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), 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-GNN, to jointly identify the message passing mechanism and fine-tune the GNN hyper-parameters on top of the elucidated message passing operators. 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.

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.

S1.1 Step-by-step derivation of min-max optimization in Section 2.2.1 By substituting Eq. 2 into Eq. 1 in the main manuscript, we can obtain the objective function of subscript z (we temporarily drop ifor clarity): J(z) = max Since z might be in high dimensional space, solving such a large system of linear equations under the constraint |z| 1is oftentimes computationally challenging. In order to find a practical solution for z that satisfies the constrained minimization problem in Eq. By setting zl as point of coincidence, we can find a separable majorizer of M(z) by adding the non-negative function (z zl) (βI Gx Gx)(z zl) (S6) 37th Conference on Neural Information Processing Systems (NeurIPS 2023). Note, to unify the format, we use the matrix transpose property in Eq. Then, the next step is to find z RN that minimizes z z 2bz subject to the constraint |z| 1. Let's first consider the simplest case where z is a scalar: argmin If b 1, then the solution is z = b.

Graphs are ubiquitous in various domains, such as social networks and biological systems. Despite the great successes of graph neural networks (GNNs) in modeling and analyzing complex graph data, the inductive bias of locality assumption, which involves exchanging information only within neighboring connected nodes, restricts GNNs in capturing long-range dependencies and global patterns in graphs. Inspired by the classic Brachistochrone problem, we seek how to devise a new inductive bias for cutting-edge graph application and present a general framework through the lens of variational analysis. The backbone of our framework is a two-way mapping between the discrete GNN model and continuous diffusion functional, which allows us to design application-specific objective function in the continuous domain and engineer discrete deep model with mathematical guarantees. First, we address over-smoothing in current GNNs.

Despite enormous successful applications of graph neural networks (GNNs), theoretical understanding of their generalization ability, especially for node-level tasks where data are not independent and identically-distributed (IID), has been sparse. The theoretical investigation of the generalization performance is beneficial for understanding fundamental issues (such as fairness) of GNN models and designing better learning methods. In this paper, we present a novel PAC-Bayesian analysis for GNNs under a non-IID semi-supervised learning setup. Moreover, we analyze the generalization performances on different subgroups of unlabeled nodes, which allows us to further study an accuracy-(dis)parity-style (un)fairness of GNNs from a theoretical perspective. Under reasonable assumptions, we demonstrate that the distance between a test subgroup and the training set can be a key factor affecting the GNN performance on that subgroup, which calls special attention to the training node selection for fair learning. Experiments across multiple GNN models and datasets support our theoretical results4.

Graph neural networks (GNNs) are recognized for their strong performance across various applications, with the backpropagation (BP) algorithm playing a central role in the development of most GNN models. However, despite its effectiveness, BP has limitations that challenge its biological plausibility and affect the efficiency, scalability and parallelism of training neural networks for graph-based tasks. While several non-backpropagation (non-BP) training algorithms, such as the direct feedback alignment (DFA), have been successfully applied to fully-connected and convolutional network components for handling Euclidean data, directly adapting these non-BP frameworks to manage non-Euclidean graph data in GNN models presents significant challenges. These challenges primarily arise from the violation of the independent and identically distributed (i.i.d.) assumption in graph data and the difficulty in accessing prediction errors for all samples (nodes) within the graph. To overcome these obstacles, in this paper we propose DFA-GNN, a novel forward learning framework tailored for GNNs with a case study of semi-supervised learning.

GNNs to mitigate the bias as well as boost their utility in an end-to-end manner.

Since the PDE in Eq. 5 in the main manuscript is equivalent to the E-L equation of the quadratic