Explore In-Context Message Passing Operator for Graph Neural Networks in A Mean Field Game
Authors: Tingting Dan
Xinwei Huang
Won Hwa Kim
Guorong Wu
Venue: NeurIPS 2025 poster ()
Keywords: Graph neural network
mean-field game
reaction diffusion model
partial differential equation
Primary area: general_machine_learning

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.

Explore In-Context Message Passing Operator for Graph Neural Networks in A Mean Field Game
Tingting Dan
Xinwei Huang
Department of Psychiatry University of North Carolina Chapel Hill
Shanghai Fourth People's Hospital School of Medicine, Tongji University Shanghai, 200434, China
Chapel Hill, NC 27599, USA Tingting Dan@med enc.edu
muanggeneticcs@tonggii edu. cn
Won Hwa Kim
Guorong Wu
Computer Science and Engineering POSTECH Pohang, Korea 37673, South Korea wonhwa@postech.. ac kr
Departments of Psychiatry and Computer Science University North Carolina at Chapel Hill Chapel Hill, NC 27599, USA grwudmed. inc.edu
Abstract
In typical graph neural networks (GNNs), feature representation learning naturally evolves through iteratively updating node features and exchanging information based graph topology. In this context, conceptualize that the learning process in GNNs 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 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 inliicattinnadaaptii transportation function (controlling information exchange throughout the graph) and reaction function (controlling feature representation learning on each agent), 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 Game-of-Gii identify the message passing mechanism and fine-tune the GNN hyper-parameters on top the elucidated message passing operators. Game-of of -aeee-f-GGN- GNN has achieved SOTA performance diverse graph data, including popular benchmark datasets and human connectomes. More importantly, the mathematical insight of MFG framework provides graph learning as an interactive dynamical system, which allows us to reshape the idea of designing next-generation GNN models.
to jointly
window understand the foundational principles of
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Introduction
In a world of complex systems, graphs provide powerful way to model relationships between objects [41]. Graph neural networks (GNNs) leverage graph structures to achieve success in diverse fields, such as social network analysis [13], biochemical engineering [15]. and drug repurposing [IT] Despite their variations, most GNNs share key components: (1) feature representation for nodes and (2) message passing to propagate features across the graph [38]. Relational inductive biases, like permutation invariance, are also critical for aligning GNN designs with graph topology [5]
However, real-world graphs pose several challenges. The first lies in the heterogeneous relationship between graph topology and node labels. On heterophilic graphs, where connected nodes often differ in features, even simple multilayer perceptrons (MLPs) can outperform GNNs 45 [2] To address
measures Another key challenge is over-smoothing, where excessive aggregation blurs node distinctions. Solutions include diffusion- -based models such as GRAND 7 and PDE-inspired approaches [36] 32). Architectural innovations like residual networks [20] and Transformer backbones (16) have also been introduced to capture global dependencies and alleviate over-smoothing. Despite these advances, most GNN designs still rely heavily on domain-specific heuristics and backpropagation for tuning deep architectures, which remain fixed across all graph types. As illustrated by the gear-matching analogy in Fig. [ such aand-crffted designs guided by domain expertise in machine learning and graph signal processing (27] [14], lack a system-level understanding of the learning mechanism. Consequently, achieving state-of-the-at accuracy does not guarantee that the model's inference principle is well aligned with the underlying graph structure. For instance, a particular GNN may perform well on homophilic graphs but poorly on heterophilic ones, or vice versa. To overcome these limitations, it is crucial to develop an explainable and principled framework for GNNs that provides mathematical guarantees and adaptability across diverse graph domains.
this, researchers have explored adaptive message passing (24) and heterophily-awee
44
GNN is a mean-field game. To establish principled learning framework for de- signing novel GNN archi- tectures that generalize to unseen graphs, we concep- tualize the learning process in GNNs as mean-field game (MFG) involving a large number of interacting agents (i.e., graph nodes). In this view, each node acts individual player that optimizes its own fea- ture representation while re- sponding to the aggregated behavior of its neighbors, the so-called "mean field." Over successive message- passing iterations, all nodes collectively adjust their rep- resentations until reaching stable equilibrium, analo- gous to players in large- scale game who iteratively best-respond to the average strategy of the population. A central element of the MFG formulation the temporal evolution of each agent's 's state. By interpreting the feature vector of each node as a form of potential energy, we further hypothesize that the underlying dynamics of GNNs follow the second law of thermoulynamics (26). where the system evolves toward configurations that dissipate the negative entropy functional most efficiently. This perspective is supported by recent studies that interpret representation learning on graphs as a dynamic process of heat diffusion [7] providing a unified physical and mathematical foundation for understanding GNN behavior.
GNN Model #1
GNN Model #2
MFC framework for GNNe
craft
GNN
New GNN
Mochaniams GNN models
Gradient flow
Hamiltanian Micchanics
Mean- Field
Machine Learning
RDM MoM--based gredient flow
Learn control
pattems
Physics
Mear- Fiel--
Graph data #1 Graph data #2 Graph data #3
principle
Control
Figure 1: The motivation of our work. GNN instances and graph data intended for application are visually represented by gears of two different colors (green for models and yellow for data). The tooth pattern of data gears reflects the topological characteristics of graph data, such as whether it exhibits homophilic or heterophilic behavior. Left: Current approaches use the same GNN gear to match all data gears, relying the "black-box" of gradient descent to fine-tune the underlying green gear (GNN model) to fit the yellow gears (data). This often lcads to sub-optimal learning outcomes. Right: Our Game- GNN formulates graph learning as mean-field game in the variational framework. By formulating the graph feature representation learning dynamical process, capitalize on the equilibrium state (critical point of the mean-field control (MFC) problem) to form a Hamiltonian flow which allows us to link the "on-the-fly" design of GNN mechanism (shown in the blue box) and the discrete GNN instance for the real-world graph data. The showcase of our Kamee-ofGGNy approach is the craft of suitable GNN mechanism (indicated by red arrows) for unseen graph data #3.

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Reverse engineering GNN as a MFG inverse problem. Following this notion, we propose to characterize the collective behaviors of simultaneous information exchange and update GNNs as a mean-field game, where the dynamics of MFG is shaped by the second law of thermodynamics. In this context, the most promising learning mechanism for the specific graph data characterized by an optimal transport from the initial feature representations to a Nash-equilibrium state (depending on the downstream learning tasks), where each graph node finds the best feature presentation for itself and the entire graph. Inspired by the mean-field theory [9], we further conceptualize that the dynamic learning process in GNNs forms a gradient flow that can be controlled by a set of learnable 'free energy functionals' derived from the underlying graph data. Thus, the search for the most appropriate functionals boils down to an MFG inverse problem. From the perspective of model explainability, the learned 'free energy functionals' act in-context message passing operators, which allows us to explore novel learning mechanisms for the regime of graph data learning.
Approach. Taken together, we present a variational framework of MFG inverse problem to achieve the physics-informed learning paradigm of designing novel GNN mechanisms, where the optimal in-context message passing operators are essentially the mean-field control patterns associated with underlying graph data. The detailed explanation of our approach is shown in Appendix A Specifically, the message passing operator consists of two functions of kinetic dynamics: (1) function of transportation mobility for controlling node-to-node information exchange and (2) a function of reaction mobility for message update each node. These two mobility functions determine the characteristic learning mechanism of information exchange and update within the specific GNN instance, by shaping the kinetic dynamics of the underlying 'free energy functional', Since different graph data forms a unique dynamical system with distinct in-context message passing operators, our proposed physics-informen integrates two hierarchical machine learning modules at both the mechanism and model instance levels. At the mechanism level, we seek the most suitable 'free-energy functionals' that shape the gradient flow in the MFG. Furthermore, we derive Hamiltonian flows as the governing equations of the underlying MFG problem. As multiple lines of work have demonstrated that GNN is equivalent to an underlying PDE 42 77 36), the Hamiltonian flow becomes a stepping stone, which allows us to link the high-level learning mechanism in the variational framework of MFG and the fine-tuning hyperparameters at the GNN model instance level. The outcome of our work is an end-to-end deep model that jointly identifies the most suitable message passing operators and refines hyperparameters for the corresponding GNN instance.
learning framework for GNN, coined Game-of- GNN,
The major technical contributions are four-fold. (1) We present a physics-informed learning frame- work for GNN that crafts the most suitable GNN model while performing machine learning on graph data. (2) We integrate the theory of mean-field game into graph neural networks which not only offers an in-depth understanding of GNNs but also provides a general guideline for developing deep models for unseen graph data. (3) We present a practical end-to-end solution, based on Hamiltonian flows, to customize the best GNN model for the underlying graph data. (4) In addition to the comprehensive evaluation on graph benchmark datasets, we explore the foundational principles of graph learning as an interactive dynamical system, which is valuable for the conceptual framework of developing future GNN models.
2 Methods
2.1
Background on Mean-Field Game
2.1.1 Gradient flow for optimizing mean-field game
Suppose we have an undirected, weighted graph g = (V, P) with V= (u;li= N) is a finite set of N vertices and P C V V denotes the set of edges. The adjacency matrix is denoted as A = [am]] where [ E P. Suppose x; (z(r, f) E V} E R xd represent the distribution of graph feature embeddings (aka. potential energy) associated at time t. The continuity equation describes the evolution of the distribution I, can be formulated as a 4 -dvv( where div(-) denotes the divergence operator and % is the latent velocity field. By constraining the evolution of à
I, being the gradient flow that minimizes the energy functional E(xx) Ig G(z,)du there exists
unique potential function 4 such that 7t- VΦ, 121.
3

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We remark that if ((r(v,t) = =(v, t) logæ(v,)) t), then the gradient flow Or 8t -iv((>V@) satisfies the heat equation dr AA, where ( t) G" (v, ((g + Mounting evidence shows that the message-passing mechanism
Remark 1.
Ar graph convolutional
networks can be formulated as neural graph diffusion process
2.1.2 Variational framework of MFG constrained by reaction diffusion model
Suppose (v) is situated in the continuous domain of smooth positive density space. First, we construct a Lyapunov functional (v)( = Sg ())(d)) where G : IR + is convex with
G"(V)> 0. If the gradient flow satisfies the reaction diffusion model (RDM)
de 0F
(F( R(V).
minimizing (()()) forms a dynamical system:
at E(v,) -(V)) a au E(V.) o(V) ov)) (-div-
F(V G"
RV)
(I)
G'(V)
where g(V) is the weighted elliptic operator (Proof in Appendix Sec.
A.2) ). To foreshadow the
motivation for introducing the notion of control pattern in the MFG framework (in Section 2.3.2).
define the following two functionals to simplify the analytic expression of g( g
Definition 1. Free-energy functional for transportation and reaction. Given F, R. and G, we define the transportation mobility function (V) R(V O2(V) V) = ('())'
F and reaction mobility function
The derivation of g not only links objective functional EV) and RDM-based gradient flow, but also allows us define the mean-field information metrics [3] follows.
Definition 2. Mean-field information metric. Denote a as a smooth, positive density function. Given the elliptic operator g (in Eq. D the metric between two densities 01 and 02 is:
((01, 02) = (Vo1, Vo (01, 02).22
(2)
Remark 2. In the special case of θ; v and O₂ = 0 (first term in Eq. 2). the variational
MFG problem seeks the optimal transport 1 to move the mass from Vo V by minimizing L2- Sa-Wasserstein metric. In another special case that θ = 0 and θ₂ = V/ (second term in Eq. B]. the variational problem is corresponding to the Fisher-Rao metric, which has been well studied in information geometry [2]. It is clear there are different choices of operator lead to different mean-field information metrics.
Definition 3. Variational framework of mean-field game. Given I and O2, a variational problem
J.'' [.22 (1, t)ll (V(,.)) + W/a(v, t) O2(V dt 1P1:
inf
where the infimum is taken among all density functions (u), vector fields vh, and reaction rate functions 2/2, such that
O,V t) + V (V) with fixed initial and terminal density functions Vo, V1. Remark 3.
(4)
The example in Remark 1 indicates that the dynamic process of graph learning can be framed as a variational problem governed by predefined gradient flow. Furthermore, it is possible to identify the most appropriate combination of energy function E(D) (Eq. 3 and gradient flow क (Eq. 4 using machine learning technique, which sets the stage for 'meta-learning' paradigm for unseen graph data. what follows, we first unify existing GNN models into a reaction-diffusic model (RDM): 84
AF(W) + R(V)), where F(-) and R(-) are diffusion and reaction functions, respectively. By constraining the gradient flow to follow the characteristics of RDM, we further formulate the design of GNN instance into a variational framework of MFG inverse problem, seeking to uncover the 'free-energy functionals' (determining the kinetics of information exchange and update) that yield the best graph learning outcome downstream applications. Together, we present -inforss- approach to jointly perform machine learning using the most suitable message passing operators and fine-tune the GNN model instance through the Hamiltonian flows derived from the associated RDM.
Bt

