Title: Re-Think and Re-Design Graph Neural Networks in Spaces of Continuous Graph Diffusion Functionals
Venue: NeurIPS 2023
Accepted: yes
Keywords: graph neural networks (GNNs)

total variation (TV)

Euler–Lagrange equation

calculus of variations

over-smoothing

min-max optimization
Abstract: 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. Specifically, our inference reveals that the existing layer-by-layer models of graph embedding learning are equivalent to a ${\ell _2}$-norm integral functional of graph gradients, which is the underlying cause of the over-smoothing problem. Similar to edge-preserving filters in image denoising, we introduce the total variation (TV) to promote alignment of the graph diffusion pattern with the global information present in community topologies. On top of this, we devise a new selective mechanism for inductive bias that can be easily integrated into existing GNNs and effectively address the trade-off between model depth and over-smoothing. Second, we devise a novel generative adversarial network (GAN) to predict the spreading flows in the graph through a neural transport equation. To avoid the potential issue of vanishing flows, we tailor the objective function to minimize the transportation within each community while maximizing the inter-community flows. Our new GNN models achieve state-of-the-art (SOTA) performance on graph learning benchmarks such as Cora, Citeseer, and Pubmed.

Re-Think and Re-Design Graph Neural Networks in Spaces of Continuous Graph Diffusion Functionals

Abstract

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. Specifically, our inference reveals that the existing layer-by-layer models of graph embedding learning are equivalent to a ℓ 2 -norm integral functional of graph gradients, which is the underlying cause of the over-smoothing problem. Similar to edge-preserving filters in image denoising, we introduce the total variation (TV) to promote alignment of the graph diffusion pattern with the global information present in community topologies. On top of this, we devise a new selective mechanism for inductive bias that can be easily integrated into existing GNNs and effectively address the trade-off between model depth and over-smoothing. Second , we devise a novel generative adversarial network (GAN) to predict the spreading flows in the graph through a neural transport equation. To avoid the potential issue of vanishing flows, we tailor the objective function to minimize the transportation within each community while maximizing the inter-community flows. Our new GNN models achieve state-of-the-art (SOTA) performance on graph learning benchmarks such as Cora , Citeseer , and Pubmed .

1 Introduction

Graph is a fundamental data structure that arises in various domains, including social network analysis [42], natural language processing [36], computer vision [30], recommender systems [37], and knowledge graphs [21] among others. Tremendous efforts have been made to operate machine learning on graph data (called graph neural networks, or GNNs) at the node [23], link [41], and graph level [27]. The common inductive bias used in GNNs is the homophily assumption that nodes that are connected in a graph are more likely to have similar features or labels. In this context, most GNN models deploy a collection of fully-connected layers to progressively learn graph embeddings by aggregating the nodal feature representations from its topologically-connected neighbors throughout the graph [19].

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Under the hood of GNNs, the graph representation learning process is achieved by various learnable operations, such as message passing [5] or graph convolution [23]. Due to the nature of exchanging information in a local graph neighborhood, however, it is challenging to capture global graph representations, which go beyond node-to-node relationship, by leveraging the deep architecture in GNNs while being free of overly smoothing the feature representations for the closely-connected nodes. Fig. 1 demonstrates the root cause of over-smoothing issue in current GNNs, where node color denotes the group label (no color means unlabeled) and edge thickness indicates connection strength. It is clear that nodes #1 and #2 are located at the boundary of two communities. The inductive bias of GNNs (i.e., locality assumption) enforces the node embedding vectors on node #1 and #2 becoming similar due to being strongly connected (highlighted in red), even though the insight of global topology suggests that their node embeddings should be distinct. As additional layers are added to GNNs, the node embeddings become capable of capturing global feature representations that underlie the entire graph topology. However, this comes at the cost of over-smoothing node embeddings across graph nodes due to (1) an increased number of node-to-node information exchanges, and (2) a greater degree of common topology within larger graph neighborhoods. In this regard, current GNNs only deploy a few layers (typically two or three) [26], which might be insufficient to characterize the complex feature representations on the graph.

It is evident that mitigating the over-smoothing problem in GNNs will enable training deeper models. From a network architecture perspective, skip connections [16; 39], residual connections [25; 18], and graph attention mechanisms [34; 33] have been proposed to alleviate the information loss in GNNs, by either preserving the local feature representation or making information exchange adaptive to the importance of nodes in the graph. Although these techniques are effective to patch the over-smoothing issue in some applications, the lack of an in-depth understanding of the root cause of the problem poses the challenge of finding a generalized solution that can be scaled up to current graph learning applications.

Inspired by the success of neural ordinary differential equations in computer vision [10], research focus has recently shifted to link the discrete model in GNNs with partial differential equation (PDE) based numerical recipes [38; 29; 6; 14]. For example, graph neural diffusion (GRAND) formulates GNNs as a continuous diffusion process [6]. In their framework, the layer structure of GNNs corresponds to a specific discretization choice of temporal operators. Since PDE-based model does not revolutionize the underlying inductive bias in current GNNs, it is still unable to prevent the excessive information change between adjacent nodes as in nodes #1 and #2 in Fig. 1. In this regard, using more advanced PDE solvers only can provide marginal improvements in terms of numerical stability over the corresponding discrete GNN models, while the additional computational cost, even in the feed-forward scenario, could limit the practical applicability of PDE-based methods for large-scale graph learning tasks.

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In this regard, pioneering work on continuous approaches has prompted to re-think GNN as a graph diffusion process governed by the Euler-Lagrange (E-L) equation of the heat kernel. This formulation is reminiscent of the Brachistochrone problem 2 , which emerged over 400 years ago and established the mathematical framework of classical mechanics. The powerful calculus of variations allows us to generate solutions for various mechanics questions (e.g., the slope that yields the fastest ball sliding down the curve is given by a cycloid) through the lens of E-L equation, as shown in Fig. 2 (top).

