Title: When Do Graph Neural Networks Help with Node Classification? Investigating the Homophily Principle on Node Distinguishability
Venue: NeurIPS 2023
Accepted: yes
Keywords: Graph Neural Networks

Homophily

Heterophily

Low-pass filter

High-pass filter

Node Distinguishability

Metrics
Abstract: Homophily principle, i.e., nodes with the same labels are more likely to be connected, has been believed to be the main reason for the performance superiority of Graph Neural Networks (GNNs) over Neural Networks on node classification tasks. Recent research suggests that, even in the absence of homophily, the advantage of GNNs still exists as long as nodes from the same class share similar neighborhood patterns. However, this argument only considers intra-class Node Distinguishability (ND) but neglects inter-class ND, which provides incomplete understanding of homophily on GNNs. In this paper, we first demonstrate such deficiency with examples and argue that an ideal situation for ND is to have smaller intra-class ND than inter-class ND. To formulate this idea and study ND deeply, we propose Contextual Stochastic Block Model for Homophily (CSBM-H) and define two metrics, Probabilistic Bayes Error (PBE) and negative generalized Jeffreys divergence, to quantify ND. With the metrics, we visualize and analyze how graph filters, node degree distributions and class variances influence ND, and investigate the combined effect of intra- and inter-class ND. Besides, we discovered the mid-homophily pitfall, which occurs widely in graph datasets. Furthermore, we verified that, in real-work tasks, the superiority of GNNs is indeed closely related to both intra- and inter-class ND regardless of homophily levels. Grounded in this observation, we propose a new hypothesis-testing based performance metric beyond homophily, which is non-linear, feature-based and can provide statistical threshold value for GNNs' the superiority. Experiments indicate that it is significantly more effective than the existing homophily metrics on revealing the advantage and disadvantage of graph-aware modes on both synthetic and benchmark real-world datasets.

When Do Graph Neural Networks Help with Node Classification? Investigating the Impact of Homophily Principle on Node Distinguishability

Abstract

Homophily principle, i.e., nodes with the same labels are more likely to be connected, has been believed to be the main reason for the performance superiority of Graph Neural Networks (GNNs) over Neural Networks on node classification tasks. Recent research suggests that, even in the absence of homophily, the advantage of GNNs still exists as long as nodes from the same class share similar neighborhood patterns [38]. However, this argument only considers intra-class Node Distinguishability (ND) but neglects inter-class ND, which provides incomplete understanding of homophily on GNNs. In this paper, we first demonstrate such deficiency with examples and argue that an ideal situation for ND is to have smaller intra-class ND than inter-class ND. To formulate this idea and study ND deeply, we propose Contextual Stochastic Block Model for Homophily (CSBM-H) and define two metrics, Probabilistic Bayes Error (PBE) and negative generalized Jeffreys divergence, to quantify ND. With the metrics, we visualize and analyze how graph filters, node degree distributions and class variances influence ND, and investigate the combined effect of intra- and inter-class ND. Besides, we discovered the mid-homophily pitfall, which occurs widely in graph datasets. Furthermore, we verified that, in real-work tasks, the superiority of GNNs is indeed closely related to both intraand inter-class ND regardless of homophily levels. Grounded in this observation, we propose a new hypothesis-testing based performance metric beyond homophily, which is non-linear, feature-based and can provide statistical threshold value for GNNs' the superiority. Experiments indicate that it is significantly more effective than the existing homophily metrics on revealing the advantage and disadvantage of graph-aware modes on both synthetic and benchmark real-world datasets.

1 Introduction

Graph Neural Networks (GNNs) have gained popularity in recent years as a powerful tool for graphbased machine learning tasks. By combining graph signal processing and convolutional neural networks, various GNN architectures have been proposed [27, 19, 50, 36, 23], and have been shown to outperform traditional neural networks (NNs) in tasks such as node classification ( NC ), graph classification, link prediction and graph generation. The success of GNNs is believed to be rooted in the homophily principle (assumption) [42], which states that connected nodes tend to have similar attributes [18], providing extra useful information to the aggregated features over the original node features. Such relational inductive bias is thought to be a major contributor to the superiority of GNNs over NNs on various tasks [4]. On the other hand, the lack of homophily, i.e., heterophily, is considered as the main cause of the inferiority of GNNs on heterophilic graphs, because nodes from different classes are connected and mixed, which can lead to indistinguishable node embeddings, making the classification task more difficult for GNNs [57, 55, 37]. Numerous models have been proposed to address the heterophily challenge lately [46, 57, 55, 37, 5, 32, 7, 54, 21, 34, 31, 51, 35].

Recently, both empirical and theoretical studies indicate that the relationship between homophily principle and GNN performance is much more complicated than "homophily wins, heterophily loses" and the existing homophily metrics cannot accurately indicate the superiority of GNNs [38, 35, 37]. For example, the authors in [38] stated that, as long as nodes within the same class share similar neighborhood patterns, their embeddings will be similar after aggregation. They provided experimental evidence and theoretical analysis, and concluded that homophily may not be necessary for GNNs to distinguish nodes. The paper [35] studied homophily/heterophily from post-aggregation node similarity perspective and found that heterophily is not always harmful, which is consistent with [38]. Besides, the authors in [37] have proposed to use high-pass filter to address some heterophily cases, which is adopted in [7, 5] as well. They have also proposed aggregation homophily, which is a linear feature-independent performance metric and is verified to be better at revealing the performance advantages and disadvantages of GNNs than the existing homophily metrics [46, 57, 32]. Moreover, [6] has investigated heterophily from a neighbor identifiable perspective and stated that heterophily can be helpful for NC when the neighbor distributions of intra-class nodes are identifiable.

