Revisiting Heterophily For Graph Neural Networks
Abstract
Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using graph structures based on the relational inductive bias (homophily assumption).
While GNNs have been commonly believed to outperform NNs in real-world tasks, recent work has identiﬁed a non-trivial set of datasets where their performance compared to NNs is not satisfactory.
Heterophily has been considered the main cause of this empirical observation and numerous works have been put forward to address it.
In this paper, we ﬁrst revisit the widely used homophily metrics and point out that their consideration of only graph-label consistency is a shortcoming.
Then, we study heterophily from the perspective of post-aggregation node similarity and deﬁne new homophily metrics, which are potentially advantageous compared to existing ones.
Based on this investigation, we prove that some harmful cases of heterophily can be effectively addressed by local diversiﬁcation operation.
Then, we propose the Adaptive Channel Mixing (ACM), a framework to adaptively exploit aggregation, diversiﬁcation and identity channels node-wisely to extract richer localized information for diverse node heterophily situations.
ACM is more powerful than the commonly used uni-channel framework for node classiﬁcation tasks on heterophilic graphs and is easy to be implemented in baseline GNN layers.
When evaluated on 10 benchmark node classiﬁcation tasks, ACM-augmented baselines consistently achieve signiﬁcant performance gain, exceeding state-of-theart GNNs on most tasks without incurring signiﬁcant computational burden.
Introduction
Deep Neural Networks (NNs)
[22] have revolutionized many machine learning areas, including image recognition [21], speech recognition [13] and natural language processing [2], due to their effectiveness in learning latent representations from Euclidean data.
Recent research has shifted focus on non-Euclidean data [6], e.g., relational data or graphs.
Combining graph signal processing and convolutional neural networks [23], numerous Graph Neural Network (GNN) architectures have been proposed [38, 10, 15, 40, 19, 29], which empirically outperform traditional NNs on graph-based machine learning tasks such as node classiﬁcation, graph classiﬁcation, link prediction and graph generation, etc.
GNNs are built on the homophily assumption [34]: connected nodes tend to share similar attributes with each other [14], which offers additional information besides node features.
This relational inductive bias [3] is believed to be a key factor leading to GNNs’ superior performance over NNs’ in many tasks.
However, growing empirical evidence suggests that GNNs are not always advantageous compared to traditional NNs.
In some cases, even simple Multi-Layer Perceptrons (MLPs) can outperform GNNs by a large margin on relational data [45, 28, 31, 8].
An important reason for this is believed to be the heterophily problem: the homophily assumption does not always hold, so connected nodes may in fact have different attributes.
Heterophily has received lots of attention recently and an increasing number of models have been put forward to address this problem [45, 28, 31, 8, 44, 43, 32, 16, 24].
In this paper, we ﬁrst show that by only considering graph-label consistency, existing homophily metrics are not able to describe the effect of some cases of heterophily on aggregation-based GNNs.
We propose a post-aggregation node similarity matrix, and based on it, we derive new homophily metrics, whose advantages are illustrated on synthetic graphs (Sec. 3).
Then, we prove that diversiﬁcation operation can help to address some harmful cases of heterophily (Sec. 4).
Based on this, we propose the Adaptive Channel Mixing (ACM)
GNN framework which augments uni-channel baseline GNNs, allowing them to exploit aggregation, diversiﬁcation and identity channels adaptively, node-wisely and locally in each layer.
ACM signiﬁcantly boosts the performance of 3 uni-channel baseline GNNs by 2.04% ∼ 27.5% for node classiﬁcation tasks on 7 widely used benchmark heterophilic graphs, exceeding SOTA models (Sec. 6) on all of them.
For 3 homophilic graphs, ACM-augmented GNNs can perform at least as well as the uni-channel baselines and are competitive compared with SOTA.
Contributions 1.
To our knowledge, we are the ﬁrst to analyze heterophily from post-aggregation node similarity perspective.
The proposed ACM framework is highly different from adaptive ﬁlterbank with multiple channels and existing GNNs for heterophily: 1) the traditional adaptive ﬁlterbank channels [39] uses a scalar weight for each ﬁlter and this weight is shared by all nodes.
In contrast, ACM provides a mechanism so that different nodes can learn different weights to utilize information from different channels to account for diverse local heterophily; 2) Unlike existing methods that leverage the high-order ﬁlters and global property of high-frequency signals [45, 28, 8, 16] which require more computational resources, ACM successfully addresses heterophily by considering only the nodewise local information adaptively .
Unlike existing methods that try to facilitate learning ﬁlters with high expressive power [45, 44, 8, 16], ACM aims that, when given a ﬁlter with certain expressive power, we can extract richer information from additional channels in a certain way to address heterophily.
This makes ACM more ﬂexible and easier to be implemented.
Preliminaries
In this section, we introduce notation and background knowledge.
We use bold font for vectors ( e.g., v ).