In a similar vein, the question that arises in the context of community detection is: What graph diffusion pattern is best suited for preserving community organizations? The question for graph classification would be: What graph diffusion pattern works best for capturing the system-level characteristics of graph topology? Following the spirit of Brachistochrone problem, we present a general research framework to customize application-specific GNNs in a continuous space of graph diffusion functionals. As shown in Fig. 2 (bottom), we have established a fundamental structure for our framework that involves a two-way mapping between a discrete GNN model and a continuous graph diffusion functional. This allows us to develop application-specific objective functions (with an explainable regularization term) in the continuous domain and construct a discrete deep model with mathematical guarantee. We demonstrate two novel GNN models, one for addressing over-smoothing and one for predicting the flows from longitudinal nodal features, both achieving state-of-the-art performance ( Cora : 85.6%, Citeseer : 73.9%, Pubmed : 80.10%, even in 128 network layers).

We have made four major contributions. (1) We establish a connection between the discrete model of GNNs and the continuous functional of inductive bias in graph learning by using the E-L equation as a stepping stone to bridge the discrete and continuous domains. (2) We introduce a general framework to re-think and re-design new GNNs that is less 'black-box'. (3) We devise a novel selective mechanism upon inductive bias to address the over-smoothing issue in current GNNs and achieve state-of-the-art performance on graph learning benchmarks. (4) We construct a novel GNN in the form of a generative adversarial network (GAN) to predict the flow dynamics in the graph by a neural transport equation.

2 Methods

In the following, we first elucidate the relationship between GNN, PDE, and calculus of variations (COV), which sets the stage for the GNN-PDE-COV framework for new GNN models in Section 2.2.

2.1 Re-think GNNs: Connecting dots across graph neural networks, graph diffusion process, Euler-Lagrange equation, and Lagrangian mechanics

Graph diffusion process. Given graph data G = ( V, W ) with N nodes V = { v i | i = 1 , . . . , N } , the adjacency matrix W = [ w ij ] N i,j =1 ∈ R N × N describes connectivity strength between any two nodes. For each node v i , we have a graph embedding vector x i ∈ R m . In the context of graph topology, the graph gradient ( ∇ G x ) ij = w ij ( x i -x j ) indicates the feature difference between v i and v j weighted

by the connectivity strength w ij , where ∇ G is a R N → R N × N operator. Thus, the graph diffusion process can be formulated as ∂x ( t ) ∂t = div ( ∇ G x ( t )) , where the evolution of embedding vectors x = [ x i ] N i =1 is steered by the graph divergence operator.

Connecting GNN to graph diffusion. In the regime of GNN, the regularization in the loss function often measures the smoothness of embeddings x over the graph by x T ∆ x , where ∆ = div ( ∇ G ) is the graph Laplacian operator [23]. To that end, the graph smoothness penalty encourages two connected nodes to have similar embeddings by information exchange in each GNN layer. Specifically, the new graph embedding x l in the l th layer is essentially a linear combination of the graph embedding x l -1 in the previous layer, i.e., x l = A W, Θ x l -1 , where the matrix A depends on graph adjacency matrix W and trainable GNN parameter Θ . After rewriting x l = Ax l -1 into x l -x l -1 = ( A -I ) x l -1 , updating graph embeddings x in GNN falls into a discrete graph diffusion process, where the time parameter t acts as a continuous analog of the layers in the spirit of Neural ODEs [10]. It has been shown in [6] that running the graph diffusion process for multiple iterations is equivalent to applying a GNN layer multiple times.

GNN is a discrete model of Lagrangian mechanics via E-L equation. The diffusion process ∂x ( t ) ∂t = div ( ∇ G x ( t )) has been heavily studied in image processing for decades ago, which is the E-L equation of the functional min x ∫ Ω |∇ x | 2 dx . By replacing the 1D gradient operator defined in the Euclidean space Ω with the graph gradient ( ∇ G x ) ij , it is straightforward to find that the equation governing the graph diffusion process ∂x ( t ) ∂t = div ( ∇ G x ( t )) is the E-L equation of the functional min x ∫ G |∇ G x | 2 dx over the graph topology. Since the heat kernel diffusion is essentially the mathematical description of the inductive bias in current GNNs [6; 14], we have established a mapping between the mechanics of GNN models and the functional of graph diffusion patterns in a continuous domain.

Tracing the smoking gun of over-smoothing in GNNs. In Fig. 1, we observed that the inductive bias of link-wise propagation is the major reason for excessive information exchange, which is attributed to the over-smoothing problem in GNNs. An intuitive approach is to align the diffusion process with high-level properties associated with graph topology, such as network communities. After connecting the GNN inductive bias to the functional of graph diffusion process, we postulate that the root cause of over-smoothing is the isotropic regularization mechanism encoded by the ℓ 2 -norm. More importantly, connecting GNN to the calculus of variations offers a more principled way to design new deep models with mathematics guarantees and model mechanistic explainability.

2.2 Re-design GNNs: Revolutionize inductive bias, derive new E-L equation, and construct deeper GNN

The general roadmap for re-designing GNNs typically involves three major steps: (1) formulating inductive bias into the functional of graph diffusion patterns; (2) deriving the corresponding E-L equation; and then (3) developing a new deep model of GNN based on the finite difference solution of E-L equation. Since the graph diffusion functional is application-specific, we demonstrate the construction of new GNN models in the following two graph learning applications.