Inspite that the current literature on studying homophily principle provide the profound insights, they are still deficient: 1. [38, 6] only consider intra-class node distinguishability ( ND ), but ignore inter-class ND; 2. [35] does not show when and how high-pass filter can help with heterophily problem; 3. There is a lack of a non-linear, feature-based performance metric which can leverage richer information to provide an accurate statistical threshold value to indicate whether GNNs are really needed on certain task or not.

To address those issues, in this paper: 1. We show that, to comprehensively study the impact of homophily on ND, one needs to consider both intra- and inter-class ND and an ideal case is to have smaller intra-class ND than inter-class ND; 2. To formulate this idea, we propose Contextual Stochastic Block Model for Homophily (CSBM-H) as the graph generative model. It incorporates an explicit parameter to manage homophily levels, alongside class variance parameters to control intra-class ND, and node degree parameters which are important to study homophily [38, 54]; 3. To quantify ND of CSBM-H, we propose and compute two ND metrics, Probabilistic Bayes Error ( PBE ) and Negative Generalized Jeffreys Divergence ( D NGJ), for the optimal Bayes classifier of CSBM-H. Based on the metrics, we can analytically study how intra- and inter-class ND impact ND together. We visualize the relationship between PBE, D NGJ and homophily levels and discuss how different graph filters (full-, low- and high-pass filters), class variances and node degree distributions will influence ND in details; 4. In practice, we verify through hypothesis testing that the performance superiority of GNNs is indeed related to whether intra-class ND is smaller than inter-class ND, regardless of homophily levels. Based on this conclusion and the p-values of hypothesis testing, we propose Classifier-based Performance Metric (CPM), a new non-linear feature-based metric that can provide statistical threshold values. Experiments show that CPM is significantly more effective than the existing homophily metrics on predicting the superiority of graph-aware models over graph-agnostic.

2 Preliminaries

We use bold font for vectors ( e.g., v ) and define a connected graph G = ( V , E ) , where V is the set of nodes with a total of N elements, E is the set of edges without self-loops. A is the symmetric adjacency matrix with A i,j = 1 if there is an edge between nodes i and j , otherwise A i,j = 0 . D is the diagonal degree matrix of the graph, with D i,i = d i = ∑ j A i,j . The neighborhood set N i of node i is defined as N i = { j : e ij ∈ E} . A graph signal is a vector in R N , whose i -th entry is a feature of node i . Additionally, we use X ∈ R N × F h to denote the feature matrix, whose columns are graph signals and i -th row X i, : = x /latticetop i is the feature vector of node i ( i.e., the full-pass (FP) filtered signal). The label encoding matrix is Z ∈ R N × C , where C is the number of classes, and its i -th row Z i, : is the one-hot encoding of the label of node i . We denote z i = arg max j Z i,j ∈ { 1 , 2 , . . . C } . The indicator function 1 B equals 1 when event B happens and 0 otherwise.

/negationslash

For nodes i, j ∈ V , if z i = z j , then they are considered as intra-class nodes ; if z i = z j , then they are considered to be inter-class nodes . Similarly, an edge e i,j ∈ E is considered to be an intra-class edge if z i = z j , and an inter-class edge if z i = z j .

/negationslash

2.1 Graph-aware Models and Graph-agnostic Models

A network that includes the feature aggregation step according to graph structure is called graphaware ( G-aware ) model, e.g., GCN [27], SGC [53]; A network that does not use graph structure information is called graph-agnostic ( G-agnostic ) model, such as Multi-Layer Perceptron with 2

layers (MLP-2) and MLP-1. A G-aware model is often coupled with a G-agnostic model because when we remove the aggregation step in G-aware model, it becomes exactly the same as its coupled G-agnostic model, e.g., GCN is coupled with MLP-2 and SGC-1 is coupled with MLP-1 as below,

where ˆ A sym = ˜ D -1 / 2 ˜ A ˜ D -1 / 2 , ˜ A ≡ A + I and ˜ D ≡ D + I ; W 0 ∈ R F 0 × F 1 and W 1 ∈ R F 1 × O are learnable parameter matrices. For simplicity, we denote y i = arg max j Y i,j ∈ { 1 , 2 , . . . C } . The random walk renormalized matrix ˆ A rw = ˜ D -1 ˜ A can also be applied to GCN, which is essentially a mean aggregator commonly used in some spatial-based GNNs [19]. To bridge spectral and spatial methods, we use ˆ A rw in the theoretical analysis, but self-loops are not added to the adjacency matrix to maintain consistency with previous literature [38, 35].

To address the heterophily challenge, high-pass (HP) filter [14], such as I -ˆ A rw, is often used to replace low-pass (LP) filter [39] ˆ A rw in GCN [5, 7, 35]. In this paper, we use ˆ A rw and I -ˆ A rw as the LP and HP operators, respectively. The LP and HP filtered feature matrices are represented as H = ˆ A rw X and H HP = ( I -ˆ A rw ) X . For simplicity, we denote h i = ( H i, : ) /latticetop , h HP i = ( H HP i, : ) /latticetop .

To measure how likely the G-aware model can outperform its coupled G-agnostic model before training them ( i.e., if the aggregation step according to graph structure is helpful for node classification or not), a lot of homophily metrics have been proposed and we will introduce the most commonly used ones in the following subsection.

2.2 Homophily Metrics

The homophily metric is a way to describe the relationship between node labels and graph structure. We introduce five commonly used homophily metrics: edge homophily [1, 57], node homophily [46], class homophily [32], generalized edge homophily [26] and aggregation homophily [35], adjusted homophily [47] and label informativeness [47] as follows:

/negationslash where H v node is the local homophily value for node v ; [ a ] + = max(0 , a ) ; h k is the class-wise homophily metric [32]; Mean u ( {·} ) takes the average over u of a given multiset of values or variables and S ( ˆ A,Z ) = ˆ AZ ( ˆ AZ ) /latticetop is the post-aggregation node similarity matrix; D c = ∑ v : z v = c d v , ¯ p c = D c 2 |E| , p c 1 ,c 2 = ∑ ( u,v ) ∈ E 1 { z u = c 1 ,z v = c 2 } 2 |E| , c, c 1 , c 2 ∈ { 1 , . . . , C } .