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2.2 Unifying GNNs into the Dynamics Shaped by Reaction Diffusion Model
In this section, we briefly review some representative GNNs and unify them in the umbrella of RDM. More details are shown in the Appendix A
Graph neural diffusion (GRAND) draws inspiration from the heat diffusion equation, offering unified mathematical framework for vanilla message-passinggg laws on graphs by är div[cVV], where F = V/ and R(V) ) O. To simplify the problem formulation here, we assume c a homogeneous and time- inverinvari diffusion function. Thus, the intrinsic diffusion- can be further simplified as @4e A after dropping c.
GRAND 77.
pattern
2t
GraphBel [1]. Extended from GRAND, the Beltrami diffusion on graph (GraphBel) proposed to VV
| where [ is a discrete analogue of the mean curvature operator. Without changing the diffusiio--eaactinn property, we use A to indicate the normalized graph Laplacian operator here. Thus, the evolution of Y becomes av = || | VV, where || | = ( V is time-invariant magnitude of graph gradient.
use Beltrami flow to normalize the graph gradient as
0% at = -र:ा div
Vy
or
Since || is decoupled with the divergence operator, it is straightforward to derive F F(V) = V and VV 0 in GraphBel.
ACMP 36 Inspired by interacting particle dynamics, Allen -Chn message-passinng (ACMP) graph
neural network models both attractive and repulsive forces between two connected nodes during message-passing process using a predefined Allen-Cahn double-well well potential function f M.
2% The ACMP- GNN models can be formulated as 8t my [div (s V f'( where me is a sypeer-aaameter for the mobility and Ev a constant. Since my and ev do not determine the reaction-diffusio property. the PDE in ACMP can be simplified to scenario, ((V) = Y and R(V)= = VVV.
2 ग Av (VV)). In this
It is apparent that the characteristic of information exchange and update in a particular GNN instance determined by gradient flow associated with the RDM. As we conceptualize that GNN is a av
mean-field game, gradient flow at AF( + R(W) eventually leads the minimization of an energy functional E( ((V) = fg G which describes the collective behavior of agents in the MFG.
Remark 4. Current GNN models apply the same energy functional {(V) across all graph data, regardless of whether the graph exhibits homophilic or heterophilic properties. It is analogous to using an identical strategy in variety of games, irrespective of their differing rules. However, the gradient flow exhibits distinct dynamics depending on the choice of energy functional. In light of this, is appealing use the most appropriate energy functional (VV) that promotes the highest reward in each mean-field game of GNN. Building on this idea, we seek to reshape the kinetic dynamics by introducing control patterns, yielding the in-context message passing operator.
2.3 A Variational Framework of MFG Inverse Problem for Designing Novel GNN Models
Following the spirit of mean-field theory, we frame the dynamic process of message passing in GNNs as mean-field game, where optimal features emerge as the system reaches Nash equilibrium governed by the control patterns learned from the underlying graph data.
2.3.1 Graph neural network is a mean-field game
We set up miel-field game with N players in a continuum of con-cooperative rational agents (graph nodes) distributed spatially in the graph G and temporally in [0, 1J. For agent v starting at ()( the evolution of (1, v) is completely determined by Eq. [4] To play the game over a time interval [0, 1), each agent seeks to minimize the objective functional in Eq. where the transportation cost 11 and reaction mobility cost O2 are incurred by each agent's 's own action. In addition, the Hamilton Hamilton-Jacobii given below.
(HJB) equation equivalent to the variational formulation in Definition 3 is
Propasition 1. HJB equation for mean-field game. Assume (v: f) > 0 for t E 10, 1]. Then there exists a function tb G x (0, 1] - R. such that the critical points of variational problem Eq. satisfy w (v. = (v. ()2(v,)) = (vv
(5)
with
( -v
+B 0.0(,,))+ + V (2 [
0,
(6)
5

---

and (e, 0) - (o(()) (e, = (2))
Sketch of proof. We introduce + as the Lagrange multiplier of variational problem (Eq. 3 constrained by the gradient flow in Eq. 4 Then we derive the solution of vector field wh, reaction function 4/2,
and Hamiltonian flow in Eq. 6 by following the schema of saddle point problem. The detailed proof is shown in Appendix Sec.
A.3
Remark 5. If I = V and 0₂ = O, the above formulation corresponds to the well-known Benamou- Brenier formula [6] in optimal transport. If O2 and O2 are positive functions then the objective functional in Eq. 3 is convex, making the derived gradient flow in Eq. o minimizer of the variational
MFC problem 21.
Remark 6. Suppose Vo is the initial graph representations. Given O and 22 Proposition indicates that, at the mechanism level, the dynamical mechanics of feature representation learning from Wo to 1 is characterized by a Hamiltonian flow (Eq. (). while at the model instance level, the alignment between the learned features 1 (terminal state) and the downstream task can be fine-tuned using a PDE-based GNN approach [42 which is governed by the Hamiltonian flow.
Given Vo and 11 a natural question is: What the most efficient way to transport Vo to 1 The key to answering this optimal transport question is to study the critical point of the objective functional (V)) with respect to 11 and 22 which leads to the MFG inverse problem.
2.3.2 Discover in-context message passing operators through MFG inverse problem
The variational MFG framework provides a potential optimal solution for the objective functional in Eq. by examining the saddle point. Recall GNN is mean-field game. The physical principle is characterized by the pre-selected transportation functional θ and reaction functional 22 In contrast to the special cases of Wassersstn distance (where O₂ (T) V O₂ (T) 0) and Fish-Rao metric (where ②1 (V) = 0, Oz )) θ1 and O₂ are essentially the weighted functions on each location v, acting as the expected in-context message passing operator that allow us to regulate the local message exchange and update during the evolution of graph representations V,, Naturally, we are motivated learn the in-context message passing operators (Oi, O2). from the underlying graph data to improve the performance of GNN models.
In light of this, we present the following meta- meta-learning paradigm that derives the most suitable learning mechanism from MFG inverse problem and meanwhile optimizes model parameters using GNN backbones. By doing so, we expect to (I) enhance graph data learning performance top of the existing GNN models and (2) establish an in-depth understanding of how individual node learns the best feature representations for themselves and the entire graph. Specifically, we introduce the functional commitonnJJaoboe equations in positive density space (i.e., graph space) and define a Hamilton functional H. G x G + R as follows:
H( 4) = 1. ( 11) 1 [10POs(V)) du.
where the density function serves as the state variable (akin position), while the potential function 4 acts as the momentum variable in graph space.
2.4 Reverse Engineering GNNs by In-context Message Passing Operators
In Sec. 2.2 we have shown the relationship between GNN model instance and paction-diffusion equation. Despite many GNN models being fundamentally linked to the same PDE, they exhibit varied learning behaviors, yielding distinct learned feature representations. Within the variational framework of the MFG inverse problem, such diversity can be attributed to the fact that different GNNs are driven by distinct objective functionals (V)), each governed by unique physical principles. In Table I we summarize the energy variational functional ε, mobility functions θ₁ and O2 fraction-diffusii equation, and the corresponding Hamiltonian flows.
It is clear that the objective functional ((V)= = Sg G(D)du (for crafting GNN mechanism) and the associated gradient flow in 8 (F( + R(V) (for optimizing GNN instance) are both related ăt
to transportation mobility function O and reaction mobility function O2. By capitalizing on this property, our Game-of-GNN emerges as a meta-learning graph learning approach. For clarity, we summarize how GNN is formulated as mean-ficld games in Appendix A.4

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Table Variational functionals E(V) fg G(V) diffusion function F(-), reaction function R(-), mobility functions (.(+) and (2(-), and Hamiltonian equations.
Model ((W) Io G(V)dv F R(W) (v))) = =
Hamiltonian Equation
a + (VV4)
GRAND
Ig Volg 1)de
0,4 - $ [0]]
0, (VV))
GraphBel
V
0
0 =0
0,4
( =0
ACMP
f(V)dv
(V) ("(D)-
0.4 } ||
("
Network architecture for Game-of-GNNN. Inspired by 42). we propose agnostic end-to-end deep model based on Hamiltonian mechanics, which characterizes information propagation in graph networks using Hamiltonian-liks structure. The implemented details are shown in Algorithm Specifically, we regard the potential energy and latent function & (V is a flow vector field) in Eq. as the position and momentum variables, respectively, in the Hamiltonian system, where the phase space (V, Φ) characterizes the system's evolution. Prior to (O), (0)). we deploy a set of fully-connected layers F to project the observed nodal features x to the energy function. There two major inter-connected network components in Game-of-GNN: (1) meta-learning component O for generating pesssage-pssiing operators O₃ and Og based on the current estimation of phase space (L, Q): and (2) -bsed GNN instance M for solving the evolution of Hamiltonian flow, where the terminal state of Hamiltonian flow used plug-in with the down-stream learning task. The connection between M and H. is the learned message passing operators O and θ2
Algorithm 1: Game-of-GNN algorithm
Input: Graph G PV, node features zft), adjacency matrix A Output: The mobilities O1, O2, the evolved node feature representation (TT)
for i 11=1... 1.../. do
Construct phase space by (V, p.) t F();(t));
for ( 1... do
Learn mobilities , O₂ by θ1 θ₂ + ( ((VVV))) +)):
Construct Hamiltonian function (I),,(), (() by Eq.
Build PDE of the evolution of system state on graph by Eq.
Derive the trajectory (t), (t)) by PDE solver;
end
Yield the evolved node feature representation (T)) by (TT) - ((T)))
10 end
O: Generating θ and O2 by input convex neural network (ICNN). Since energy function H(V, ②) is completely determined by mobility function θ₃ and O₂ (shown in Eq. 7 we propose to use neural network O to establish the implicit mapping between the input (vr +) and output (02. O2). As a crucial prerequisite for deriving the Hamiltonian flow outlined in Eq. 6 the objective
function needs to be convex. Therefore, use input convex neural network 4 the backbone of O, which yields the convex function instance in recursive manner:
(k+1) - (k) (k) ((4) + w (W, (W, 4b) 6(4)
(8)
where 24) denotes the output of thh layer. Each layer consists of two MLPs which project (1) the output from the previous layer - (parameterized by wx(k) and (2) the current phase-space (V, @) (parameterized by ((( and concatenate the output of two MLPs into 2(*+1) by applying a non-linear activation (4) with a bias vector b(R) Thus, the output of meta-learning component O is the transportation function O and reaction function O2, which allows to define the message passing operator for each graph node based on the phase space (V, Φ).
M: GNN model based on Hamiltonian flow. In physics, systems evolve according to fundamental physical laws, with (pre-defined) conserved quantity function H(V, b) that remains constant along the system's trajectory of evolution. This conserved quantity is commonly interpreted as the 'system
7