2.2.1 Develop VERY deep GNNs with a selective mechanism for link-adaptive inductive bias

Problem formulation. Taking the feature learning component (learnable parameters Θ ) out of GNNs, the graph embeddings x L (output of an L -layer GNN) can be regarded as the output of an iterative smoothing process ( L times) underlying the graph topology G , constrained by the data fidelity ∥ ∥ x L -x 0 ∥ ∥ 2 2 (w.r.t. the initial graph embeddings x 0 ) and graph smoothness term ∫ G |∇ G x | 2 dx . Inspired by the great success of total variation (TV) for preserving edges in image denoising [31], reconstruction [35] and restoration [8], we proposed to address the over-smoothing issue in current GNN by replacing the quadratic Laplacian regularizer with TV on graph gradients, i.e., J TV ( x ) = ∫ |∇ G x | dx . Thus, the TV-based objective function for graph diffusion becomes: min x ( ∥ ∥ x -x 0 ∥ ∥ 2 2 + J TV ( x )) .

However, the ℓ 1 -norm function, denoted by | · | in the definition of the total variation functional J TV , is not differentiable at zero. Following the dual-optimization schema [4; 7], we introduce the latent

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auxiliary matrix z ∈ R N × N and reformulate the TV-based functional as min x max z J TV ( x, z ) = max z min x ∫ ( z ⊗ ∇ G x ) dx , subject to | z | ≤ 1 N × N , where ⊗ is Hadamard operation between two matrices. Furthermore, we use an engineering trick of element-wise operation z ij ( ∇ G x ) ij to keep the degree always non-negative (same as we take the absolute value), which makes the problem solvable. In the end, we reformulate the minimization of J TV ( x ) into a dual min-max functional as J TV ( x, z ) , where we maximize z ( z → 1 N × N ) such that J TV ( x, z ) is close enough to J TV ( x ) . Therefore, the new objective function is reformulated as:

where λ is a scalar balancing the data fidelity term and regularization term. Essentially, Eq. 1 is the dual formulation with min-max property for the TV distillation problem [44].

Constructing E-L equations. To solve Eq. 1, we present the following two-step alternating optimization schema. First , the inner minimization problem (solving for x i ) in Eq. 1 can be solved by letting ∂ ∂x i J ( x i , z i ) = 0 :

Replacing ( ∇ G x ) ij with w ij ( x i -x j ) , the intuition of Eq. 2 is that each element in ˆ x i is essentially the combination between the corresponding initial value in x 0 i and the overall graph gradients z i ∇ G x i = ∑ j ∈N i w ij ( x i -x j ) z i within its graph neighborhood N i . In this regard, Eq. 2 characterizes the dynamic information exchange on the graph, which is not only steered by graph topology but also moderated by the attenuation factor z i at each node.

Second , by substituting Eq. 2 back into Eq. 1, the objective function of z i becomes J ( z i ) = max | z i |≤ 1 ∥ ∥ λ 2 z i ∇ G x i ∥ ∥ 2 2 + λz i ∇ G ( x 0 i -λ 2 z i ∇ G x i ) . With simplification (in Eq. S1 to Eq. S3 of Supple- mentary), the optimization of each z i is achieved by arg min | z i |≤ 1 z i ∇ G x i z i ∇ G x i -4 λ z i ∇ G x 0 i .

Specifically, we employ the majorization-minimization (MM) method [15] to optimize z i by solving this constrained minimization problem (the detailed derivation process is given in Eq. S4 to S12 of Section S1.1 of Supplementary), where z i can be iteratively refined by:

β is a hyper-parameter that is required to be no less than the largest eigenvalue of ( ∇ G x i )( ∇ G x i ) ⊺ .

Develop new GNN network architecture with a selective inductive bias. The building block in vanilla GNN [23] is a FC (fully-connected) layer where the input is the embedding vectors after isotropic graph diffusion (in ℓ 2 -norm). Since the estimation of graph embeddings x in Eq. 2 depends on the latest estimation of z ( l ) , such recursive min-max solution for Eq. 1 allows us to devise a new network architecture that disentangles the building block in vanilla GNN into the feature representation learning and graph diffusion underlying TV. As shown in Fig. 3, we first deploy a FC layer to update the graph embeddings x ( l ) . After that, we concatenate a diffusion-gathering ( DG ) layer for selective graph diffusion, which sequentially applies (1) node-adaptive graph diffusion (blue

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Remarks. Eq. 3 indicates that larger connective degree results in larger value of z . Thus, the DG layer shifts the diffusion patterns by penalizing the inter-community information exchange (due to strong connections) while remaining the heat-kernel diffusion within the community. The preference of such link-adaptive diffusion can be adjusted by the hyper-parameter λ 4 in Eq. 1. Recall our intuitive solution for over-smoothing problem in Fig. 1, the DG layer offers the exact global insight of graph topology to keep the node embeddings distinct between nodes #1 and #2. We demonstrate the effect of DG layer on the real-world graph data in Fig. S3 of Supplementary document.

2.2.2 Predict flow dynamics through graph neural transport equation

Problem formulation. We live in a world of complex systems, where everything is intricately connected in multiple ways. A holistic insight of how the system's components interact with each other and how changes in one part of the system can affect the behavior of the whole sheds new light on the dynamic behaviors of these complex systems over time. However, oftentimes it is an ill-posed problem. Taking the toy system in Fig. 4(a) as an example, while it is simple to calculate the future focal patterns based on the focal patterns at the current time point and the node-to-node flow information, determining flow dynamics based on longitudinal nodal observations is computationally hard since the solution is not unique.

The naïve solution to predict the spreading flow is to (1) train a GNN to learn the intrinsic node embeddings and (2) predict the flow based on the difference of learned embeddings. However, this two-step approach might suffer from vanishing flow due to over-smoothing in GNNs. Following the spirit of Brachistochrone problem, we ask the question "What flow field f ( t ) = [ f ij ( t )] N i,j =1 underlines the system mechanics to the extent that it is able to predict the behaviors in the future?"