These metrics all fall within the range of [0 , 1] , with a value closer to 1 indicating strong homophily and implying that G-aware models are more likely to outperform its coupled G-agnostic model, and vice versa. However, the current homophily metrics are almost all linear, feature-independent metrics which cannot provide a threshold value [35] for the superiority of G-aware model and fail to give an accurate measurement of node distinguishability (ND) . In the following section, we focus on quantifying the ND of graph models with with homophily levels and analyzing their relations.

3 Analysis of Homophily on Node Distinguishability (ND)

3.1 Motivation

The Problem in Current Literature Recent research has shown that heterophily does not always negatively impact the embeddings of intra-class nodes, as long as their neighborhood patterns "corrupt in the same way" [38, 6]. For example, in Figure 1, nodes {1,2} are from class blue and both have the same heterophilic neighborhood patterns. As a result, their aggregated features will still be similar and they can be classified into the same class.

However, this is only partially true for ND if we forget to discuss inter-class ND, e.g., node 3 in Figure 1 is from class green and has the same neighborhood pattern ( 1 / 3 orange, 1 / 3 yellow and 1 / 3 green) as nodes {1,2}, which means the inter-class ND will be lost after aggregation. This highlights the necessity for careful consideration of both intra- and inter-class ND when evaluating the impact of homophily on the performance of GNNs and an ideal case for NC would be node {1,2,4}, where we have smaller intra-class "distance" than inter-class "distance". We will formulate the above idea in this section and verify if it really relates to the performance of GNNs in section 4. In the following subsection, we will propose a toy graph model, on which we can study the relationship between homophily and ND directly and intuitively, and analyze intra- and inter-class ND analytically.

3.2 CSBM-H and Optimal Bayes Classifier

In order to have more control over the assumptions on the node embeddings, we consider the Contextual Stochastic Block Model (CSBM) [11]. It is a generative model that is commonly used to create graphs and node features, and it has been widely adopted to study the behavior of GNNs [49, 3, 52]. To investigate the impact of homophily on ND, the authors in [38] simplify CSBM to the two-normal setting, where the node features X are assumed to be sampled from two normal distributions and intra- and inter-class edges are generated according to two separate parameters. This simplification does not lose much information about CSBM, but 1. it does not include an explicit homophily parameter to study homophily directly and intuitively; 2. it does not include class variance parameters to study intra-class ND; 3. the authors do not rigorously quantify ND.

In this section, we introduce the Contextual Stochastic Block Model for Homophily/Heterophily (CSBM-H), which is a variation of CSBM that incorporates an explicit homophily parameter h for the two-normal setting and also has class variance parameters σ 2 0 , σ 2 1 to describe the intra-class ND. We then derive the optimal Bayes classifier (CLBayes) and negative generalized Jeffreys divergence for CSBM-H, based on which we can quantify and investigate ND for CSBM-H.

CSBM-H( µ 0 , µ 1 , σ 2 0 I, σ 2 1 I, d 0 , d 1 , h ) 1 The generated graph consists of two disjoint sets of nodes, i ∈ C 0 and j ∈ C 1 , corresponding to the two classes. The features of each node are generated independently, with x i generated from N ( µ 0 , σ 2 0 I ) and x j generated from N ( µ 1 , σ 2 1 I ) , where µ 0 , µ 1 ∈ R F h and F h is the dimension of the embeddings. The degree of nodes in C 0 and C 1 are d 0 , d 1 ∈ N respectively. For i ∈ C 0 , its neighbors are generated by independently sampling from h · d 0 intra-class nodes and (1 -h ) · d 0 inter-class nodes 2 . The neighbors of j ∈ C 1 are generated in the same way. As a result, the FP (full-pass), LP and HP filtered features are generated as follows, where ˜ µ 0 = h ( µ 0 -µ 1 ) + µ 1 , ˜ µ 1 = h ( µ 1 -µ 0 ) + µ 0 , ˜ µ HP 0 = (1 -h )( µ 0 -µ 1 ) , ˜ µ HP 1 = (1 -h )( µ 1 -µ 0 ) , ˜ σ 2 0 = ( h ( σ 2 0 -σ 2 1 )+ σ 2 1 ) d 0 , ˜ σ 2 1 = ( h ( σ 2 1 -σ 2 0 )+ σ 2 0 ) d 1 , (˜ σ HP 0 ) 2 = σ 2 0 + ( h ( σ 2 0 -σ 2 1 )+ σ 2 1 ) d 0 , (˜ σ HP 1 ) 2 = σ 2 1 + ( h ( σ 2 1 -σ 2 0 )+ σ 2 0 ) d 1 . If σ 2 0 < σ 2 1 , we refer to C 0 as the low variation class and C 1 as the high variation class. The variance of each class can reflect the intra-class ND. We abuse the notation x i ∈ C 0 for i ∈ C 0 and x j ∈ C 1 for j ∈ C 1 .

To quantify the ND of CSBM-H, we first compute the optimal Bayes classifier in the following theorem. The theorem is about the original features, but the results are applicable to the filtered features when the parameters are replaced according to equation 3.