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100
G Data Classification
Brain Data Disease Diagnosis
0.05
Accurnty (%)
70
60
50
20
Tools
Wisconsin
Actor
Squirrll
Chameleon
Comnell
Citeseer
Pubmed
ABIDE
OAGES
PPMI
Datasets
GraphšaGE
GraphioN
Graphßel
Game-of-GMn
Figure 2: Left: Test accuracies (%) on nine graph networks for node classification task. Statistical significance assessed based on 20 resampling tests conducted using randomized seed. '*' means statistically significance with p I 0.05. Right: Diagnosis accuracies (%) on disease-based datasets.
energy'. We model the evolution of graph feature representations by following Hamiltonian equation:
6 0, V
5
H(V. 4). 0,0
H 1), 8y
with the initial features ( (0), (()) at time t 0 being the vectors of potential energies. Supposing (V) E Rd and ((v) Rd, we generate a 2dddimmnsiioal vectors that then split into two equal halves: the first half serves as the feature (position) vector V/ while the second half represents the momentum vector guiding the system's 's evolution. Assuming terminal time point & T, the solution of the system is represented by (T") and (T'). obtained through integration to derive the trajectory (I (t), (t)) described in Eq. 9 After that, we apply the canonical projection function II extract the concatenated feature vector V of the nodes from ( (T),, o (T)), yielding (T), ((T)) - (T)), which can then be utilized for downstream tasks such as node classification.
3 Experiments
Experiment setup. The evaluation on Game-of-GNN not only includes benchmark with respect
to existing state-of-the-an GNN models but also a proof-of-concep exploration to uncover novel insights into graph learning- Specifically, benchmark tests include (1) node classification and (2) graph classification. For graph node classification, we apply our method to both heterophilic and homophilic datasets (sorted by homophily ratio h (5)):: Texas (h 0.11), Wisconsin (h = 0.21), Actor (h = 0.22), Squirrel (h- 0.22), Chameleon (h 0.23), Cornell (h = 0.3). Citeseer (h 0.74). Pubmed (h- 0.8) and Cora (h- 0.81), where h indicate the fraction of edges that connect nodes with the same label. For graph classification, we first conduct an experiment on the benchmark results on TUDataset 25 including MUTAG, NCI1, ENZYMES D&D,
IMDB- B and PROTEINS. To demonstrate the generality and scalability of our proposed model, we then apply the Game-of- of-GNN to human connectomes for disease diagnosis, we use the processed neuroimaging data in the published datasets [39] ABIDE (Autism), ADNI (Alzheimer)
[37]. OASIS (Alzheimer) 18] PPMI (Parkinson), where we use regional BOLD (blood oxygenation level-dependent) time series as the graph embedding and functional connectivity (FC) [9] with automated anatomical labeling (ALL.) atlas (116 regions) 33 as the adjacency matrix. The data description is shown in Sec.
B.1
We compare the performance with various benchmark GNN models, including vanilla GCN GraphS [3] GraphCON [30]. GraphBel [1. GRAND [7], ACMP (36] HANG 42
GAT 34
GIN
40
and AM-GCN (35). For conventional graph data node classification experiments, we
follow challenging data-splittig method published in 43 (graph robustness benchmark), with 60% for training, 10% for validation, and the rest of the nodes for the testing set. For TUDataset, we
report the 10-fold cross-vale validation (follow (2)) results on different models. For human brain data, we report the 5-fold cross- voss-validation results.
8

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p 0.05
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0.055
0.55
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GIN
644
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Figure 3: Performance on TUDataset across seven methods.
3.1 Benchmark Evaluations
Performance on graph node classification. Fig. 2 (left) lists the comparison results for nine classic graph datasets on nine methods. Gfme-of-GNN achieves SOTA performance on both heterophilic and homophilic over the existing hand band-designed GNN models. Moreover, we perform an ablation study in terms of depth of ICNN, the result shown in Appendix Discussion. These results provide evidence that our variational framework is able to customize the most suitable mobility functions for different graph data, which contributes to enhanced learning performance compared to other 'one- bize-fits-all' approaches. This improvement primarily stems from the design of θ and O2, which enable the model to effectively adapt heterophilic structures. Specifically, O, ( (V) acts like a feature- dependent attention weight, where heterophilic nodes, Vi shows high neighbor variance leading to residual gate,
θ L and hence less smoothing. In contrast, 22(V) behaves like a feeture--dppender where high neighbor variance results in 02 1, indicating stronger self-correction. By making both transport (+t) and reaction (92) depend on local feature statistics V, our framework automatically throttles neighbor averaging and boosts self-repulsion in heterophilic regions, providing a principled mechanism for handling heterophily.
Performance on graph classification. We include a 'global maa__maa_ap____aaaaa pool' function and fully con- nected layer achieve the graph classification task. Fig. presents the benchmark results for six classic methods and our Game-of-GNN on the popular TUDataset. Our Game-of- ofNNN, demonstrates decent performance across various types of graph data, including molecules, bioinformatics, and social networks, outperforming several existing hand-designed GNN models. Discussion. These results provide compelling evidence that our variational framework well-suited for various types of graph datasets, resulting in improved learning performance compared to approaches tailored to specific datasets.
Benchmark on human connectomes. Fig. (right) summarizes the diagnostic performance across four dased datasets, where we predict the risk of developing neurological disease in unseen subjects using graph data. The experimental findings demonstrate that our method exhibits significant effectiveness in disease diagnosis, suggesting the promising clinical value of deploying our approach in disease early diagnosis. Discussion. Along with atypical neuron growth/loss (23 many neurological diseases manifest network dysfunction syndromes (8). In addition to the standard attention mechanism in GNN R4. the variational framework in Game-of-GNN allows us to uncover the dynamic mechanism of disease progress from a system perspective, as shown below.
3.2 Method Exploration: New Insight of Graph Learning Beyond Attention
In this section, we put the spotlight on the transport function O since this mobility function is intuitively relevant to the message-exchanging mechanism in GNNs. For each graph dataset, one of the outputs of Game-of-GNN the learned O each graph node, where we essentially employ ICNN backbone (Eq. in the learning module O) to generate convex function based on the flow information V. Assuming the latent convex function is polynomial function, we compute the mean polynomial power at each graph node by applying uni uni-variate polynomial fitting for each element and then averaging the degrees of polynomial power. After that, we conduct several post-hoe analyses at graph level and node level, respectively. First, we use the averaged polynomial power (across nodes) to express the graph homophily ratio h, to uncover the new insight into how the dynamies of information exchange in GNN correlates with the properties of the graph data. Second, we extend this global analysis to each graph node with the hypothesis that mobility of spreading node embeddings (related to neuropathology burdens) underlies the biological mechanism in discase progression.

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Figure 4: Left: Correlation between graph homophily ratio (y-axis) the learned control pattern θ1 (VV = To (x-axis). h is a glohal measurement where a large degree indicates a better alignment between graph topology and the consistency of labels across connected nodes. We polynomial function to fit θ for each node, where the degree of is inversely proportional to the freedom of local information exchange. It clear that the learned control pattern highly correlated with the heuristic measurement. Right: Top ten significant brain regions associated with the pathophysiological mechanism of AD, Autism, and PD.
Results. As the white curve shown in Fig. 4left, the a ~ h relationship across nine graph dataset indicates notable anti-correlation (more detailed analysis is provided in Appendix (). It implies that the effective way to perform graph learning is to promote information exchange on homophilic graphs (such as Cora and Pubmed) while constraining the diffusion of information between connected nodes with different labels heterophilic graphs (such as Texas and Wisconsin). The reason behind is rooted in the MFG objective functional (Eq, 3 that larger degree of O encourages the optimization process favoring smaller flows v which is aligned with the heuristic of penalizing information exchange in heterophilic graphs. Furthermore, we display the learned θ at node level for Texas, Wisconsin, Chamelcon, and Pubmed in Fig. Aleft, where bright yellow and dark red denote for small and large degree of O₁ (v)), respectively. Importantly, we find that global homophily h alone does not fully explain the performance differences across datasets. As shown in Texas, Chameleon, and Pubmed (as shown in Fig. 2 left), datasets with similar h can still exhibit markedly different gains due to variations local structure and feature quality, further underscoring the importance of adapting mobility functions beyond global homophily.
Discussion. Similarly, we conduct the same post-hoe analysis to investigate the biological underpin- ning between the learned node-wise transport mobility degree and pathophysiological mechanism of disease progression. Fig. right, we use large size node to indicate the larger mobility of the underlying node (associated with smaller degree of O (v)). It is interesting find that the brain regions with high dynamics for pathology propagation are closely associated with our current findings on disease etiology. Take Alzheimer's 's discase (AD) for example, resting-state fMRI studies have identified significant alterations in BOLD signal dynamics within the default mode network (DMN), which may indicate abnormalities in functional connectivity [[2). Here,
machine learning techniques to provide another piece data-driven evidence to support this finding as most of the lazge-size nodes are located in DMN. Additionally, our findings reveal that (1) increased mobility of pathological factors in the cerebellum correlates with the progression of Parkinson's is disease, and (2) accelerated neuron overgrowth in the dorsal attention and limbic networks, well as the cerebellum, may potentially be a contributing factor to autism. These promising results underscore the new window answer neuroscience questions using explainable machine learning techniques.
Conclusions
In this work, we integrate the theory of wrean-field game into GNNs to enhance our understanding and guide the development of deep models for new graph datasets. We also provide an end- to-end solution using Hamiltonian flows to jointly learn suitable message passing operators for GNN model and fit the customized GNN model to the underlying graph data. Our approach thoroughly evaluated on standard benchmark datasets, and we explore fundamental principles of graph learning as an interactive dynamical system, which not only advances GNN understanding but also contributes to the broader field of graph-based machine learning.
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Acknowledgement
This work was supported by the National Institutes of Health (AG091653, AG068399, AG084375) and the Foundation of Hope. The views and conclusions contained in this document are those of the authors and should not be interpreted representing the official policies, either expressed implied, of the NIH.
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Beyond homophily in graph
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Answer: [Yes]
Justification:
We have the code of our model and have uploaded it to GitHub.
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20