In this paper, we focus on the conservative system of energy transportation [2]. The system mechanics is formulated as:

where q = [ q ij ] N i,j =1 is the flux field which propagates the potential energy u ( t ) = [ u i ( t )] N i =1 (conserved quantity) over time. Similar to a gravity field driving water flow, the intuition of Eq. 4 is that the change of energy density u (we assume there is a non-linear mapping ϕ from external force x to u , i.e., u i = ϕ ( x i ) ) leads to energy transport throughout the entire graph. As flux is closely related to the difference of energy ∇ G u underlying the graph topology, we assume the energy flux q is regulated by the potential energy field ∇ G u , i.e., q = α ⊗∇ G u , where α = [ α ij ] N i,j =1 is a learnable matrix characterizing the link-wise contribution of each energy potential ∇ G u ij to the potential energy flux q ij .

By plugging q = α ⊗∇ G u into Eq. 4, the energy transport process can be reformulated as:

where ∆ = div ( ∇ G ) is the graph Laplacian operator. Since the PDE in Eq. 5 is equivalent to the E-L equation of the quadratic functional J ( u ) = min u ∫ G α ⊗ |∇ G u | 2 du (after taking ϕ

away), a major issue is the over-smoothness in u that might result in vanishing flows. In this context, we propose to replace the ℓ 2 -norm integral functional J ( u ) with TV-based counterpart J TV ( u ) = min u ∫ G α ⊗|∇ G u | du .

Renovate new E-L equation - graph neural transport equations. Following the min-max optimization schema in Eq. 1-3, we introduce an auxiliary matrix f to lift the undifferentialable barrier. After that, the minimization of J TV ( u ) boils down into a dual min-max functional J TV ( u, f ) = min u max f ∫ G α ⊗ f ( ∇ G u ) du . Since u ( t ) is a time series, it is difficult to derive the deterministic solutions (as Eq. 2-3) by letting ∂ ∂u J TV = 0 and ∂ ∂f J TV = 0 . Instead, we use G ˆ ateaux variations to optimize J TV ( u, f ) via the following two coupled time-dependent PDEs (please see Section S1.2, Eq. S14 to Eq. S19, in Supplementary for details):

Remarks. The solution to Eq. 6 is known as continuous max-flow and constitutes a continuous version of a graph-cut [1]. Since α is a latent variable and potential energy u is given, the maximization of f opts towards maximizing the spreading flow through the lens of α . In this regard, the mechanistic role of auxiliary matrix f is essentially the latent (maximized) spreading flows that satisfy u ( t +1) i = u ( t ) i + ∑ N j =1 f ij ( t ) . The potential energy ˆ u can be solved via a wave equation ( u tt = div ( f t ) = α 2 ⊗ ∆ u ), where the system dynamics is predominated by the adjusted Lagrangian mechanics α 2 ⊗ ∆ u . By determining α at a granularity of graph links, we devise a novel GAN model to predict the spreading flows f which not only offers explainability underlying the min-max optimization mechanism in Eq. 6 but also sets the stage to understand system dynamics through machine learning. Develop a GAN model of flow prediction with TV-based Lagrangian Mechanics. The overall network architecture is shown in Fig. 4 (b), which consists of a generator (red solid box) and a discriminator module (blue solid box). Specifically, the generator ( G ) consists of (1) a GCN component [14] to optimize ˆ u through the wave equation u tt = α 2 ⊗ ∆ u and (2) a FC layer to characterize the non-linear mapping function ˆ x ( t +1) = ϕ -1 (ˆ u ( t )) . In contrast, the discriminator ( D ) is designed to (1) synthesize α and (2) construct the future ˜ u t +1 based on the current u t and current estimation of spreading flow f = α ⊗ ∇ G u (orange dash box). To make the network architecture consistent between generator and discriminator modules, we include another GCN to map the synthesized ˜ u ( t +1) to the external force ˜ x ( t +1) . Note, since the working mechanism of this adversarial model underlines the min-max optimization in the energy transport equation, the nature of predicted spreading flows is carved by the characteristics of max-flow .

The driving force of our network is to minimize (1) the MSE (mean square error) between the output of the generator ˆ x t +1 and the observed regional features, (2) the distance between the synthesized regional features ˜ x t +1 (from the discriminator) and the output of generator ˆ x t +1 (from the generator). In the spirit of probabilistic GAN [43], we use one loss function L D to train the discriminator ( D ) and another one L G to train the generator ( G ):

where ξ denotes the positive margin and the operator [ · ] + = max(0 , · ) . Minimizing L G is similar to maximizing the second term of L D except for the non-zero gradient when D ( G ( x t )) ≥ ξ .

3 Experiments

In this section, we evaluate the performance of the proposed GNN-PDE-COV framework with comparison to six graph learning benchmark methods on a wide variety of open graph datasets [32], as well as a proof-of-concept application of uncovering the propagation pathway of pathological events in Alzheimer's disease (AD) from the longitudinal neuroimages.

3.1 Datasets and experimental setup

Dataset and benchmark methods. We evaluate the new GNN models derived from our proposed GNN framework in two different applications. First , we use three standard citation networks, namely

Cora , Citeseer , and Pubmed [32] for node classification (the detailed data statistic is shown in Table S2 of Supplementary). We adopt the public fixed split [40] to separate these datasets into training, validation, and test sets. We follow the experimental setup of [9] for a fair comparison with six benchmark GNN models (vanilla GCN [23], GAT [34], GCNII [9], ResGCN [25], DenseGCN [25], GRAND [6]). Since our DG -layer can be seamlessly integrated into existing GNNs as a plug-in. The corresponding new GNN models (with DG -layer) are denoted GCN+, GAT+, GCNII+, ResGCN+, DenseGCN+, and GRAND+, respectively.