/negationslash

Theorem 1. Suppose σ 2 0 = σ 2 1 and σ 2 0 , σ 2 1 > 0 , the prior distribution for x is P ( x ∈ C 0 ) = P ( x ∈ C 1 ) = 1 / 2 , then the optimal Bayes Classifier (CLBayes) for CSBM-H ( µ 0 , µ 1 , σ 2 0 I, σ 2 1 I, d 0 , d 1 , h ) is

Advantages of CLBayes Over the Fixed Linear Classifier in [38] The decision boundary in [38] is defined as P = { x | w /latticetop x -w /latticetop ( µ 0 + µ 1 ) / 2 } where w = ( µ 0 -µ 1 ) / || µ 0 -µ 1 || 2 is a fixed parameter. This classifier only depends on µ 0 , µ 1 and is independent of h . However, as h changes, the "separability" of the two normal distributions should be different. The fixed classifier cannot capture this difference, and thus is not qualified to measure ND for different h . Besides, we cannot investigate how variances σ 2 0 , σ 2 1 and node degrees d 0 , d 1 affect ND with the fixed classifier in [38].

In the following subsection, we will define two methods to quantify ND of CSBM-H, one is based on CLBayes, which is a precise measure but hard to be explainable; another is based on KL-divergence, which can give us more intuitive understanding of how intra- and inter-class ND will impact ND at different homophily levels. These two measurements can be used together to analyze ND.

3.3 Measure Node Distinguishability of CSBM-H

The Bayes error rate (BE) of the data distribution is the probability of a node being mis-classified when the true class probabilities are known given the predictors [20]. It can be used to measure the distinguishability of node embeddings and the BE for CLBayes is defined as follows,

Definition 1 (Bayes Error Rate) . The Bayes error rate [20] for CLBayes is defined as

/negationslash

Specifically, the BE for CSBM-H can be written as

To estimate the above value, we compute Probabilistic Bayes Error (PBE) for CSBM-H as follows.

Probabilistic Bayes Error (PBE) The random variable in each dimension of x is independently normally distributed. As a result, Q ( x ) defined in Theorem 1 follows a generalized χ 2 distribution [9, 10](See the calculation in Appendix C). Specifically,

where w 0 = a σ 2 0 , w 1 = a σ 2 1 , the degree of freedom is F h , λ 0 = ( µ 0 σ 0 + b 2 a σ 0 ) /latticetop ( µ 0 σ 0 + b 2 a σ 0 ) , λ 1 = ( µ 1 σ 1 + b 2 a σ 1 ) /latticetop ( µ 1 σ 1 + b 2 a σ 1 ) and ξ = c -b /latticetop b 4 a . Then, by using the Cumulative Distribution Function (CDF) of ˜ χ 2 , we can calculate the predicted probabilities directly as,

Suppose we have a balanced prior distribution ( x ) = ( x ) = 1 / 2 . Then, PBE is,

To investigate the impact of homophily on the ND for LP and HP filtered embeddings, we just need to replace ( µ 0 , σ 2 0 , µ 1 , σ 2 1 ) by ( ˜ µ 0 , ˜ σ 2 0 , ˜ µ 1 , ˜ σ 2 1 ) and ( ˜ µ HP 0 , (˜ σ HP 0 ) 2 , ˜ µ HP 1 , (˜ σ HP 1 ) 2 ) as equation 3.

PBE can be numerically calculated and visualized to show the relationship between h and ND precisely. However, we do not have an analytic expression for PBE, which makes it less explainable and intuitive. To address this issue, we define another metric for ND in the following paragraphs.

Generalized Jeffreys Divergence The KL-divergence is a statistical measure of how a probability distribution P is different from another distribution Q [8]. It offers us a tool to define an explainable ND measure, generalized Jeffreys divergence.

Definition 2 (Generalized Jeffreys Divergence) . For a random variable x which has either the distribution P ( x ) or the distribution Q ( x ) , the generalized Jeffreys divergence 4 is defined as

With P ( x ∼ P ) = P ( x ∼ Q ) = 1 / 2 5 , the negative generalized Jeffreys divergence for the two-normal setting in CSBM-H can be computed by (See Appendix B for the calculation)

where d 2 X = ( µ 0 -µ 1 ) /latticetop ( µ 0 -µ 1 ) is the squared Euclidean distance between centers; ρ = σ 0 σ 1 and since we assume σ 2 0 < σ 2 1 , we have 0 < ρ < 1 . For h and h HP , we have d 2 H = (2 h -1) 2 d 2 X , d 2 HP = 4(1 -h ) 2 d 2 X . The smaller D NGJ the CSBM-H has, the more distinguishable the embeddings are.

From equation 5, we can see that D NGJ implies that ND relies on two terms, Expected Negative Normalized Distance (ENND) and the Negative Variance Ratio (NVR): 1. ENND depends on how large is the inter-class ND d 2 X compared with the normalization term 1 4 σ 2 1 + 1 4 σ 2 0 , which is determined by intra-class ND (variances σ 0 , σ 1 ); NVR depends on how different the two intra-class NDs are, i.e., when the intra-class ND of high-variation class is significantly larger than that of low-variation class ( ρ is close to 0), NVR is small which means the nodes are more distinguishable and vice versa.

Now, we can investigate the impact of homophily on ND through the lens of PBE and D NGJ 6 . Specifically, we set the standard CSBM-H as µ 0 = [ -1 , 0] , µ 1 = [0 , 1] , σ 2 0 = 1 , σ 2 1 = 2 , d 0 = 5 , d 1 = 5 . And as shown in Figure 2, its PBE and D NGJ curves for LP filtered feature h are bellshaped 7 . This indicates that, contrary to the prevalent belief that heterophily has the most negative impact on ND, a medium level of homophily actually has a more detrimental effect on ND than extremely low levels of homophily. We refer to this phenomenon as the mid-homophily pitfall .

The PBE and D NGJ curves for h HP are monotonically increasing, which means that the high-pass filter works better in heterophily areas than in homophily areas. Moreover, it is observed that x , h , and h HP will get the lowest PBE and D NGJ in different homophily intervals, which we refer to as the "FP regime (black) ", "LP regime (green) ", and "HP regime (red) " respectively. This indicates that LP filter works better at very low and very high homophily intervals (two ends), HP filter works better at low to medium homophily interval 8 , the original ( i.e., full-pass or FP filtered) features works betters at medium to high homophily area.