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A Proof of Propositions And Detailed Formulations And Explanations
A.1 The Explanation of Definition 1
The motivation of formulating Q (()) and θ = - is to facilitate (1) linking the diffusion
R(V) G
(P]
term FVV) and reaction term RR( v) RDM energy functional ( Sg G( (v))d and (2)
generalizing existing GNNs into the RDM framework. As described in Sec. (.. current PDE- DDE-based
GNN models can be regarded as the RDM with empirically defined diffusion functional instance F(V) and reaction functional instance R(V). In our approach, the behavior of diffusion F and reaction R is not fixed for all graph data. Instead, we learn the most appropriate diffusion and reaction functions through the transport functional θ and reaction mobility functional O2
A.2 Proof of Eq. (1)
To prove Eq. (1), we consider constructing a Lyapunov functional E g - R thus considering EVV) ((V())),o, where G R + R is convex function with (V) > 0. In such cases, we have
to study the RDM,
d (((((t,))) | ( a, -. CCVVC / R(V ot)
di
-- -- (G((()
Ul((t du / G'(V(t,))) C' u) du ("(V(L,
(( U))
(10)
where we apply
Eq. 10 Herein, we assume
0 and F' > 0 for
V/ >> O, thus we have E S O, indicting that functional EVV) is not increasing along flow. The decay behavior described above suggests gradient flow formulation for the dynamics outlined in RDM. To refine and clarify this concept, introduce inverse of the weighted elliptic operator
F(V)
R(V)
((V) = -V.
V
(11)
(1)
G'v))
Thus we have
9(0))=((- --(--- F'(V)
R(V) 8
0,V V- -g(VV-- C(V) = -(v)=
(T) v)
(VV)
G'(V))
5y
(12)
( v + R(V)
= V.
( G'(D) = (F + R(D)
where sw denotes the I first variation w.r.t V E (ME). Based the above notation, the dissipation
of Lyapunov functional along RDM satisfies d. de ( - (. (VV) (V)) é ((V) du S 0.
A.3 Proof of Proposition
To prove Proposition I, we first rewrite the variables in variational problem Eq. of the main text as
q v) = (1 (t) (e(t,v) = O2 (t,,,,
(13)
Thus the variational problem Eq. forms
{ Jh 1 w1) + 1))( 20, (t. (O. v)) dudt: V an (1. v) (2(1, v), fixed Vo V1
inf mi 12
(14)
a (L,
Denote the Lagrange multiplier of Eq. 14 by 4. We consider the following saddle point problem
inf sup L (q), q2, M, 2) n w 4
(15)
21

---

with
L (Q11,92 v. [. J. (e,
(v((t + w2
20,(
(16)
+ ((,, u) (O, v (1(,,)) 92(1, v))} dudt.
By finding the saddle point of C, we have
6 L O,
91 VΦ,
ēyu óạa [ O,
9
92 $
6 O, 84 0,
ll 2
0
14.l 2 83
0, 0, 0
(17)
0f
0.4 + Q1
0.
where 50g Kgg" R6 are L first variations w.r. functions Q1 92. V 2, respectively. After that, by substituting the above two equations into the last two row equations of Eq. [7] derive the PDE pair Eq. i in the main text.
5 6 6 5
A.4 GNN Is A Mean Filed Game
In the following, we emphasize the explanation of the principle of how GNN is formulated mean-field game.
GNN is a dynamical system. Simply put, GNN is a black box that converts the initial feature repre- sentations into a latent subspace by set of information exchanges (constrained by graph topology) and projection (using mapping function shared by all graph nodes). As GNNs often consist of multiple layers, the evolution of feature representation from the initial state (input graph embeddings) to the terminal state (last layer of GNN) can be regarded as a time-dependent dynamical system, where the dynamics is determined by a governing equation (in the form of PDE). the reminiscent of the Brachistochrone problem (a classic physics problem that involves finding the curve down which bead sliding under the influence of gravity will travel in the least amount of time between two points), the powerful calculus of variations (COV) allows us to generate various governing equations, providing a necessary condition that a function must satisfy in order to be an extremum of a given functional.
GNN-PDE-COV interplay. Inspired by recent PDE-based deep models such as Neural ODE and GRAND, we frame the aayer-by-layer feed-forward process ((++) (AWxx!) as a dynamical system, where the time-evolving mechanics determined by the graph heat equation 0z(t) Az (t). øt
Here, A and A denote the normalized adjacency matrix and Laplacian matrix, W is the learnable mapping parameters, and denotes the nonlinear activation function. Indeed, the evolution of the heat equation forms the gradient flow of the Dirichlet energy <(x) = V Thus, we have established a connection between the GNN model instance in the discrete domain and the equivalent variation functional in the continuous domain, where the governing equation is acting as a stepping stone.
Setup of mean-field game for GNN. Following the notion of mean-field game, each graph node is acting as an agent. The game is to find the best feature presentations for all graph nodes that minimize the loss function in GNN. In a mean field game, each agent (aka. graph node) makes decisions based on both their individual state and the aggregate effect of the states and actions of all other agents, often referred to as the "mean field." The primary goal is to find Nash equilibrium, which is strategic decision-makiin benefit by changing their strategy while others keep theirs unchanged. Mathematically. mean field games often involve solving coupled partial differential equations such as Hamilton-Jacee Bacl---Blaaa (HJB) equation which describes the optimal control problem for the evolution of the distribution of agents' states over time. In our work, we introduce mean-field game and mean-field control (MFC) into GNN, as described below.
in very large populations of interacting agents such that no agent can
A MFC framework for designing novel GNN model. First, we extend the heat equation a graph- based reaction-dillluiin model, where the system belivvior is determined by a mobility functional
22

---

(01) and reaction functional (92). Second, we follow the recent work of MFC 21 to define the main variational problem, consisting of metric space (Eq. 3 and gradient flow (Eq. 4 After that, we study the critical point of the variational problem, yielding a Hamiltonian flow in MFC problem (Proposition 2).
Takeaway. We formulate the dynamic process of graph feature representation as a mean-field game where the game policy defined in mean-field control perspective. the real application, our model simultaneously (1) crafts GNN model instance by identifying the most appropriate game policy (i.e.. derive the instance of mobility functional O and reaction functional (2), and (2) optimizes GNN instance using Hamiltonian flow.
B Experimental Details
B.1 Datasets and Hyperparameters
Classic graph data for node classification. We summarize the data information in the following Table
2
Table 2: Data description for node classification.
Texas
Wisconsin
Actor
Squirrel
Chameleon
Cornell
Citeseer
Pubmed
Cora
Hom. ratio
0.11
0.21
0.22
0.22
0.23
0.3
0.57
0.74
0.81
#Nodes [V]
183
7,600
5,201
2,277
183
3,327
19,717
2.708
#Edges [P)
295
466
26,752
198.493
31,421
280
4.676
44.327
5.278
#Classes y
Classic graph data for graph classification. We summarize the involved TUDdataset in the following
Table 3
Table 3: TUDataset description.
MUTAGE
NC11
ENZYMES
D&D
PTC_FM
IMDB
PROTEINS
#Graphs V
188
4,110
600
1,178
349
1,000
1,113
#Classes (P)
2
2
6
2
2
AvgeNodes y
17.93
29.87
32.63
284.32
14.11
19.77
39.06
AvgdEdges y
19.79
32.30
62.14
715.66
14.48
96.53
72.82
Disease-based human connectome data. We summarize the data information in the following Table
Note, Destrieux atlas [10] (160 brain regions) used in OASIS to verify the scalability of the
models.
Table 4: Disease-based human connectome data statistics.
Dataset
Condition
# of Subjects
of Classes
Regions/Nodes
Avg # of Node Features
ABIDE
Autism
1025
2
116
201
ADNI
Alzheimer
250
5
116
177
OASIS
Alzheimer
1475
2
160
330
PPMI
Parkinson
209
116
For binary dataset consisting of two classes, one representing a disease group and the other a normal control group. For ADNI dataset, following clinical outcomes, we categorized subjects into distinct groups representing different cognitive statuses. These groups include: cognitive normal (CN), Subjective memory concern (SMC), ctrly-stage mild cognitive impairment (EMCI), late-stage mild cognitive impairment (LMCI), and Alzheimer's is Disease (AD) groups. To facilitate population counts, we regard CN, SMC and EMCI as "CN-like" group, while LMCI and AD as "AD-like" groups This partitioning allows for the analysis and comparison of individuals across varying levels of cognitive function, providing valuable insights into disease progression and cognitive decline within the study population. For the PPMI dataset, which encompasses four distinct classes, including normal control, scans without evidence of dopaminergic deficit (SWEDD), prodromal, and Parkinson's 's disease (PD).
Hyperpprameters. We use the Adam optimizer with learning rate of 0.01, and the epoch is set as 250. Most hidden dimensions are set to 128 (Squirrel and Chameleon are set to 64, Cora and ABIDE set to 32). The code is released at GitHub: https: wttptthhw....o/aa/////////////// Game of GNN
23

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B.2 Comparison Methods
Graph neural networks (GNNs) have emerged as powerful tools for learning from graph-structured data, achieving state-of-the-art performance in various domains such as social networks, biological networks, and recommendation systems. In this work, we compare our method against diverse set of benchmark GNN models that represent key advancements in the field:
The Graph Convolutional Network (GCN) introduced the foundational concept of convolutional operations on graph-structured data, leveraging spectral graph theory to propagate node features across the graph. Despite its simplicity, GCN remains a widely used baseline in GNN research.
Vanilla GCN T
GAT [4]: : The Graph Attention Network (GAT) improved upon GCN by incorporating an attention mechanism to adaptively weigh neighbor contributions, enabling the model to capture more nuanced patterns in heterogencous and large-scale graphs
This inductive framework generates embeddings by sampling and aggregating features from node neighborhoods, making it particularly effective for large and dynamic graphs where new nodes can be introduced.
GraphSAGE 13:
GraphCON 30] GraphCON leverages neural ODEs and skip connections to improve gradient flow during training, enabling it to address the oversmoothing problem in deep GNN architectures.
GraphBel 31 This model focuses enhancing robustness against adversarial attacks by learning more resilient graph representations through belief propagation mechanisms.
GRAND 77: GRAND introduces random diffusion processes to improve message passing, focusing on long-range dependencies and reducing the oversquashing issue commonly seen deep GNNs.
The Adversarial Contrastive Message Passing (ACMP) framework utilizes contrastive learning to enhance node representations, particularly in the presence of noisy or incomplete graphs
ACMP [6)]
HANG 42: HANG employs adversarial training to learn robust graph embeddings, effectively tackling challenges posed by graph perturbations and adversarial noise.
The Graph Isomorphism Network (GIN) is designed be as powerful as the Weisfeiler- Lehman graph isomorphism test, achieving high expressiveness by using learnable aggregation function.
GIN 40
AM-GCN 351: This model integrates both node features and graph topology in balanced way, enhancing its ability learn from graphs with highly diverse connectivity patterns.
These models collectively capture a wide range of design principles, from improved aggregation mechanisms (e.g., GAT, GIN) and inductive capabilities (e.g., GraphSAGE) adversarial robustness (e.g., HANG, GraphBel) and advanced training techniques (e.g., GraphCON, GRAND). By bench- marking against these ofte-of-theear GNNs, we provide comprehensive evaluation of our method's performance, highlighting its strengths and contributions to the field.
C Discussion and Limitations
Discussion. To systematically investigate the impact of the ICNN depth model performance, we conducted a comprehensive ablation study with ICNN depths of [2, 4, 8, 16,32, 64) across nine benchmark datasets. reports both the mean test accuracy and the corresponding a for each configuration, with homophily ratios (h) indicated in parentheses.
Two major trends emerge from these ablation results. First, datasets with smaller homophily ra- tios (h) generally require deeper ICNN architectures achieve satisfactory performance. This observation suggests that heterophilic graphs exhibit more complex and non-local dependencies between connected nodes, thereby necessitating deeper neural representations effectively capture the underlying feature- struccurr relationships. Second, the scaling factor a is primarily determined by the homophily ratio h and remains largely insensitive to network depth. In other words, the intrinsic coupling between a and /i persists regardless of ICNN depth, indicating that reflects structural characteristics of the underlying graph rather than architectural complexity. Taken together, our findings demonstrate that while deeper ICNN modules can better capture the intricate edge
24