Second , we apply the GAN model in Section 2.2.2 to predict the concentration level of AD-related pathological burdens and their spreading pathways from longitudinal neuroimages. Currently, there is no in-vivo imaging technique that can directly measure the flow of information across brain regions. Here, our computational approach holds great clinical value in understanding the pathophysiological mechanism involved in disease progression [22]. Specifically, we parcellate each brain into 148 cortical surface regions and 12 sub-cortical regions using Destrieux atlas [12]. The wiring topology of these 160 brain regions is measured by diffusion-weighted imaging [3] and tractography techniques [17]. The regional concentration levels AD pathology including amyloid, tau, and fluorodeoxyglucose (FDG) and cortical thickness (CoTh) are measured from PET (positron emission tomography) and MRI (magnetic resonance imaging) scans [20]. We use a total of M = 1 , 291 subjects from ADNI [28], each having longitudinal imaging data (2-5 time points). The details of image statistics and pre-processing are shown in Sec. S2.1.2. Since we apply the flow prediction model to each modality separately, we differentiate them with X-FlowNet ( X stands for amyloid, tau, FGD, and CoTh).

Experimental setup . In the node classification task, we verify the effectiveness and generality of DG -layer in various number of layers ( L = 2 , 4 , 8 , 16 , 32 , 64 , 128 ). All baselines use their default parameter settings. Evaluation metrics include accuracy, precision and F1-score. To validate the performance of X-FlowNet , we examine (1) prediction accuracy (MAE) of follow-up concentration level, (2) prediction of the risk of developing AD using the baseline scan, and (3) understand the propagation mechanism in AD by revealing the node-to-node spreading flows of neuropathologies.

The main results of graph node classification and flow prediction are demonstrated in Section 3.2 and 3.3, respectively. Other supporting results such as ablation study and hyper-parameter setting are shown in Section S2 of the Supplementary document.

3.2 Experimental results on graph node classification

We postulate that by mitigating the over-smoothing issue, we can leverage the depth of GNN models to effectively capture complex feature representations in graph data. As shown in Table 1, we investigate the graph node classification accuracy as we increase the number of GNN layers by six benchmark GNN models and their corresponding plug-in models (indicated by '+' at the end of each GNN model name) with the DG -layer. The results demonstrate that: (1) the new GNN models generated from the GNN-PDE-COV framework have achieved SOTA in Cora (86.30% by GCNII+), Citeseer (75.65% by GRAND+), and Pubmed (80.10 % by GCNII+); (2) all of new GNN models outperforms their original counterparts with significant improvement in accuracy; (3) the new GNN models exhibit less sensitivity to the increase of model depth compared to current GNN models; (4) the new GNN models are also effective in resolving the gradient explosion problem [26] (e.g, the gradient explosion occurs when training GAT on all involved datasets with deeper than 16 layers, while our GAT+ can maintain reasonable learning performance even reaching 128 layers.)

It is important to note that due to the nature of the graph diffusion process, graph embeddings from all GNN models (including ours) will eventually become identical after a sufficiently large number of layers [11]. However, the selective diffusion mechanism (i.e., penalizing excessive diffusion across communities) provided by our GNN-PDE-COV framework allows us to control the diffusion patterns and optimize them for specific graph learning applications.

3.3 Application for uncovering the propagation mechanism of pathological events in AD

First , we evaluate the prediction accuracy between the ground truth and the estimated concentration values by our X-FlowNet and six benchmark GNN methods. The statistics of MAE (mean absolute error) by X-FlowNet , GCN, GAT, GRAND, ResGCN, DenseGCN and GCNII, at different noise levels on the observed concentration levels, are shown in Fig. 5 (a). It is clear that our X-FlowNet consistently outperforms the other GCN-based models in all imaging modalities.

Second , we have evaluated the potential of disease risk prediction, which can be regarded as a graph classification problem. We assume that we have baseline amyloid, tau, FDG, and CoTh scans, and evaluate the prediction accuracy, precision and F1-score of various models in forecasting the risk of developing AD. We consider two dichotomous cases: one included only AD vs. CN groups and the other involved AD/LMCI vs. CN/EMCI. The results of the mean of 5-fold cross-validation are shown in Table 2. Our GNN-PDE-COV model not only achieved the highest diagnostic accuracy but also demonstrated a significant improvement (paired t -test p < 0 . 001 ) in disease risk prediction compared to other methods. These results suggest that our approach holds great clinical value for disease early diagnosis.

Third , we examine the spreading flows of tau aggregates in CN (cognitively normal) and AD groups. As the inward and outward flows shown in Fig. 5(b), it is evident that there are significantly larger amount of tau spreading between sub-cortical regions and entorhinal cortex in CN (early sign of AD onset) while the volume of subcortical-entorhinal tau spreading is greatly reduced in the late stage of AD. This is consistent with current clinical findings that tau pathology starts from sub-cortical regions and then switches to cortical-cortical propagation as disease progresses [24]. However, our Tau-FlowNet offers a fine-granularity brain mapping of region-to-region spreading flows over time, which provides a new window to understand the tau propagation mechanism in AD etiology [13].