Researchers have always been interested in exploring how node degree relates to the effect of homophily [38, 54]. In the upcoming subsection, besides node degree, we will also take a deeper look at the impact of class variances via the homophily-ND curves and the FP, LP and HP regimes.

(a) PBE

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Negative Generalized Jeffreys Divergence

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3.4 Ablation Study on CSBM-H

Increase the Variance of High-variation Class ( σ 2 0 = 1 , σ 2 1 = 5 ) From Figure 3, it is observed that as the variance in C 1 increases and the variance between C 0 and C 1 becomes more imbalanced, the PBE and D NGJ of the three curves all go up which means the node embeddings become less distinguishable under HP, LP and FP filters. The significant shrinkage of the HP regimes and the expansion of the FP regime indicates that the original features are more robust to imbalanced variances

(a) PBE

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especially in the low homophily area. From Figure 3 (d), we can see that the main cause is that the NVR of the 3 curves all move down but the HP curve moves less in low homophily area than other 2 curves. This implies that the HP curve exhibits less sensitivity to ρ within the area of low homophily.

Increase the Variance of Low-variation Class ( σ 2 0 = 1 . 9 , σ 2 1 = 2 ) As shown in Figure 4, when the variance in C 0 increases and the variance between C 0 and C 1 becomes more balanced, PBE and D NGJ curves go up, which means the node embeddings become less distinguishable. The LP, HP and the FP regimes almost stays the same because the magnitude of NVR becomes too small that it almost has no effect to ND as shown in Figure 4 (d).

Interestingly, we found the change of variances cause less differences of the 3 regimes in ENND than that in NVR 9 and HP filter is less sensitive to ρ changes in low homophily area than LP and FP filters. This insensitivity will have significant impact to the 3 regimes when ρ is close to 0 and have trivial effect when ρ is close to 1 because the magnitude of NVR is too small.

Increase the Node Degree of High-variation Class ( d 0 = 5 , d 1 = 25 ) From Figure 5, it can be observed that as the node degree of the high-variation class increases, the PBE and D NGJ curves of FP and HP filters almost stay the same while the curves of LP filters go down with a large margin. This leads to a substantial expansion of LP regime and shrinkage of FP and HP regime. This is mainly due to the decrease of ENND of LP filters and the decrease of its NVR in low homophily area also plays an important role.

Increase the Node Degree of Low-variation Class ( d 0 = 25 , d 1 = 5 ) From Figure 6, we have the similar observation as when we increase the node degree of high-variation class. The difference is that the expansion of LP regime and shrinkage of FP and HP regimes are not as significant as before.

From ˜ σ 2 0 , ˜ σ 2 1 we can see that increasing node degree can help LP filter reduce variances of the aggregated features so that the ENND will decrease, especially for high-variation class while HP filter is less sensitive to the change of variances and node degree.

3.5 More General Theoretical Analysis

Besides the toy example, in this subsection, we aim to gain a deeper understanding of how LP and HP filters affect ND in a broader context beyond the two-normal settings. To be consistent with previous literature, we follow the assumptions outlined in [38], which are: 1. The features of node i are sampled from distribution F z i , i.e., , x i ∼ F z i , with mean µ z i ∈ R F h ; 2. Dimensions of x i are independent to each other; 3. Each dimension in feature x i is bounded, i.e., a ≤ x i,k ≤ b ; 4. For node i , the labels of its neighbors are independently sampled from neighborhood distribution

(a) PBE

(b)

D

NGJ

(c) ENND

(d) NVR

(a) PBE

(b)

D

NGJ

(c) ENND

(d) NVR

Probabilistic Bayes Error

0.5

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0.0 -

0.0

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30

=5

0

0.0

Negative Generalized Jeffreys Divergence

0.2

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Homophily h

I

0.6

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1.0

H_high

H

X

10

-15

20

25

=5

0

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Expected Negative Normalized Euclidean Distance

0.2

I

0.4

Homophily h

0.6

1

1

1

0.8

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H_high

X

H

=0,5

1,0

1.5

2,0

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=2,5

0,0

0.0

Probabilistic Bayes Error

0.5

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0.3

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0.1

0.0 :

0.0

0.2

0.4

Homophily h

9°0

一

0.8

× error

h error

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10

15

20

S2-

30

S-

0

0.0

Negative Generalized Jeffreys Divergence

0.2

↓

1

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Homophily h

I

=

0.6

0.8

1.0

H_high

X

H

15

20

10

25

-5

0

0.0

Expected Negative Normalized Euclidean Distance

0.2

I

I

I

0.4

Homophily h

I

1

1

1

0.6

0.8

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X

H

0,5

=1,0

1.5

2.0

2.5

3.0

3,5

4,0

0,C

D z i and repeated for d i times. We refer to a graph that follows the above assumptions as G = {V , E , {F c , c ∈ C} , {D c , c ∈ C}} , C = { 1 , . . . , C } and ( b -a ) 2 reflects how variation the features are. The authors in [38] analyze the distance between the aggregated node embedding and its expectation, i.e., ‖ h i -E ( h i ) ‖ 2 , which only considers the intra-class ND and has been shown to be inadequate for a comprehensive understanding of ND. Instead, we investigate how significant the intra-class embedding distance is smaller than the inter-class embedding distance in the following theorem, which is a better way to understand ND.

