solver to evolve node representations, and perform downstream tasks at the final state. The experiments cover 9 small - to - medium - scale datasets, 2 large graphs from OGB, 7 TU graph classification datasets, and 4 groups of human brain connectomes, achieving state - of - the - art (SOTA) or comparable performance. Meanwhile, an interpretability analysis is provided (the negative correlation between Θ₁ - α and homogeneity h).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths：\n- Unifying GNNs within the mean - field game framework can theoretically cover existing diffusion/reaction - diffusion - style models. \n\n\nWeaknesses：\n- **High computational cost**: Solving with ICNN + ODE/PDE is significantly slower than conventional models, resulting in high deployment costs. The paper only briefly mentions this but lacks systematic profiling.\n- **Hyperparameter sensitivity**: Θ is generated by ICNN. The depth and regularization have a significant impact on stability, yet the paper does not provide sufficient details on hyperparameter tuning. \n- **Strong theoretical assumptions**: It is necessary to assume that Θ₁ and Θ₂ are positive and smooth, Ψ > 0, etc. Whether these assumptions hold for real - world graphs (with discrete features and isolated nodes) is not discussed.\n\nQUESTIONS:\n1. **Convergence Guarantee**: During the training process, does the Hamilton energy remain conserved for Θ₁ and Θ₂ generated by ICNN? Can you provide a numerical stability analysis or details of the soft - constraint implementation?\n2. **Computational Complexity**: Please provide the training/inference time - memory curves under different graph scales, and discuss the comparison of the search costs with GraphNAS and AutoGNN. \n3. **Dynamic Graph/Time - Varying Attributes**: The current derivation is based on static graphs. If the topology changes over time, do Θ₁ and Θ₂ need to explicitly model the dependence on t?\n4. **ICNN Depth Ablation**: The paper mentions that heterogeneous graphs require deeper ICNN. Can you present systematic experiments on α - depth - performance to guide practical choices?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel, physics-informed framework that models GNN as a mean-field game (MFG), allowing each node to adapt its message passing strategy according to the graph structure. By formulating the GNN learning process as a variational MFG inverse problem and developing the Game-of-GNN deep model, the proposed model achieves state-of-the-art performance across diverse graph datasets.\n\nSTRENGTHS AND WEAKNESSES:\nQuality: good\nClarity: medium\nSignificance: good\nOriginality: good\n\nQUESTIONS:\n1. Could you please explain how does your proposed model address heterophily problem more intuitively?\n\n2. Can you provide a deeper interpretability of the role of $\\theta_2$ for learning on a heterophilic graph? How do $\\theta_1, \\theta_2$ relate to or differ from existing operators?\n\n3. More experiments and comparisons on the challenging heterophilic datasets and large-scale datasets as listed in [1].\n\n4. \"the $\\alpha \\sim h$ relationship across nine graph dataset indicates a notable anti-correlation.\" It seems that the information flow and homophily level shouldn't be linearly correlated, i.e. the most information-constrained area should be at the middle homophily area. Does this mean that your model miss something (probably the reason of the underperformance on Chameleon and Squirrel compared to some SOTA models [3,4], but I won't blame it.)? Or maybe you need to add more components into your analysis and modeling, e.g. the synergy among features, graph structures and labels [5].\n\n5. In Figure 2, Chameleon and Texas has similar level of homophily but significantly different $\\alpha$; in addition, Chameleon and Pubmed has similar level of $\\alpha$ but significantly different $h$, what is the reason behind it?\n\n\n\n[1] The heterophilic graph learning handbook: Benchmarks, models, theoretical analysis, applications and challenges. arXiv preprint arXiv:2407.09618. 2024 Jul 12.\n\n[2] When do graph neural networks help with node classification? investigating the homophily principle on node distinguishability. Advances in Neural Information Processing Systems. 2024 Feb 13;36.\n\n[3] Simplifying approach to node classification in graph neural networks. Journal of Computational Science, 62, 101695.\n\n[4] Finding global homophily in graph neural networks when meeting heterophily. In International Conference on Machine Learning (pp. 13242-13256). PMLR.\n\n[5] What is missing for graph homophily? disentangling graph homophily for graph neural networks. In The Thirty-eighth Annual Conference on Neural Information Processing Systems.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper integrates the notion of Graph Neural Networks into the idea of Mean Field Games. This integration makes sense in many aspects, and indeed, the paper provides several examples where this integration has improved performance. What the paper lacks is a theoretical foundation for why this integration works. I would very much like to see a theorem showing that, under some well-defined conditions, the integration of Graph Neural Networks and Mean Field Game improves performance, or at least does not harm it, under some well-defined notion of performance assessment. If this is impossible, then I would at least like to see some heuristic discussion about the contexts in which this method is likely to succeed.

STRENGTHS AND WEAKNESSES:
.

QUESTIONS:
.

LIMITATIONS:
.
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-16

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper offers and interesting perspective on GNNs being instances of Mean Field Games (MFG) and the authors demonstrated that several continuous graph neural networks models (GRAND, GraphBel, etc) are examples that fit the MFG framework. Building on this perspective, the paper proposes Game-of-GNN, a framework that re-formulates GNNs as MFGs.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n- **Theoretical Depth**: This paper displays a solid theoretical depth that bridges mean-field game theory with graph neural networks, offering a physics-inspired framework for message passing.\n\n- **Adaptive Message Passing**: To the best of my understanding (the authors may help correct me if I am wrong), the authors propose that we can dynamically regulate message passing based on graph topology. In their experiment in Figure 2, the operator $\\\\Theta_1$ is inversely proportional to the homophily ratio, which discourages information exchange for heterophilic graphs and encourages exchange for homophilic graphs. This is a practical advantage to their framework which also adds some degree of interpretability.\n\n---\n\n**Weaknesses**:\n\n- **Efficiency**: From table 11, it seems like there can be a reasonable concern regarding computational resources. Game-of-GNN runs at 33ms while other dynamical GNNs architectures like GRAND, GraphBel and ACMP runs at 10ms, 24ms and 27ms, respectively. I would imagine that the discrepancy will be dilated even more on large node datasets like OGBN-ARXIV.\n\n- **Accessibility**: I am personally not familiar with the work itself as well as the related works. Though I can understand the idea of the paper at a high level. I would imagine that a typical GNN audience will have a hard time understanding all the materials.\n\nQUESTIONS:\n**Questions**: Building upon the concern about computational efficiency. Did the authors conduct some experiments to show how the difference in computing time of Game-of-GNN scales compared to other architectures for increasingly larger node datasets?\n\nLIMITATIONS:\nLimitations: Please refer to the “Weaknesses” Section.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-23"}