Tau-PET

Amyloid-PET

FDG-PET

CoTh-PET

MAE

(a)

MAE

Amygdala

Pallidus

Putamen

Thalamus

Hippocampus

Caudate

1.5391

2.8801

5.5074

2.1690

15.1162

0.8062

15.5130

1.2906

19.2119

0.1282

22.1058

0.1493

total flow volume

(b)

CN/AD

0.0391

1.4532

0.7349

0.6788

1.7045

0.1859

2.1807

0.1152

0.2693

0.3422

0.6108

0.6733

78.99/7.04

7.42/1.96

flow degree

'inward'

'outward'

FDG-FlowNet

GCN

GAT

GRAND

ResGCN

DenseGCN

std=1

std=0.1

std=0.08

std=0.04

std=0.02

no noise

CoTh-FlowNet

GCN

GAT

GRAND

ResGCN

DenseGCN

Tau-FlowNet

GCN

GAT

GRAND

ResGCN

DenseGCN

Amyloid-FlowNet

GCN

GAT

GRAND

ResGCN

DenseGCN

2.5

1

0.5

0.5

0

0

D

一

0.4

0.2

0

Table 2: The performance of disease risk prediction. Note: RGCN denotes ResGCN, DGCN denotes DenseGCN. '*' denotes the significant improvement (paired t -test: p < 0 . 001 ). Blue: Tau, red: amyloid, orange: FDG, green: CoTh.

4 Conclusion

In this work, we present the GNN-PDE-COV framework to re-think and re-design GNN models with great mathematical insight. On top of this, we devise the selective inductive bias to address the over-smoothing problem in GNN and develop new GNN model to predict the pathology flows in-vivo via longitudinal neuroimages. Future work may involve exploring innovative graph regularization techniques and conducting further validation on a broader range of graph-based learning tasks.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on devising a new inductive bias for cutting-edge graph application and present a general framework through the lens of variational analysis. To this end, the authors first introduce a new selective mechanism that can be easily integrated into existing GNNs and effectively address the trade-off between model depth and over-smoothing, and then devise a novel generative adversarial network to predict the spreading flows in the graph through a neural transport equation. Extensive experiments show that the proposed GNN models can achieve state-of-the-art performance on graph learning benchmarks such as Cora, Citeseer, and Pubmed.\n\nSTRENGTHS:\n1. The perspective of the article is innovative, as it creatively utilizes heuristic problems from traditional physics to inspire the design of GNN structures. \n2. The description of the proposed GNN architecture is clear, and the experimental setup is well-defined. \n3. The experimental results validate the superiority of the method proposed by the author and demonstrate the rationality of the proposed theory.\n\nWEAKNESSES:\n1. The majority of the content in the article is based on the Euler-Lagrange (E-L) equation of the heat kernel. However, I don't believe that the detailed knowledge of the EL equation is familiar to every graph neural network researcher. Therefore, I think it is important to briefly introduce it when it first appears (Line 74) or mention it in the appendix. This is crucial for maintaining the readability and coherence of the article. In fact, it took me a lot of time to consult relevant materials for the subsequent introduction of the Euler-Lagrange (E-L) equation to LaGrangian Mechanics.\n2. The issue of over-smoothing in GNNs has been extensively studied since 2020, but the theoretical reasons behind this problem have not been conclusively determined [1, 2, 3]. In line 135-136, the authors claim that after connecting the GNN inductive bias to the function of the graph diffusion process, we can postulate that the root cause of over-smoothing is the isotropic regularization mechanism encoded by the ℓ2-norm. Treating this theory as a conclusion is certainly not a problem. However, due to the fact that many subsequent formula derivations are based on this theory, I have to consider whether this theory has been somewhat hasty and overclaimed. I think the authors should provide some deductions before presenting this theory to maintain logical rigor.\n3. In line 166, the authors claim that Eq. 1 is the dual formulation with min-max property for the TV distillation problem. It is well known that the dual problem is strictly mathematically defined. I do not fully understand how the duality problem here is derived. Please provide further explanation.\n4. In line 185-186, I am confused about how the recursive min-max solution for Eq. 1 is obtained by disentangling the building block in vanilla GNN into the feature representation learning and graph diffusion underling TV. I do not intuitively perceive the connection between the two. Please explain. \n\nreference:\n1. DropEdge: Towards Deep Graph Convolutional Networks on Node Classification. ICLR 2020\n2. Towards Deeper Graph Neural Networks. KDD 2020\n3. Beyond Low-frequency Information in Graph Convolutional Networks. AAAI 2021\n\nQUESTIONS:\nPlease see the weak points.\n\nLIMITATIONS:\nPlease see the weak points.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies a novel design of GNN leveraging the connection with graph diffusion. Specifically, this work considers the discrete GNN model in view of continuous graph diffusion functional formulated as the Euler-Lagrange equation. This work proposes a new design of GNN with selective inductive bias which alleviates the over-smoothing in GNNs. Further, this work proposes a new approach for predicting the flow dynamics in the graph via a neural transport equation using the GAN model to predict the spreading flow.\n\nSTRENGTHS:\n- The paper is well-written with clear motivation and justification. The writing is easy to follow with summarized questions and the paper's approach (re-think and re-design) helps to understand this work.\n\n- The proposed architecture is novel yet simple based on theoretical justifications leveraging the connection with the graph diffusion (although this was studied widely in other papers). Further, the GAN model for flow prediction is also new, using neural transport equations.\n\n- The experimental results show superior performance compared to the baselines in node classification, and the results in node classification demonstrate that the proposed architecture mitigates the over-smoothing issue.\n\nWEAKNESSES:\n- Related work section could greatly help the readers for understanding this work as it deals with new GNN as well as flow prediction in graphs. I assume that the related work section was omitted due to the page limit, and recommend adding it if possible.\n\n- The reason for the superior performance of the GAN model is not clear.\n\nQUESTIONS:\n- What is the purpose of using total variation? Is it simply from the inspiration explained in line 151?\n\n- Although the extra training time is described in section S2.4, comparing it with the training time without DC layer would show that the computational burden is not significant.\n\nLIMITATIONS:\nThe limitations are discussed in the supplementary file section S2.4.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

comments: SUMMARY:
The authors connects graph neural networks with discretizations of PDEs and connects their limiting behavior with E-L equations. By changing regularization / smoothing term from second order to first order, the authors designed a new E-L equation and its corresponding GNN. The author then present numerical experiments to show their improvement in performance.