---

Table 5: Ablation results ICNN depth. Each entry reports test accuracy (%) + standard
deviation, along with the corresponding a.
Tesas (8.11)
Wisconsin (0.21)
Squirrel (0.22)
Actor (0,
Chameleun (0.23)
Cornell (0.30)
Citeseer (0.57)
Pubmed (0.74)
Cora (0.81)
$6.23 + 3.2
58.22
37.98 = 0.7
32.82
0.27
57.37 1.5
85.76 = 5.6
75.17
8.6
86.33 + 1.0
87.53
64,00 11.8
88.64
39.32 0.7
32.53
0,000
61.26 1.5
83.59 2.2
77.61
90,75 1 0.3
87.30
K3,79 11.3
$4,64
36.18 1.2
16.38 1.4
63.48
15.5
8393 2 4.5
77.96
K2,96 1:
87.09 1.5
85. 52 3.7
57.17
40.18 = 0.6
16.60 6.2
60.14
83.32 = 3.7
75.82
6.9
64.59 + 144
87.31
0.77
4: 5.6
38,64
0.9
4.4.
63.85
83:01
13:12
57,40
86.04
94,00 2.2
90,66
37.39 0.5
33,55
4.6
59.43
15.2
$1.88 1
65,46
83,03
85.10
L:1355
0,8509
0.8651
0.8425
0,8129
0.7041
0.4578
0.1315
0.1153
1.1476
1,0462
0.9514
0.8815
0.7616
0.6690
0.3623
0.1312
0.1076
1,0699
1,0188
0.9431
0.9032
0.7391
0.6918
0.5043
0.1426
0,1100
L
1,0055
0.9496
0.9006
0.7656
0.6857
0.4198
0.1310
0.1091
L,055]
0,9312
0.3848
0.8781
0.8283
0.6767
0.7757
0,1422
0.111
1,259]
1.1059
019013
0.7025
0.6563
0.4566
0,1428
relationships present in heterophilic data, the learned a parameter remains a stable descriptor of graph homophily, independent of the model's depth.
Limitations. Although Game-of-Gan exhibits slightly higher inference time on small graphs (e.g., 0.0331 s on Cora compared with 0.0027 for GCN), its performance large-scale datasets reveals an unexpected yet encouraging trend Table 6 On the OGBN-PRODUCTI dataset (~~.445 45 M nodes), Gf achieves inference time of 0.2971 s. which is comparable to that of GCN (0.2700 s), despite its dynamical system formulation. In contrast, other dynamical GNN architectures such GraphBel, ACMP, GraphCON, and HANG encounter out-of out-of-memory (OOM) errors under the same data loading protocol PrygNodderoPPPedDtttaeee equipped with six NVIDIA RTX 6000 Ada GPUs (48 GB each), and no subgraph- based mini-batchin techniques were employed.
All experiments were conducted on a system
This observation can be attributed to three key factors. (1) Lightweight iterative solver. Although Game-of- GNN performs step-wise integration of a learned PDE system via an explicit Fuler scheme, each update step is computationally inexpensive and involves only local, sparse neighborhood aaaiiioos-aaallgouu to the message passing in GCN, Unlike attention-based models such as GAT or GraphBel, our update rule consists of deterministic flow steps with fixed per-node computational cost. (2) Effective hardware utilization on large graphs. On small graphs, GPU resources are underutilized, making minor per-node overheads (e.g.. from PDE state updates) more noticeable. On large graphs, however, the abundant computational workload fully occupies the GPU, thereby amortizing the integration cost across many nodes and yielding inference times close to GCN.
Table 6: Training and inference time comparison (in seconds per epoch and milliseconds per inference) across datasets. "OMM" indicates out memory.
Cora (2,708 nodes)
Pubmed (19,717 nodes)
OGBN-ARXIV (~170K)
OGBN-PRODUCTS (>-2.5MM)
Model
Train (slep)
Infer (ms)
Train (slep)
Infer (ms)
Train (slep)
Infer (ms)
Train (s/ep)
Infer (ms)
GCN
0.0098
0.0027
0.0097
0.0046
0.0147
0.0076
0.3602
0,2700
GAT
0.0201
0.0050
0,0202
0.0053
0.0779
0.0319
OMM
OMM
0.0059
0.0015
0.0064
0.0016
0.0184
0.0071
0.4149
0.1757
GraphCON
0.0394
0.0026
0.2044
0.0177
0.0900
0,0480
OMM
OMM
GraphBe]
0.1943
0.0241
0,2022
0.0326
0.7426
0.1638
OMM
OMM
ACMP
0.2172
0.0288
0.3678
0.0635
0.7193
0.1970
OMM
OMM
0.0417
Gome-of-GNN
0.0755
0.0331
0.0924
0.0324
0.1023
0.0376
0,8145
0.2971
D Impact Statement
Our major contribution to the machine learning field is that we introduce principled approach to optimize GNNs for fitting diverse graph datasets. Through the integration of mean-field control theory and Hamiltonian flows into GNN abstract learning, developed novel methodology that enhances our understanding deep learning models applied to graph datasets.
From the application perspective, our deep model represents a promising approach to bridge the gap between graph-based machine learning and neuroscience research, offering new avenues for studying disease processes.
25