/negationslash

Theorem 2. Suppose a graph G = {V , E , {F c , c ∈ C} , {D c , c ∈ C}} meets all the above assumptions (1-4). For nodes i, j, v ∈ V , suppose z i = z j and z i = z v , then for constants t x , t h , t HP that satisfy t F D ( i, j ) , t F D ( i, j ) , t F D ( i, j ) we have x ≥ √ h x h ≥ √ h h HP ≥ √ h HP P ( ‖ x i -x j ‖ 2 ≥ ‖ x i -x v ‖ 2 + t x ) ≤ 2 F h exp ( -( D x ( v,j ) -t x √ F h ) 2 V x ( v,j ) ) , P ( ‖ h i -h j ‖ 2 ≥ ‖ h i -h v ‖ 2 + t h ) ≤ 2 F h exp ( -( D h ( v,j ) -t h √ F h ) 2 V h ( v,j ) ) , P ( ∥ ∥ h HP i -h HP j ∥ ∥ 2 ≥ ∥ ∥ h HP i -h HP v ∥ ∥ 2 + t HP ) ≤ 2 F h exp   -( D HP ( v,j ) -t HP √ F h ) 2 V HP ( v,j )   , (6) where D x ( v, j ) = ∥ ∥ µ z v -µ z j ∥ ∥ 2 , V x ( v, j ) = ( b -a ) 2 , D h ( v, j ) = ∥ ∥ ˜ µ z v -˜ µ z j ∥ ∥ 2 , V h ( v, j ) = ( 1 2 d v + 1 2 d j ) ( b -a ) 2 , D HP ( v, j ) = ∥ ∥ ∥ µ z v -˜ µ z v -( µ z j -˜ µ z j )∥ ∥ ∥ 2 , V HP ( v, j ) = ( 1 + 1 2 d v + 1 2 d j ) ( b -a ) 2 , ˜ µ z v = ∑ u ∈ N ( v ) E z u ∼ D zv , x u ∼ F zu [ 1 d v x u ] . See the proof in Appendix G

We can see that, the probability upper bound mainly depends on a distance term (inter-class ND) and normalized variance term (intra-class ND). The normalized variance term of HP filter is less sensitive to the changes of node degree than that of LP filter because there is an additional 1 in the constant term. Moreover, we show that the distance term of HP filter actually depends on the relative center distance , which is a novel discovery. As shown in Figure 7, when homophily decreases, the aggregated centers will move away from the original centers, and the relative center distance (purple) will get larger which means the embedding distance of nodes from different classes will have larger probability to be big. This explains how HP filter work for some heterophily cases. Overall, in a more general setting with weaker assumptions, we can see that ND is also described by the intra- and inter-class ND terms together rather than intra-class ND only, which is consistent with CSBM-H.

4 Empirical Study of Node Distinguishability

Besides theoretical analysis, in this section, we will conduct experiments to verify whether the effect of homophily on the performance of GNNs really relates to its effect on ND. If a strong relation can be verified, then it indicates that we can design new training-free ND-based performance metrics beyond homophily metrics, to evaluate the superiority and inferiority of G-aware models against its coupled G-agnostic models.

4.1 Hypothesis Testing on Real-world Datasets

To test whether "intra-class embedding distance is smaller than the inter-class embedding distance" strongly relates to the superiority of G-aware models to their coupled G-agnostic models in practice, we conduct the following hypothesis testing 10 .

Experimental Setup We first train two G-aware models GCN, SGC-1 and their coupled G-agnostic models MLP-2 and MLP-1 with fine-tuned hyperparameters provided by [35]. For each trained model, we calculate the pairwise Euclidean distance of the node embeddings in output layers. Next, we compute the proportion of nodes whose intra-class node distance is significantly smaller than inter-class node distance 11 e.g., we obtain Prop ( GCN ) for GCN. We use Prop to quantify ND and we train the models multiple times for samples to conduct the following hypothesis tests:

H 0 : Prop ( G-aware model ) ≥ Prop ( G-agnostic model ); H 1 : Prop ( G-aware model ) < Prop ( G-agnostic model )

Specifically, we compare GCN vs. MLP-2 and SGC-1 vs. MLP-1 on 9 widely used benchmark datasets with different homophily values for 100 times. In each time, we randomly split the data into training/validation/test sets with a ratio of 60%/20%/20%. For the 100 samples, we conduct T-test for the means of two independent samples of scores , and obtain the corresponding p-values. The test results and model performance comparisons are shown in Table 1 (See more experimental tests on state-of-the-art model in Appendix H).

It is observed that, in most cases (except for GCN vs. MLP-2 on PubMed 12 ), when H 1 significantly holds, G-aware models will underperform the coupled G-agnostic models and vice versa. This supports our claim that the performance of G-aware models is closely related to "intra-class vs. inter-class node embedding distances", no matter the homophily levels. It reminds us that the p-value can be a better performance metric for GNNs beyond homophily. Moreover, the p-value can provide a statistical threshold, such as p ≤ 0 . 05 . This property is not present in existing homophily metrics.

However, it is required to train and fine-tune the models to obtain the p-values, which make it less practical because of computational costs. To overcome this issue, in the next subsection, we propose a classifier-based performance metric that can provide p-values without training.

4.2 Towards A Better Metric Beyond Homophily: Classifier-based Performance Metric

A qualified classifier should not require iterative training. In this paper, we choose Gaussian Naïve Bayes (GNB)[20] and Kernel Regression (KR) with Neural Network Gaussian Process (NNGP) [30, 2, 16, 41] to capture the feature-based linear or non-linear information.

To get the p-value efficiently, we first randomly sample 500 labeled nodes from V and splits them into 60%/40% as "training" and "test" data. The original features X and aggregated features H of the sampled training and test nodes can be calculated and are then fed into a given classifier. The predicted results and prediction accuracy of the test nodes will be computed directly with feedforward method. We repeat this process for 100 times to get 100 samples of prediction accuracy for X and H . Then, for the given classifier, we compute the p-value of the following hypothesis testing,

The p-value can provide a statistical threshold value, such as 0.05, to indicate whether H is significantly better than X for node classification. As seen in Table 1, KR and GNB based metrics significantly outperform the existing homophily metrics, reducing the errors from at least 5 down to just 1 out of 18 cases. Besides, we only need a small set of the labels to calculate the p-value, which makes it better for sparse label scenario. Table 2 summarizes its advantages over the existing metrics. (See Appendix H for more details on classifier-based performance metrics, experiments on synthetic datasets, more detailed comparisons on small-scale and large-scale datasets, discrepancy between linear and non-linear models, results for symmetric renormalized affinity matrix and running time.)