STRENGTHS:
- The authors tackles the problem of oversmoothing in GNN, which is an essential problem GNN cannot get as deep as other neural networks.
- The authors provided solid mathematical support on their method and great experimental performance.
- It is an original work connecting TV-minimization from the old school machine learning, with the SOTA design of GNNs.

WEAKNESSES:
- The authors should do an table outlining the steps of their algorithm. It will make it easier for audience who prefer testing model before going through the math.
- The infomation in figure 5 is too dense. It would be better if the authors could split it into two figures.

QUESTIONS:
- Equation 3 sounds similar to the MBO scheme, which is basically diffusion + clip. Are there any connections in between?
- Does your model transfer to other types of neural networks like convolutional neural networks?
- Does your model requires more computation time? How much more compared to the origional run time of GCN, GAT, and GRAND?

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a framework based on the Euler-Lagrange equation to derive specialized GNNs by discretizing continuous diffusion functionals. By deriving a new GNN layer from the Total Variation functional, they manage to control the oversmoothing problem in existing GNN architectures and improve node classification performance of six existing architectures with up to 128 layers. Additionally, the authors introduce a new GAN to learn flow problems on graphs and evaluate it on longitudinal data from Alzheimer's disease.\n\nSTRENGTHS:\nThe prominent strengths of this paper are the thorough evaluation and the great results on node classification showing that the proposed DC layer can improve different architectures and enable deeper graph neural networks. The proposed GAN also performs very well on flow prediction in an Alzheimer's disease dataset though I cannot judge the subject-specific interpretation of those results. In the derivation of their method, the authors combine many techniques though it is in parts difficult to follow (see Weaknesses).\n\nWEAKNESSES:\nThe one major weakness of this paper is its (lack of) clarity in writing. Many ideas are insufficiently explained and notation is often inconsistent and confusing. In particular, I refer to the following:\n\n1. Line 90: The numbers for Cora and Citeseer are not in Table 1.\n1. Line 110: Graph divergence operator is never defined and no reference is given.\n1. Line 111-121: Unclear writing that mixes two different ideas. The first two sentences talk about an often-used (citations missing) regularization term and its effect, but the remainder of the paragraph is about interpreting GNN structures as neural ODEs on graphs.\n1. Line 127-130: Drawing the conclusion that you have established this mapping requires either further explanation or at least a citation.\n1. Line 136: Which L2 norm does this refer to? The one in the functional in line 127?\n1. Line 158-159: The definition of $J_{\\mathrm{TV}}$ is completely unclear. $x$ is triple bound (parameter of $J_{\\mathrm{TV}}$, parameter of $\\min$ operator and integrated over) and $z$ is double bound.\n1. Line 160-161: What exactly does this \"trick\" consist of and in which sense is \"degree\" used in this sentence?\n1. Equation (1): Same issue as line 158-159.\n1. Equation (3): Is the clipping elementwise?\n1. Line 190: Why are the $x_i$ being clipped when Equation (3) referes to the $z_i$?\n1. Line 191: How does large degree increase $z_i$? An arbitrary number of edges of identical nodes would leave $z_i$ still unchanged, wouldn't it?\n1. Line 192-193: What does it mean to \"shift the diffusion pattern\"? Which cases of the case distinction in Equation (3) correspond to \"heat-diffusion within community\" and \"penalizing inter-community exchange\"?\n1. Line 211: How is future predictability related to the Brachistochrone problem that asks about the shortest path?\n1. Section 2.2.2: The whole section is difficult to follow because it introduces yet another problem and solution in little space. While I understand the importance of this section as an example for an alternative GNN architecture derived from the proposed method, for the overall clarity of the paper the space might be better used to clarify the main method.\n1. Overall, the importance of the Brachistochrone problem is overemphasized and diverts attention from the main contribution of the paper. It is a nice example application of the E-L equation but even in Figure 1 the analogy between the mechanics and ML case is weak.\n\nThis paper leaves the impression of solid work, though the opaque presentation prevents me from saying so with any certainty. A thorough rewrite of Section 2 with a focus on the reader could make this a great paper.\n\nQUESTIONS:\n1. In how far does GNN-PDE-COV rely on the homophily assumption? Part of the problem illustrated in Figure 1 is that the strong connection between nodes of separate classes breaks the homophily assumption of many GNN models.\n1. How does the runtime of your model scale in the number of nodes $N$ and number of edges $E$?\n\nLIMITATIONS:\nThe appendix mentions briefly limitations of the mode of evaluation. I would be interested in a point on how feasible it is to derive specialized architectures based on the GNN-PDE-COV framework given that the optimization problem in the proposed GAN is completely different from the optimization problem in Section 2.2.1.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors develop a framework for linking discrete (message passing) GNNs to continuous graph diffusion functional networks using Euler-Lagrange equations of heat kernels.  Via this framework, they analyze the causes of oversmoothing in current GNNs. By noting that the main cause of oversmoothing is the minimization of the quadratic graph smoothness term in the diffusion equation, their main contribution is to replace this by a Total Variation (TV) term (as used in image reconstruction, restoration etc.), yielding a new objective function. Further since the $l_1$ norm term of TV is non-differentiable at 0, they develop a dual min-max approach for solving this objective function iteratively by first minimizing for X and then maximizing for Z. This solution technique then results in a new GNN architecture with a separate feature learning layer (Fully Connected layer) for $l^{th}$ layer embeddings $X^l$ followed by a diffusion clip layer for generating the Z terms. They then apply their method for a specific application of predicting spreading flow dynamics. Experimental results are presented on 3 benchmark citation datasets and 6 GNN models including Vanilla GCN and  one other diffusion GCN (GRAND).