Reviews:
[{"review_id": "XNmfSmhcJH", "reviewer": "Reviewer_AGYb", "summary": "This 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.", "questions": "**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?", "limitations": "Limitations: Please refer to the “Weaknesses” Section.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "The authors have proposed a novel perspective of viewing GNN as Mean-field game. Additionally, they have mostly addressed my concern about the computational cost of Game-of-GNN. Therefore, I would like to maintain my positive rating.", "rebuttal": "**We appreciate the reviewer’s thoughtful comments and are encouraged that you found both the theoretical contributions and the practical adaptivity of our method compelling. In the following, we address each of your concerns and suggestions in detail.**\n\n**Q1: 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\n**A1:** Thank you for raising this important concern. While we acknowledge that Game‑of‑GNN has slightly higher inference time on small graphs (e.g., 0.0331s on Cora vs. 0.0027s for GCN), our empirical results on large-scale datasets—particularly OGBN‑PRODUCTS  ($\\sim$2.45 M) —yielded an unexpected yet encouraging observation: Game‑of‑GNN achieves inference times that are nearly identical to those of the classic GCN, despite its dynamical system formulation.  As shown in the following table, Game‑of‑GNN achieves an inference time of 0.2971s on OGBN‑PRODUCTS, which is comparable to GCN ( 0.2700s, respectively). In contrast, other dynamical GNN architectures such as GraphBel, ACMP, GraphCON, and HANG fail to run on these datasets due to out of memory (OOM) under the same dataloader manner (*PygNodePropPredDataset*). Note, this experiment was conducted on 1× NVIDIA RTX 6000 Ada GPU (48 GB each, 6 GPUs in total). No subgraph-based mini-batching techniques were used in this benchmark.\n\n\n| Model           | Cora Train | Cora Infer | Pubmed Train | Pubmed Infer | ARXIV Train | ARXIV Infer | Products Train | Products Infer |\n|-----------------|------------|------------|--------------|--------------|-------------|-------------|----------------|----------------|\n| GCN             | 0.0098     | 0.0027     | 0.0097       | 0.0046       | 0.0147      | 0.0076      | 0.3602         | 0.2700         |\n| GAT             | 0.0201     | 0.0050     | 0.0202       | 0.0053       | 0.0779      | 0.0319      | OOM            | OOM            |\n| GraphSAGE       | 0.0059     | 0.0015     | 0.0064       | 0.0016       | 0.0184      | 0.0071      | 0.4149         | 0.1757         |\n| GraphCON        | 0.0394     | 0.0026     | 0.2044       | 0.0177       | 0.0900      | 0.0480      | OOM            | OOM            |\n| GraphBel        | 0.1943     | 0.0241     | 0.2022       | 0.0326       | 0.7426      | 0.1638      | OOM            | OOM            |\n| ACMP            | 0.2172     | 0.0288     | 0.3678       | 0.0635       | 0.7193      | 0.1970      | OOM            | OOM            |\n| HANG            | 0.0794     | 0.0335     | 0.1103       | 0.0417       | 0.1638      | 0.0951      | OOM            | OOM            |\n| Game-of-GNN     | 0.0755     | 0.0331     | 0.0924       | 0.0324       | 0.1023      | 0.0376      | 0.8145         | 0.2971         |\n\nWe attribute this finding to the following key factors: (1) Lightweight iterative solver: Although Game‑of‑GNN performs step-wise integration of a learned PDE system during inference (via explicit Euler method), each update step is computationally inexpensive and only involves local, sparse neighborhood operations—similar to GCN’s message passing. In contrast to models like GAT or GraphBel that incur high attention-related overhead, our update rule consists of deterministic flow steps with fixed per-node cost.  (2) Effective hardware utilization on large graphs: On small graphs, GPU resources are underutilized, making even minor per-node overheads (e.g., from PDE state updates) more pronounced. However, on large graphs, the computational workload is sufficient to fully occupy the GPU, thereby amortizing the cost of each integration step across a large number of nodes. This results in overall inference times that closely match those of GCN. (3) No attention or explicit aggregation during inference: While our model implicitly incorporates structural information through the PDE dynamics, it does not perform explicit attention-based or neighbor aggregation operations at test time. Instead, inference consists of fixed-time Euler integration steps involving only sparse, local interactions, further contributing to its efficiency. \n\nTo further address concerns about inference scalability, we additionally benchmarked Game-of-GNN on a higher-end GPU platform (NVIDIA H100 NVL, 94 GB, 8 GPUs in total). As shown in the following, the inference time decreased from 0.2971 s (on RTX 6000 Ada) to 0.1375 s, surpassing GCN (0.2162 s). This demonstrates that Game-of-GNN not only remains scalable on large graphs like OGBN‑PRODUCTS, but also benefits more from modern hardware acceleration. Game-of-GNN is inherently parallelizable due to its local, sparse update structure. Each Euler integration step involves independent, node-wise computations that can be efficiently scheduled across GPU threads. The observed speedup on H100, despite identical code and batch size, highlights that our model structure is well-aligned with hardware-level parallel execution, enabling efficient utilization of modern high-throughput accelerators.\n\n|           | **GCN**     | **GraphSAGE** | **Game-of-GNN** |\n|-----------|-------------|---------------|-----------------|\n| **train** | 0.3277s | 0.3069s | 0.4223s   |\n| **infer** | 0.2162s | 0.1177s | 0.1375s   |\n\nWe also test the cost on a large-scale dataset-**Pokec** ($\\sim$1.6M nodes), the training time is 0.19988s and inference time is 0.0620s.\n\nThese results suggest that while some methods may appear efficient on small-scale datasets, they may not scale to industrial-level graphs. Game‑of‑GNN demonstrates both theoretical grounding and empirical robustness in such settings. We are eager to further validate the scalability of Game‑of‑GNN on even larger datasets in future work.\n\nIn addition, when training on OGBN datasets using neighbor sampling (e.g., PyG’s NeighborSampler), each model processes subgraphs of comparable size, regardless of the full graph scale. Under this setup, we observe that the wall-clock training time is reduced by approximately 4–6× compared to single-GPU execution.  Such scalability highlights the practical applicability of Game‑of‑GNN to large-scale real-world graphs. We will include this discussion in the final version.\n\n**Q2: Accessibility**\n\n**A2:** Thank you for your honest feedback. We fully acknowledge that the integration of mean-field game theory with GNNs may introduce conceptual and notational complexity, especially for readers more familiar with traditional GNN literature.\n\nTo improve accessibility, we will revise the manuscript to:\n\n- Add a high-level intuitive summary of the Game-of-GNN framework at the beginning of Sec. 2 (Appendix A.5 in the original manuscript);\n- Include a brief background paragraph on MFGs and their relevance to graph learning in the Introduction section. –”You can think of a GNN as a “mean‑field game” by imagining each node as a player who repeatedly updates its own feature vector (strategy) based on its current state and the average behavior of its neighbors (the “mean field”). Over multiple rounds, all nodes jointly adapt until they reach a stable equilibrium—just like players in a large game who best‑respond to the average crowd behavior.”\n\nWe hope these changes will make the paper more approachable to the broader GNN community."}, {"review_id": "MzUWjl0K3m", "reviewer": "Reviewer_WHbx", "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.", "questions": ".", "limitations": ".", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": ".", "was_revised": false, "final_justification": "", "rebuttal": "**We sincerely thank the reviewer for their thoughtful and constructive feedback. We especially appreciate the recognition of the motivation behind integrating Graph Neural Networks with Mean Field Games, as well as the insightful suggestion to provide a stronger theoretical foundation. Your comments have significantly helped us improve the clarity and depth of the manuscript.**\n\n**Q: 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.**\n\n**A:** We thank the reviewer for highlighting the importance of rigorous theoretical justification. To address your insightful comments, we have expanded our manuscript with both rigorous theoretical foundations (Propositions 1 in Sections 2.3 and Appendix A.3) and a clear heuristic regime analysis, explicitly describing under what circumstances our *Game-of-GNN* integration with Mean Field Game (MFG) theory improves performance:\n\n- **1. Theoretical Foundations (Sec. 2.3, Appendix A.2–A.3)**\n\nWe rigorously derived the dissipation property and saddle-point equations characterizing our proposed variational formulation. Although general analytical guarantees comparing Game-of-GNN directly with standard GCN methods remain challenging due to inherent nonlinearity, our established framework is internally consistent and well-grounded theoretically.\n\n- **2. Heuristic Regime Analysis (Sec. 2.1.2 & Sec. 2.4)**\n\nBuilding explicitly upon our variational definitions (Defs. 1–3) and recognizing existing GNNs as special cases within a reaction-diffusion PDE framework (Table 1), we now provide a targeted discussion clarifying when Game-of-GNN is expected to excel:\n\n$\\bigstar$ **High Homophily ($h \\to 1$)**: Under conditions of strong within-class node similarity, the transportation mobility term $\\Theta_{1}(\\Psi)$ naturally enhances intra-class diffusion, sharpening and clearly delineating class clusters within the embedding space.\n\n$\\bigstar$ **Mixed Homophily/Heterophily Regimes:** The adaptively learned reaction term $\\Theta_{2}(\\Psi)$ selectively suppresses excessive smoothing across distinct classes. This mechanism ensures that node embeddings remain discriminative and prevents undesirable mixing within heterophilic regions of the graph.\n\n$\\bigstar$ **Dynamic Equilibration (Eq. (9)):**  The derived Hamiltonian equations explicitly enable node embeddings and their momentum to dynamically re-equilibrate:  $ \\partial_{t}\\Psi = \\frac{\\delta \\mathcal{H}}{\\delta \\Phi}, \\quad\n       \\partial_{t}\\Phi = -\\frac{\\delta \\mathcal{H}}{\\delta \\Psi}$. Thus, when graph topology or node features evolve over time, our framework naturally adapts embeddings accordingly.\n- **3. Summary of Clarifications**\n\nThese additions deliver both a *concrete theorem* under a prototypical graph model and an *intuitive roadmap*—grounded in our MFG inverse‑problem variational view—for understanding when and why integrating GNNs with mean‑field games improves or at least preserves performance.  We have revised our manuscript accordingly, clearly incorporating these heuristic insights alongside our rigorous derivations, explicitly clarifying the contexts under which Game-of-GNN is most effective.\n\nWe appreciate your suggestions, which significantly enhance the clarity and theoretical rigor of our manuscript."}, {"review_id": "fxZfph2edD", "reviewer": "Reviewer_CKAr", "summary": "This 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.", "questions": "1. 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.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Quality: good\nClarity: medium\nSignificance: good\nOriginality: good", "was_revised": true, "final_justification": "After going through the disccusion from other reviewers, I'll keep my positive score.", "rebuttal": "**We appreciate your recognition of our MFG-based framework and the Game-of-GNN model. Below, we respond to your concerns one by one.**\n\n**Q1: Could you please explain how does your proposed model address heterophily problem more intuitively?**\n\n**A1:** Thank you for your insightful comment, we'd like to give an intuitive explanation: How Game‑of‑GNN Handles Heterophily?\n\n- 1. **Adaptive Message Strength**  \n   - **Transport vs. Reaction**  \n     - Transport term:  $ \\Theta_{1}(\\Psi)\\,\\nabla \\Phi$ blends a node with its neighbors.  \n     - Reaction term:  $\\Theta_{2}(\\Psi)\\,\\Phi$ pulls a node back toward its own state.  \n   - **Heterophily Effect**  \n     - In regions where neighbors are dissimilar, the model learns a **large** $\\Theta_{2}$, which **dampens** blending and **prevents over‑smoothing**.\n- 2. **Learned Control Patterns via ICNN**  \n   - An **Input‑Convex Neural Network** (ICNN) takes each node’s local statistics $(\\Psi(v), \\Phi(v))$ and outputs $\\Theta_{1}$ and $\\Theta_{2}$.  \n   - If a node’s neighborhood features are very diverse, the ICNN automatically sets:  \n     - **Low** $\\Theta_{1}$ → weak neighbor‑averaging  \n     - **High** $\\Theta_{2}$ → strong self‑repulsion\n- 3. **Hamiltonian Flow Ensures Separation**  \n   - The Hamilton–Jacobi–Bellman update:   $ \\partial_{t}\\Psi = \\frac{\\delta \\mathcal{H}}{\\delta \\Phi},   \\quad  \\partial_{t}\\Phi = -\\frac{\\delta \\mathcal{H}}{\\delta \\Psi}$  conserves a global “energy” $\\mathcal{H}(\\Psi,\\Phi)$, preventing all features from collapsing to the same value.  \n   - In heterophilic zones, the learned flow steers node embeddings into **different equilibrium basins**, preserving class distinctions.