5 Conclusions

In this paper, we provide a complete understanding of homophily by studying intraand inter-class ND together. To theoretically investigate ND, we study the PBE and D NGJ of the proposed CSBM-H and analyze how graph filters, class variances and node degree distributions will influence the PBE and D NGJ curves and the FP, LP, HP regimes. We extend the investigation to broader settings with weaker assumptions and theoretically prove that ND is indeed affected by both intra- and inter-class ND. We also discover that the effect of HP filter depends on the relative center distance.

Empirically, through hypothesis testing, we corroborate that the performance of GNNs versus NNs is closely related to whether intra-class node embedding "distance" is smaller than inter-class node embedding "distance". We find that the p-value is a much more effective performance metric beyond homophily metrics on revealing the advantage and disadvantage of GNNs. Based on this observation, we propose classifier-based performance metric, which is a non-linear feature-based metric and can provide statistical threshold value.

6 Reproducibility and Blogs

Code: https://github.com/SitaoLuan/When-Do-GNNs-Help .

Blog in English (on Medium): https://medium.com/SitaoLuan/ when-should-we-use-graph-neural-networks-for-node-classification-8ce77a772085 .

Blog in Chinese (on Zhihu): https://zhuanlan.zhihu.com/p/653631858 .

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the question: when do graph-aware models outperform graph-agnostic ones on node classification tasks? First, theoretical analysis is conducted. Most of the analysis assumes the two-class CSBM-H model – a generalization of the standard stochastic block model, where node features are sampled from normal distributions (with parameters depending on class labels), and parameter $h$ controls the homophily level. Then, Probabilistic Bayes Error and negative generalized Jeffreys divergence are used to quantify node distinguishability for several node representations: the original node features, aggregated node features (one step of random walk-based aggregation), and high-pass filtered features (after one step of the corresponding aggregation). The analysis shows that different representations are beneficial for different homophily regimes.\n\nIn the experiments, it is first shown that if a model has more distinguishable representation, then it usually performs better (Section 4.1). In Section 4.2, it is proposed to predict whether graph-aware models are better than graph-agnostic ones by training a simple classifier on original and aggregated features.\n\nSTRENGTHS:\nThe theoretical analysis of PBE and Jeffreys divergence for the CSBM-H and the discussion in Section 3.4 are intuitive and allow one to understand why different aggregation types are helpful for different homophily regimes.\n\nWEAKNESSES:\n- The analysis assumes a particular aggregation (random walk), and it is not clear whether conclusions may change for other aggregations.\n- The paper is limited to heterophilous datasets known to have certain drawbacks [1,2].\n- The proposed measure (Section 4.2) is quite straightforward: it suggests predicting the relative performance of a GNN by training its simplified variant.\n- The paper is hard to follow in several places.\n\nI also have the following comments and questions.\n\n1. On the definition of CSBM-H. According to the definition, $h \\cdot d_0$ and $h \\cdot d_1$ have to be integer. Also, possible values of $d_0$ and $d_1$ depend on the class sizes (the number of outgoing edges should be equal for two classes). This would not be necessary if the graphs are assumed to be directed, but this is not the case according to Section 2.\n\n2. There is another simple measure called Label Informativeness [3] that is known to better agree with GNN performance than homophily [3].\n\n3. The explanation in lines 168-172 was unclear to me; a more detailed explanation would help.\n\nMinor:\n- Definition of $h_k$ in (2): $Z_{v,k}$ does not depend on $u$, so it can be moved to the subscript of the sum.\n- L112: it is written that all homophily measures are feature-independent, but $H_{GE}$ depends on features.\n- Definition 1: notation $P(CL_{Bayes}(x))$ is not clear.\n\nSome typos:\n- L48: “Literatures”\n- L100: inconsistency of singular and plural\n- L111: “imply” → “implying”\n- L120 and below: quotes are typed incorrectly, it should be ``like this'' in latex\n- L149: “the the”\n- L170: “the fixed the classifier”\n\n[1] Lim D. et al. Large scale learning on non-homophilous graphs: New benchmarks and strong simple methods. NeurIPS 2021.\n\n[2] Platonov O. et al. A critical look at the evaluation of GNNs under heterophily: Are we really making progress? ICLR 2023.\n\n[3] Platonov O. et al. Characterizing graph datasets for node classification: Beyond homophily-heterophily dichotomy. ArXiv:2209.06177, 2022.\n\nQUESTIONS:\nThe pipeline in Section 4.1 seems overcomplicated. There are two statistical tests, and one is used to define Prop(GCN). It seems that here one can compute the proportion of nodes whose average intra-class node distance is smaller than the inter-class node distance. Would this work or statistical test on this step is necessary?\n\nLIMITATIONS:\nLimitations are not discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies when Graph Neural Networks (GNNs) can help with node classification tasks. The authors first focus on a variant of the Contextual Stochastic Block Model (CSBM) and propose new metrics for node distinguishability based on this model. The authors carry out comprehensive experiments and empirically characterize the regimes when original, low-pass filtered and high-pass filtered features are more helpful. In addition, the authors propose two additional hypothesis testing based metrics for deciding if GNNs are helpful, and empirically verify the effectiveness of the new metrics over real data. The experiments show that the new metrics are more indicative of whether GNNs can lead to better classification accuracy over baseline graph-agnostic models (i.e. MLPs).\n\nSTRENGTHS:\n- The effect of graph homophily/heterophily to the performance of GNNs is an important topic which has received a lot of interests. This paper provides a more comprehensive perspective to the subject.\n\n- The experiments are carefully designed and reasonably comprehensive. The empirical results provide a good support to the main claims of the paper.\n\n- Overall, the paper is well organized.\n\nWEAKNESSES:\n- I think the overall writing can be improved. I did not feel excited when reading the paper up to and including page 5. This is just my feeling and it does not mean the paper is not good. Moreover, I find myself sometimes lost focus when reading the first 5 pages. Maybe adding 1-2 sentences describing the emphasis of each section at the beginning of the section can be helpful.\n\n- It seems to me that Section 3 and Section 4 study completely different things and under different settings. Although they are related. I think the results of Section 3 and Section 4 can be considered as two separate contributions of the paper. It might be a good idea to make this clear from the beginning. Currently, this gap between the two sections is not clear from neither the abstract nor the introduction.\n\n- There are a few typos in the paper. For example:\n  - Line 18: it significantly -> it is significantly\n  - Line 112, the authors claim that \"the current homophily metrics are all ... feature independent\", however, the generalized edge homophily is feature dependent.\n  - Line 148: inner-class ND -> intra-class ND. Better be consistent since you used intra-class everywhere else.\n  - Most figures involving PBE, e.g. Figure 2a, 3a, have a different naming in the legend than others. Better to have a consistent naming in figure legend.\n\nQUESTIONS:\n- For Figure 2, since d_0 = d_1 = 5, how did you obtain the plot at h=0.5? It's fine to apply some smooth interpolation but the middle point is important since it is the peak for LP. Maybe d_0 = d_1 = 10 is a better choice for the base setting.\n\nLIMITATIONS:\nI could not find where the authors addressed the limitations of this work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-05"}