\n\nTL;DR takeaway of the problem setup/motivation is how to preserve community/contextual embeddings of adjacent (dissimilar) boundary nodes in different labeled communities by preventing oversmoothing through localized message passing. The proposed solution is via the Total Variation parameter Z that should penalize inter community information diffusion.\n\nSTRENGTHS:\n1.\tAvoiding local oversmoothing by replacing the quadratic graph smoothness term in the diffusion equation by an $l_1$ norm regularize which is the TV term. This seems to be a novel application of TV to diffusion GNNs.\n\n2.\tFormal derivation of an iterative dual min-max method for solving the non-differentiable objective function with the TV term. While this type of iterative technique has gained increasing popularity starting with ADMM, the detailed derivation is a good contribution. \n\n3.\tFor small layers (2 layers), their method shows performance improvement over the other baselines. This might be due to ameliorating the oversmoothing process due to the TV Z parameter.\n\nWEAKNESSES:\n1.\tI am surprised the authors do not have references to classic papers that analyze the root causes of oversmoothing in GNNs, for example, the DropEdge paper [1] “Tackling Over-Smoothing for General Graph Convolutional Networks”, W Huang∗ , Yu Rong∗  et al.. IEEE TPAMI Aug. 2015. This paper analyzes oversmoothing by looking at a spectral analysis of the underlying adjacency matrix. It would be interesting to see if there is a connection between the $beta$ parameter and degree connectivity of equation (3) and results from [1].\n\n2.\tThe baseline model comparisons are too limited - vanilla GCN, GAT, one diffusion GNN (GRAND).\n\ta. There have been several papers that either explicitly focus on deep sampling while mitigating oversmoothing e.g., [2] “Decoupling the Depth and Scope of Graph Neural Networks”, H. Zeng, M. Zhang et al. Neurips 21. \n\tb. They test their model on citation networks which are considered highly homophilic. However, dissimilar nodes (as they illustrate in fig. 1) that would ideally prove their oversmoothing claims are adjacent to each other primarily in heterophilic networks. It would be useful if they can show their results on heterophilic graphs.\nThere needs to be an extended comparison with GNN  models  such as [2] as well as others that look at heterophilic graphs e.g., $H_2GCN$ [3] “Beyond Homophily in Graph Neural Networks: Current Limitations and Effective Designs,” J. Zhu, Y. Yan et al, Arxiv. As they claim their method prevents over-smoothing, it would be very valuable to see comparative results  on heterophilic graphs and if they can show improvements over other models.\n\n3.\tExperimental Results:  [3] $H_2GCN$, [4] GeomGCN [5] GPRGNN   seem to show better results. For instance, accuracy of this paper on Pubmed is 80% while H2GCN, GeomGCN, GPRGNN show 90% accuracy. Similar results for Cora and Citeseer . \n[4] Geom-GCN: “Geometric Graph Convolutional Networks. In International Conference on Learning Representations”, Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. ICLR 2020.\n[5] GPRGNN: “Adaptive Universal Generalized PageRank Graph Neural Network”, ICLR2021.\n\n4. There may be some typos in experimental results -  published results are not consistent with published tabular data in the previous works.  E.g. GCNII results for Pubmed are 90% in Table 1 in the following nips 22 paper: https://papers.nips.cc/paper_files/paper/2022/file/75c45fca2aa416ada062b26cc4fb7641-Paper-Conference.pdf but Table 1 in this paper only shows around 79% for GCNII on Pubmed. Similarly please check Table 5 in the following nips 20 paper: https://arxiv.org/pdf/2006.11468.pdf for other discrepancies. I suggest rerunning experiments and checking for typos or otherwise undertanding why numbers are different. In general, the margin of improvement is around 2-4 points over vanilla GCNs for $H_2GCN$ and GPRGNN while the margin is also in that range for this paper, even though they show lower scores for vanilla GNN and GAT. It would definitely improve this paper if you added these papers as baselines for direct comparison.\n\n5. The diffusion flow application that is used in the paper to validate the diffusion model seems to a very niche application. It would greatly add value to the paper if you validated the model over a realistic flow model such as a traffic flow problem.\n\n6. The authors note that their iterative solution technique is not computationally intensive (appendix). However, the method was tried only on 4 limited datasets that are all citation networks. In general, for any iterative solution process the tradeoff between convergence time and accuracy requires more sophisticated and detailed evaluation. Can you improve accuracy on Pubmed (see above) at the cost of increased computation time?\n\nQUESTIONS:\nPlease consider different graph setups to validate your oversmoothing diffusion models such as heterophilic graphs in which dissimilar nodes are close by and similar nodes can be far away.\n\nMore extensive evaluation of tradeoffs between convergence time and accuracy would be helpful, especially to improve accuracy on the 4 datasets as compared to the newer baselines suggested above.\n\nConsider a more realistic diffusion flow application to validate your diffusion model such as a traffic flow problem.\n\nLIMITATIONS:\nThe authors have addressed the limitations of this work in the appendix, primarily the lack of diversity in evaluation datasets. In general, as pointed earlier, paper could be improved with more extensive comparisons on diverse datasets and models.\n\nThe authors have stated that the proposed topic doesn't have any negative societal impacts. To the contrary they state that \"From the application perspective, the new deep model for uncovering the in-vivo propagation flows has great potential to establish new underpinning of disease progression and disentangle the heterogeneity of diverse neurodegeneration trajectories.\" While technical correct, this comment in general applies to any work on sensitive datasets in the medical field.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-27"}