\n\n**Bottom line:**  \nUnlike uniform smoothing, Game‑of‑GNN **learns _where_ and _how much_ to smooth vs. repel**, so it automatically throttles neighbor‑averaging in heterophilic regions and keeps node features distinct.  We will add this discussion to the final version.\n\n\n**Q2: Can you provide a deeper interpretability of the role of  for learning on a heterophilic graph? How do $\\theta_1$, $\\theta_2$  relate to or differ from existing operators?**\n\n**A2:** That's a good question. We have provided a deeper interpretability of $\\Theta_{1}$ and $\\Theta_{2}$ on heterophilic graphs as below:\n- **$\\Theta_{1}(\\Psi)$:**  \n  - Acts like a **feature‑dependent attention weight**.  \n  - On heterophilic nodes, $\\Psi$ shows high neighbor variance → $\\Theta_{1}$ ↓ → **less smoothing**.\n\n- **$\\Theta_{2}(\\Psi)$:**  \n  - Acts like a **feature‑dependent residual gate**.  \n  - On heterophilic nodes, $\\Psi$ shows high neighbor variance → $\\Theta_{2}$ ↑ → **stronger self‑correction**.\n\n**Bottom line:**  \nBy making both transport ($\\Theta_{1}$) and reaction ($\\Theta_{2}$) depend on local feature statistics $\\Psi$, Game‑of‑GNN **automatically throttles neighbor averaging and boosts self‑repulsion** in heterophilic regions.  \n\n**Q3: More experiments and comparisons on the challenging heterophilic datasets and large-scale datasets as listed in [1].**\n\n**A3:** Thank you for your valuable suggestion. In response, we have included additional experiments on four datasets from each category mentioned in [1]. Specifically, we added roman_empire–malignant, chameleon-filtered–benign, ambiguous–ambiguous, and pokec–large-scale to Table 1 for a more comprehensive evaluation across both heterophilic and large-scale settings. Under a single, off-the-shelf parameter configuration setting, Game-of-GNN still outperforms every method listed in [1] (Table 1).\n\n| **Dataset**                    | **roman_empire (22,662)** | **chameleon-filtered (890)** | **squirrel-filtered (2,223)** | **pokec-gender (1,632,803)** |\n|--------------------------------|----------------------------|-------------------------------|-------------------------------|------------------------|\n| **Acc**                        | 73.67±0.18                 | 46.81±0.22                    | 41.57±0.14                    | 73.62±0.12             |\n\n\n**Q4: \"The a~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\n**A4:** Thank you for this insightful comment. We would like to answer this question from the following aspects:\n\n-**1. How we estimate $\\alpha$ and why a global anti-correlation is expected**\nWe fit the empirical relation: $h=\\left(\\bar{\\Theta}_1\\right)^\\alpha$ , $\\bar{\\Theta}_1=\\frac{1}{N} \\sum_v \\Theta_1\\left(\\Psi_v\\right)$, where $\\Theta_1$ is the ICNN-learned transport strength and $h$ is the label-homophily ratio of the graph.\n\nWith this log–log model, $\\alpha$ is the negative log-slope that tells us how restrictive the message flow must be to preserve label coherence:\n\n| Dataset (-> larger h) | Texas | Wisconsin | Squirrel | Actor | Chameleon | Cornell | Citeseer | Pubmed | Cora |\n| --------------------- | ----- | --------- | -------- | ----- | --------- | ------- | -------- | ------ | ---- |\n| $h$                   | 0.11  | 0.21      | 0.22     | 0.22  | 0.23      | 0.30    | 0.57     | 0.74   | 0.81 |\n| $\\alpha$       | 1.13  | 0.89      | 0.87     | 0.84  | 0.81      | 0.70    | 0.46     | 0.13   | 0.12 |\n\n  - *Interpretation*\n\n     * When $h \\to 1$  (high homophily) node labels are already coherent; the optimiser therefore pushes $\\alpha \\to 0$, allowing freer diffusion.\n     - As $h$ decreases (heterophily) the optimiser increases $\\alpha$ to throttle cross-class mixing, producing the observed monotone trend.\n\nThis global behaviour does not deny that, locally, the tightest information bottleneck can occur near intermediate homophily once degree, class imbalance, and other factors are held fixed. In the revised Appendix F we include a stratified analysis (binning by degree and class) that does reveal the expected bell-shape within each stratum, while the dataset-level average remains monotone.\n\n- **2. Depth sensitivity and heterophilic graphs**\nOur ICNN-depth sweep (2 – 64 layers) shows that low-h datasets benefit from deeper ICNNs (We kindly refer the reviewer to **A5** in our response to **Reviewer YrZr** for details):\n   - Deeper ICNNs capture the complex, non-aligned feature–label patterns typical of heterophilic graphs.\n   - Shallow ICNNs suffice once $h$ is large, so the model can be pruned to save computation. \n\nThe residual gap on Chameleon and Squirrel appears to arise from multiple factors—including data characteristics, feature–label alignment, and potential capacity limits—rather than a flaw in the $α–h$ mechanism itself.\n\n- **3. Planned extension (link to “What is Missing for Graph Homophily?” [5])**\n\nFollowing the reviewer’s suggestion, we are extending the model to disentangle feature-, structure-, and label-level homophily:\n\n| $\\lambda\\_f h\\_f$ | +$\\lambda_s h_s$ | +$\\lambda\\_{\\ell} h\\_{\\ell}$ |\n| ------------------- | ------------------- | ----------------------------- |\n| Feature loss term   | Structure loss term | Label loss term               |\n\nso that ($\\Theta_1, \\Theta_2$) learn separate sensitivities to each component.  And, we can integrate weak label or community priors into the control patterns. We will add this discussion to the final version.\n\n**Q5: In Figure 2, Chameleon and Texas has similar level of homophily but significantly different ; in addition, Chameleon and Pubmed has similar level of  but significantly different , what is the reason behind it?**: \n\n**A5:** Thank you for your detailed observation. We provided an explanation of the observed performance differences between these three datasets.\n\n- 1. Texas vs. Chameleon — similar global $h$, different local structure\n\n| Factor                     | Texas (low-$h$)                                        | Chameleon (low-$h$)                                    |\n| -------------------------- | ------------------------------------------------------ | ------------------------------------------------------ |\n| **Edge pattern**           | Sparse graph; most edges truly same-label              | Dense graph; many mixed edges                          |\n| **Risk of over-smoothing** | Low → model can set larger $\\alpha$ and diffuse freely | High → model must throttle diffusion, capping the gain |\n| **Result**                 | **+8 pp** over baseline                                | **+0.3 pp** over baseline                              |\n\n- 2.  Chameleon vs. Pubmed — similar $\\alpha$, different feature quality\n\n| Factor                      | Chameleon                                | Pubmed                                              |\n| --------------------------- | ---------------------------------------- | --------------------------------------------------- |\n| **Feature–label alignment** | Weak (noisy features)                    | Strong (word TF-IDF highly predictive)              |\n| **Baseline head-room**      | Baselines already struggle → modest gain | Baselines already near ceiling → limited extra gain |\n| **Result**                  | **60 %** (Game-of-GNN)                   | **91 %** (Game-of-GNN)                              |\n\n**Bottom line**\n\nGlobal homophily $h$ alone does not determine the head-room for improvement.\n\nPerformance also hinges on\n\n- local homophily variance (Texas < Chameleon), and\n\n- intrinsic feature quality (Pubmed > Chameleon).\n\nHence:\n\n- Texas is topologically and feature-wise “easy” → large boost.\n- Chameleon is “hard” on both counts → small boost.\n- Pubmed is feature-easy but already saturated → small boost.\n\nWe will add this discussion to the final version. Thank you so much."}, {"review_id": "xrzHrWdM9m", "reviewer": "Reviewer_YrZr", "summary": "The 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).", "questions": "1. **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?", "limitations": "yes", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths：\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.", "was_revised": true, "final_justification": "6", "rebuttal": "**We thank the reviewer for the clear and insightful summary of our Game-of-GNN framework and for highlighting both its unifying theoretical perspective and strong empirical results. We will address each of your comments and concerns in detail below.**\n\n**Q1: High computational cost**\n\n**A1:** Thank you for pointing out the need for more systematic profiling of our computational cost. We have conducted a thorough analysis—covering per‑batch runtime, scaling behavior on large benchmarks (OGBN‑ARXIV and OGBN‑PRODUCTS), and the impact of multi‑GPU parallelism—under **A1** in our response to **Reviewer AGYb**. We respectfully refer the reviewer to this answer (**A1**) for full details on how our optimizations mitigate deployment costs.  In a nutshell, under a fixed PDE integration time, reducing the ICNN depth—without significantly degrading performance—yields substantial computational savings, and our framework ensures that large-graph data fully leverages available GPU resources, achieving optimal efficiency on an 8×H100 NVL setup with 94 GB memory (per GPU). We have prepared detailed **time–memory curves** for both training and inference across different datasets and will include them in the final manuscript. \n\nWe also discussed the comparison of the search costs with GraphNAS and AutoGNN as below.\n\n| Method         | Search Strategy                     | Approx. Search Cost       | Remarks                                                       |\n|----------------|-------------------------------------|---------------------------|---------------------------------------------------------------|\n| **Game‑of‑GNN** | Meta‑learned ICNN & Hamiltonian flow | **0 GPU‑hours**           | Single end‑to‑end training—no outer‑loop NAS                  |\n| **GraphNAS**    | RL‑based NAS with parameter sharing | ≈ 7.2 GPU‑hours on 1080Ti  | 1 000 child models × 200 epochs × 0.13 s/epoch                |\n| **Auto‑GNN**    | Evolutionary/gradient NAS           | ≈ 12 GPU‑hours on 1080Ti   | ≈ 0.5 GPU‑days (without parameter sharing) |\n\n- **Game‑of‑GNN** does **not** rely on outer-loop architecture search; all components ($\\Theta_{1},\\Theta_{2}$, Hamiltonian flow) are learned **jointly** in one training run, so the *search cost* is effectively **zero** beyond the standard training time.  The only GPU cost you incur is the **standard training and inference** of our Game‑of‑GNN model, with **no additional search overhead**.\n\n**Q2: Strong theoretical assumptions**\n\n**A2:** Thank you for highlighting these important assumptions regarding the positivity and smoothness of $\\Theta_{1},\\Theta_{2}$ and the positivity of $\\Psi$. We acknowledge that real-world graphs often involve discrete or sparse features, isolated nodes, and heterogeneous connectivity patterns that may not strictly satisfy these theoretical smoothness and positivity assumptions. However, in practice, our implemented framework addresses these real-world conditions as follows:\n- 1. **Positivity and Stability**  \nNode features ($\\Psi$)  in our experimental setup undergo a normalization step ensuring numerical stability and positivity, even when the initial data are discrete or sparse.\n- 2. **ICNN Constraints**  \n   $\\Theta_1$ and $\\Theta_2$ are learned via ICNNs, which enforce positivity and smoothness by design.\n- 3. **Robustness to Irregularity**  \n   Our Hamiltonian PDE framework naturally accommodates isolated nodes and graph irregularities. Specifically, even in cases without immediate neighborhood connectivity, the self-reaction terms ($\\Theta_2$) enable stable and informative node updates.\n- 4. **Empirical Validation**\n    Our extensive experiments across diverse scenarios—including small-to-medium-scale datasets, large-scale OGB benchmarks, graph classification tasks (TUDataset), and human brain connectomes—consistently validate that our method remains robust and effective in practice, despite minor violations of theoretical assumptions.\n\nWe will provide this practical interpretation in the final version (Sec. 2.4 and Sec. 4).\n\n**Q3:Convergence Guarantee** \n\n**A3:** Thank you for raising this important point. \n- Hamiltonian Energy Conservation:\n   * In classical Hamiltonian systems, the Hamiltonian energy remains strictly conserved. But, in our framework, the Hamiltonian energy is not strictly conserved during training, as $\\Theta_1$ and $\\Theta_2$ are learned via ICNN to minimize the task-specific energy functional (Eq. 7). This controlled dissipation is consistent with our variational formulation and supports convergence toward optimal embeddings.\n- Numerical Stability Analysis and Soft-Constraint Implementation (Sec. 2.4):\n    * ICNN-induced Stability: The ICNN architecture inherently ensures convexity and positivity constraints for $\\Theta_1$, $\\Theta_2$, significantly stabilizing the training process. Convexity ensures that learned transport and reaction operators remain numerically stable and well-behaved, avoiding potential instabilities common in unconstrained neural PDE approaches. \n    * Soft-Constraint Implementation Details: 1) We parameterize ICNN outputs via smooth activation functions (e.g., ELU+softplus) to explicitly enforce positivity and differentiability, thus avoiding abrupt changes or negative values. 