comments: SUMMARY:
The authors analyze the node distinguishability in attributed graphs under the prism of homophily. They consider the binary CSBM model. They compute the classification error rate on the original features and of low- and high-frequency filters; they study the influence of inter- and intra-class variance on their performances. The authors propose new metrics based on these variances and on the comparison between GNN and NN.

STRENGTHS:
I am not very familiar with the literature needed to assess the originality of this article.

It is well written.

WEAKNESSES:
For me it seems the article mainly studies the separability of a binary Gaussian mixture. The principle it derives (l. 347, whether intra-class node embedding "distance" is smaller than inter-class node embedding "distance") just means the two Gaussians do not overlap.

QUESTIONS:
A few remarks:

L. 75 are the features in R or R^F? and l. 154, in F_h?

L. 157 what is FP? In eq. 3 the authors could remind what are the FP and LP filters among the many quantities.

Theorem 1: I would not call it the optimal Bayes classifier since it is restricted to the features only; it does not take in account the graph of the CSBM. Also, this is just the optimal classifier for a binary Gaussian mixture; maybe it is not worth a theorem.

LIMITATIONS:
The authors did not discuss possible limitations.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

comments: SUMMARY:
Recent research indicates that Graph Neural Networks (GNNs) maintain their advantage even in the absence of homophily, as long as nodes from the same class exhibit similar neighborhood patterns. This argument, however, primarily considers intra-class Node Distinguishability (ND) while overlooking inter-class ND, thus providing an incomplete understanding of homophily. Therefore, the authors in this paper propose that an ideal ND scenario entails smaller intra-class ND relative to inter-class ND. To substantiate this, the authors introduce the Contextual Stochastic Block Model for Homophily (CSBM-H) and define two ND metrics: Probabilistic Bayes Error (PBE) and negative generalized Jeffreys divergence. Experimental results reveal that the supremacy of GNNs is indeed closely tied to both intra- and inter-class ND, irrespective of homophily levels.

STRENGTHS:
1. The paper addresses a highly significant issue.
2. The proposed CSBM-H and the defined metrics illustrate that the superiority of GNNs is indeed closely linked with both intra- and inter-class ND, regardless of homophily levels. The results are interesting.
3. The paper is solidly grounded in its field.

WEAKNESSES:
N/A

QUESTIONS:
N/A

LIMITATIONS:
While I do not consider myself an expert in this area, I appreciate the content of this paper. It addresses a highly significant issue, and proposes that an ideal ND scenario would have smaller intra-class ND compared to inter-class ND. To formulate this idea, the authors introduce the Contextual Stochastic Block Model for Homophily (CSBM-H) and define two ND metrics: Probabilistic Bayes Error (PBE) and negative generalized Jeffreys divergence. The proof is robust and the paper appears to be strongly grounded in the subject matter.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-25

comments: SUMMARY:
The paper delves deeper into the topic of GNNs' superiority in node classification tasks. The authors conduct experiments to show that GNNs do better due to inter and intra class node distinguishability regardless of homophily levels as suggest by current research. They propose a new metric that sheds more light on how GNNs work.

STRENGTHS:
GNNs are a hot topic and studying how they work and what settings are conducive for GNNs to be more effective is important. This paper does that and proposes a new metric to that end.

WEAKNESSES:
1. Since the crux of the paper is about node distinguishability (ND), it would be nice to see inter and intra ND visualizations of the datasets used similar to Figure 1 (toy example).

2. I'm not sure the claim "superiority of GNNs is indeed closely related to both intra- and inter-class ND regardless of homophily levels" is sufficiently backed by the evidence in Table 1. For example, setting a different threshold than 0.5 for homophily metrics could make them look better.

QUESTIONS:
1. Anything special about the PubMed dataset causing incorrect guesses? Also, Chameleon and Squirrel datasets have more incorrect guesses than other "less homophilic" datasets. Any way to visualize ND to pin point where the difference is coming from?

2. Any comment on the time complexity trade-offs between homophily metrics and classifier based ones?

Minor:
line 18: "it is significantly" instead of "it significantly" 
line 321: "split" instead of "splits"

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-27