2) Fixed integration time and stable PDE solvers to prevent divergence.\n\nEmpirically, we consistently observed stable convergence behavior across all datasets (including large-scale benchmarks such as OGB graphs). We will include this clarification in the final version.\n\n**Q4: Dynamic Graph/Time - Varying Attributes**\n\n**A4:** You’re correct that our current formulation assumes a *static* graph $\\mathcal{G}$.  For truly *dynamic* graphs (where edges or node features evolve over time), one can naturally extend our framework by making $\\Theta_{1}$ and $\\Theta_{2}$ **time‑dependent**:\n\n- 1. **Augment ICNN Inputs with Time**  \n   - Instead of $\\Theta_{i} = f_{\\rm ICNN}(\\Psi,\\Phi)$, we can use $ \\Theta_{i}(v,t) = f_{\\rm ICNN}\\bigl(\\Psi(v,t),\\Phi(v,t)\\bigr)$, so the learned transport/reaction strengths can adapt as the graph topology or attributes change.\n- 2. **Time‐Dependent PDE on a Dynamic Graph** \n   - Let the graph $\\mathcal{G}(t)$ at time $t$ have adjacency $A(t)$ and Laplacian $\\Delta(t)$.  \n   - The density $\\Psi(v,t)$, momentum $\\Phi(v,t)$, and mobilities $\\Theta_{i}$ all vary with $t$.  \n- 3. **Practical Implementation**  \n   - **Discretize** time into snapshots $t_{0},t_{1},\\dots$, and re-compute $\\Theta_{i}$ and the Hamiltonian flow at each.  \n   - Or adopt a **neural ODE** style integrator over continuous $t$, feeding a time embedding into the ICNN.\n\nBy explicitly conditioning $\\Theta_{1}$ and $\\Theta_{2}$ on time (and on the current graph state), our MFG controller can handle **dynamic topologies** and **time‑varying attributes** without any fundamental change to the variational or Hamiltonian framework.  \n\n**Q5: Hyperparameter sensitivity** \n\n**A5:** Thanks. We have now performed a full ICNN‐depth ablation (2,4,8,16,32,64) across all nine datasets and report the corresponding accuracy and the resulting $\\alpha$ for each depth. Below is a concise summary of our findings:\n| Depth  | Texas (0.11)      | Wisconsin (0.21)  | Squirrel (0.22)   | Actor (0.22)      | Chameleon (0.23)   | Cornell (0.30)    | Citeseer (0.57)   | Pubmed (0.74)     | Cora (0.81)       |\n|:-----------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:------------------:|:-----------------:|:-----------------:|:-----------------:|:----------------:|\n| 2           | 86.23 ± 3.2       | 88.22 ± 4.3       | 37.98 ± 0.7       | 32.82 ± 0.27      | 57.77 ± 1.5        | 85.76 ± 5.6       | 75.17 ± 8.6       | 86.33 ± 1.0       | 87.53 ± 0.7      |\n| 4           | 84.40 ± 11.8      | 88.44 ± 5.5       | 39.32 ± 0.7       | 32.53 ± 0.40      | 61.26 ± 1.8        | 83.59 ± 2.2       | 77.61 ± 1.4       | 90.75 ± 0.3       | 87.80 ± 0.5      |\n| 8           | 83.79 ± 11.3      | 84.44 ± 8.3       | 38.18 ± 1.2       | 36.28 ± 1.4       | 62.48 ± 15.5       | 83.93 ± 4.5       | 77.96 ± 4.1       | 82.96 ± 1.3       | 87.09 ± 0.8      |\n| 16          | 85.32 ± 3.7       | 87.17 ± 1.8       | 40.38 ± 0.6       | 36.69 ± 6.2       | 60.14 ± 0.2        | 83.32 ± 3.7       | 73.82 ± 6.9       | 84.59 ± 1.4       | 87.21 ± 0.7      |\n| 32          | 90.12 ± 5.6       | 88.44 ± 4.2       | 41.12 ± 0.9       | 34.84 ± 5.4       | 63.85 ± 0.9        | 83.01 ± 6.7       | 73.12 ± 6.4       | 87.40 ± 0.5       | 86.04 ± 0.9      |\n| 64          | 94.00 ± 2.2       | 90.66 ± 4.1       | 37.39 ± 0.5       | 33.55 ± 4.6       | 59.43 ± 15.2       | 81.65 ± 11.1      | 65.46 ± 5.6       | 83.03 ± 1.0       | 85.10 ± 0.6      |\n| $\\alpha$ |       |       |       |       |       |       |       |       |       |\n| 2        | 1.13  | 0.89  | 0.87  | 0.84  | 0.81  | 0.70  | 0.46  | 0.13  | 0.12  |\n| 4        | 1.15  | 1.05  | 0.95  | 0.88  | 0.76  | 0.67  | 0.36  | 0.13  | 0.11  |\n| 8        | 1.07  | 1.02  | 0.94  | 0.90  | 0.74  | 0.69  | 0.50  | 0.14  | 0.11  |\n| 16       | 1.16  | 1.04  | 0.95  | 0.90  | 0.79  | 0.65  | 0.42  | 0.13  | 0.11  |\n| 32       | 1.06  | 0.93  | 0.88  | 0.88  | 0.83  | 0.68  | 0.38  | 0.14  | 0.11  |\n| 64       | 1.26  | 1.14  | 1.11  | 0.90  | 0.70  | 0.66  | 0.45  | 0.14  | 0.12  |\n\n\n**Key takeaways:**\n-  $h$: Smaller values of $h$ may require more ICNN layers to adequately capture and model the underlying data. One plausible explanation for this phenomenon is that heterophilic graph data exhibits a more complex relationship between edges compared to homophilic graph data. This complexity necessitates deeper neural network architectures to effectively learn and represent the nuanced relationships present in the data. \n-  $\\alpha$:  The value of $\\alpha$ is primarily related to the homophily ratio $h$, and the depth of ICNN does not alter the relationship between $a\\sim h$.\n\nWe will add this discussion to the final version, thanks."}, {"review_id": "mmzcb1gyfU", "reviewer": "Reviewer_RzFj", "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.", "questions": "1. The acronym for Reaction Diffusion Model (RDM) is used in section 2.1.2 without spelling out the full name.\n\n2. 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?\n\n3. 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", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength\n\n1. The formulation of GNN dynamic learning process as a mean field game is novel and interesting\n\nWeakness\n\n1. The notations can be introduced more clearly to avoid misunderstanding.\n\n2. The paper is math heavy and many concepts are not introduced but are taken for granted. This makes the paper hard to follow.\n\n3. Transitioning from theoretical framework to algorithm, the assumptions underlying are not clearly stated and discussed\n\n4. The implementation of the algorithm is not very clearly presented. In addition the Algorithm 10 (line 287) is missing in the main paper.", "was_revised": true, "final_justification": "The author has responded to my questions and concerns. The methodological work of the paper is solid and would be a good contribution to the field.", "rebuttal": "**We sincerely thank the reviewer for recognizing the novelty and interest of our formulation, which frames the GNN learning dynamics as a mean field game. We greatly appreciate your constructive feedback and will address each of your concerns in detail below.**\n\n**Q1: The notations can be introduced more clearly to avoid misunderstanding. The paper is math heavy and many concepts are not introduced but are taken for granted. This makes the paper hard to follow. Transitioning from theoretical framework to algorithm, the assumptions underlying are not clearly stated and discussed**\n\n**A1:** We agree that clearer notation, better exposition of mathematical concepts, and explicit statement of assumptions will make the paper more accessible. To respond this comment, we have applied the following changes in the final version:\n\n- 1. **Notation Table & Glossary**  \n   - Added **Table A1** (“Notation and Symbols”) in Appendix A, listing all main variables ($\\Psi,\\Phi,\\psi_{1},\\psi_{2},\\Theta_{1},\\Theta_{2}$, $G$, $\\Delta$, etc.) with one‑line definitions.\n- 2. **Gentler Theoretical Walk‑through**  \n   - Inserted a **“Key Ideas”** paragraph at the top of Section 2, whcih summarizes the three core ingredients: (1)variational RDM, (2) mean‑field game formulation, (3) Hamiltonian flow before diving into equations.  For example:  \n\n        * **Reaction‑Diffusion Model (RDM):** Imagine each node’s features as a fluid that can both spread out to neighbors (diffusion) and be generated or absorbed at the node itself (reaction). This gives a flexible way to describe how information moves and changes on the graph.\n        * **Mean‑Field Game (MFG) Formulation:** Each node acts like a player in a large game: it chooses how much to blend with neighbors versus stay true to its own features, balancing “transport” and “reaction” to minimize a global cost. The equilibrium of this game determines the final node embeddings. \n        * **Hamiltonian Flow:** To solve the MFG, we translate it into a physics‑inspired energy system (the Hamiltonian). Nodes then follow energy‑conserving dynamics—like particles moving under conservative forces—so that information propagation remains stable and interpretable. Together, these ingredients let us learn **how** and **when** to spread versus preserve information on any graph, all within a principled, physics‑inspired framework. \n   - Expanded Remarks 1–3 with concrete examples (e.g.the heat equation instantiation) so that each new concept is motivated and intuitively tied back to standard GNN message passing.\n- 3. **Added explicit Assumptions Before Algorithm**  \n   - Added a bullet‑list **“Model Assumptions”** just before Algorithm 1 in Section 2.4, stating:  \n         - **Smooth, positive density**: $\\Psi(v)>0$.  \n         - **Convexity & positivity** of $\\Theta_{i}$ (enforced via ICNN).  \n         - **Well‑posed PDE solver** (sufficiently small $\\Delta t$).  \n   - For each assumption, we briefly discuss its role and how it is enforced in practice (e.g.ELU+weight decay ensure smoothness).\n\nThese changes improve readability, bridge the theory–algorithm gap, and make the underlying assumptions crystal‑clear.  Thank you for your suggestion.\n\n**Q2: The implementation of the algorithm is not very clearly presented. In addition the Algorithm 10 (line 287) is missing in the main paper.**\n\n**A2: ** Sorry for the confusion, The reference to “Algorithm 10” is a typo—it should be **Algorithm 1** (Appendix A.6 on page 16) . We will move Algorithm 1 into the main text (Section 2.4) and update all in‑text references.  \n\n**Q3: The acronym for Reaction Diffusion Model (RDM) is used in section 2.1.2 without spelling out the full name.** \n\n**A3:** Sorry for the confusion, the full name is mentioned in Sec. 2.1.2-Remark 3 (line 141), we have updated Sec.  2.1.2 to spell out the acronym at first mention—changing the subtitle to **2.1.2 Variational framework of Mean‑Field Game constrained by Reaction‑Diffusion Model (RDM)**\n\n**Q4: Could the authors provide more explanations on how the Lyapunov functional is defined, specifically the integral over G with respect to $v \\in \\mathcal{V}$ , which is a finite set of nodes? Or is this a notational error?** \n\n**A4:** Certainly.  To keep the compact “$\\int$” notation while remaining mathematically precise, we treat the finite node set $\\mathcal{V}$ as a **measure space** endowed with the **counting measure** $\\mu$, where $\\mu(\\{v\\})=1$ for each $v\\in \\mathcal{V}$.  Specifically, we define:\n $\n    \\int_{\\mathcal{V}} f(v)\\,\\mathrm{d}v \n    =\n    \\int_{\\mathcal{V}} f(v)\\,\\mathrm{d}\\mu(v)\n    =\n    \\sum_{v\\in \\mathcal{V}} f(v).\n  $\n\n- Under this convention, all of our “integrals” over $\\mathcal{V}$ or $\\mathcal{G}$ are understood as sums over nodes. In the revised manuscript we will add a footnote or brief remark in Section 2.1.2:\n\n> *“Here and throughout, the notation $\\int_{\\mathcal{V}} f(v)\\,dv$ denotes integration against the **counting measure** on the finite node set $\\mathcal{V}$, i.e. $\\int_{\\mathcal{V}}f(v)dv = \\sum_{v\\in \\mathcal{V}}f(v)$.”*  \n\nThis preserves the unified “integral” style while making clear its discrete meaning.\n\n**Q5: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?**\n\n**A5:** We would like to clarify that there is no missing logic—both mobility learning and feature evolution happen **globally**, not in isolation per node (Fig.1 is more clear). To avoid confusion, we have removed  \"For{$i = 1 \\ldots |\\mathcal{V}|$}” loop and instead write: *\"Construct phase space by $(\\Psi_i, \\Phi_i) \\gets \\mathcal{F}(x_i(t))$ for all nodes in one shot.\"* \n\nIn practice: \n- (1) $\\Theta_{1},\\Theta_{2} \\gets \\mathcal{O}(\\Psi,\\Phi)$ is computed in one batch over **all** nodes (so each node’s output sees its neighbors’ $\\Psi,\\Phi$ indirectly through shared ICNN weights and graph‑aware inputs). \n- (2)  **PDE solver**:   Evolving $(\\Psi,\\Phi)$ with Eq.​ (9) uses the graph Laplacian/divergence operators, which **explicitly mix** each node with its neighbors. Thus, the algorithm implements the full node‑to‑node information exchange as described in the theory: first you embed all nodes into the joint phase space, then you learn global mobilities, then you solve the global PDE.  \n\nWe have amended and will incorporate all the suggestions in the final version. Thank you so much."}]

Author rebuttal:

Meta-review: This work frames GNN learning as a mean-field game (MFG), where nodes act as interacting agents. The authors propose a variational framework of the MFG inverse problem that adaptively learns task-specific message passing through transportation and reaction functions, providing an in-context selective mechanism via Hamiltonian flows. This unifies existing GNNs as special cases with different equilibria. Building on this, they introduce Game-of-GNN, an end-to-end model that discovers message passing operators while tuning hyperparameters, achieving state-of-the-art results on benchmarks and human connectomes, while offering new theoretical insights into GNNs as interactive dynamical systems.

While there are some potential issues—such as strong theoretical assumptions and occasionally unclear notation—reviewers agree that framing the GNN learning process as a mean-field game is both novel and valuable. The experiments presented in the paper and clarified in the rebuttal are comprehensive and convincing, and the theoretical guarantees are sound. Given that all reviewers have unanimously recommended acceptance, I also recommend acceptance of this paper.
Decision: Accept (poster)