Suppose we have an undirected connected graph G =
( V , E , A ) , where V is the node set with |V| = N ; E is the edge set without self-loops; A ∈ R N × N is the symmetric adjacency matrix with A i,j = 1 if e ij ∈ E
otherwise A i,j = 0 .
Let D denote the diagonal degree matrix of G , i.e., D i,i =
d i = (cid:80)
j A i,j .
Let N i denote the neighborhood set of node i , i.e., N i =
{ j : e ij ∈ E} .
A graph signal is a vector x ∈ R N deﬁned on V , where x i is associated with node i .
We also have a feature matrix X ∈ R N × F , whose columns are graph signals and whose i -th row
X i, : is a feature vector of node i .
We use Z ∈ R N × C to denote the label encoding matrix, whose i -th row Z i, : is the one-hot encoding of the label of node i .
2.1 Graph Laplacian, Afﬁnity Matrix and Variants
The (combinatorial) graph Laplacian is deﬁned as L = D − A , which is Symmetric Positive
Semi-Deﬁnite (SPSD) [9].
Its eigendecomposition is L = U Λ U T , where the columns u i of
U ∈ R N × N are orthonormal eigenvectors, namely the graph Fourier basis , Λ = diag( λ 1 , . .
. , λ N ) with λ 1 ≤ · · · ≤ λ N .
These eigenvalues are also called frequencies .
In additional to L , some variants are also commonly used, e.g., the symmetric normalized Laplacian
L sym =
D − 1 / 2 LD − 1 / 2 =
I − D − 1 / 2 AD − 1 / 2 and the random walk normalized Laplacian L
rw =
D − 1 L =
I − D − 1 A .
The graph Laplacian and its variants can be considered as high-pass ﬁlters for graph signals.
The afﬁnity (transition) matrices can be derived from the Laplacians, e.g., A rw =
I − L rw =
D − 1 A , A sym
=
I − L sym
=
D − 1 / 2 AD − 1 / 2 and are considered to be low-pass ﬁlters [33].
Their eigenvalues satisfy λ
i ( A rw )
=
λ i ( A sym )
=
1 − λ i ( L sym )
= 1 − λ i ( L rw ) ∈ ( − 1 , 1] .
Applying the renormalization trick [19] to afﬁnity and Laplacian matrices respectively leads to ˆ A sym =
˜ D − 1 / 2 ˜
A ˜ D − 1 / 2 and ˆ L sym
=
I − ˆ A sym , where ˜
A ≡ A + I and ˜ D ≡ D + I .
The renormalized afﬁnity matrix essentially adds a self-loop to each node in the graph, and is widely used in Graph
Convolutional Network (GCN) [19] as follows:
[Equation]
where W 0 ∈ R F × F 1 and W 1 ∈ R F 1 × O are learnable parameter matrices.
GCNs can be trained by minimizing the following cross entropy loss
[Equation]
where log( · ) is a component-wise logarithm operation.
The random walk renormalized matrix ˆ A rw =
˜ D − 1 ˜ A , which shares the same eigenvalues as ˆ A sym , can also be applied in GCN.
The corresponding Laplacian is deﬁned as ˆ L rw =
I − ˆ A rw .
The matrix ˆ
A rw is essentially a random walk matrix and behaves as a mean aggregator that is applied in spatial-based GNNs [15, 14].
To bridge spectral and spatial methods, we use ˆ A rw in this paper.
2.2 Metrics of Homophily
The homophily metrics are deﬁned by considering different relations between node labels and graph structures.
There are three commonly used homophily metrics: edge homophily [1, 45], node homophily [35] and class homophily [26] 1 , deﬁned as follows:
[Equation]
where H v node is the local homophily value for node v ; [ a ] + = max( a, 0) ; h k is the class-wise homophily metric [26].
All metrics are in the range of [0 , 1] ; a value close to 1 corresponds to strong homophily, while a value close to 0 indicates strong heterophily.
H edge ( G ) measures the proportion of edges that connect two nodes in the same class; H node ( G ) evaluates the average proportion of edge-label consistency of all nodes; H class ( G ) tries to avoid sensitivity to imbalanced classes, which can make H edge ( G ) misleadingly large.
The above deﬁnitions are all based on the linear featureindependent graph-label consistency .
The inconsistency relation is implied to have a negative effect to the performance of GNNs.
With this in mind, in the following section, we give an example to illustrate the shortcomings of the above metrics and propose new feature-independent metrics that are deﬁned from post-aggregation node similarity perspective, which is novel.
Analysis of Heterophily 3.1 Motivation and Aggregation Homophily
[Figure]
Heterophily is widely believed to be harmful for message-passing based GNNs [45, 35, 8] because, intuitively, features of nodes in different classes will be falsely mixed, leading nodes to be indistinguishable [45].
Nevertheless, it is not always the case, e.g., the bipartite graph 2 shown in Figure 1 is highly heterophilic according to the existing homophily metrics in equation 3, but after mean aggregation, the nodes in classes 1 and 2 just exchange colors and are still distinguishable 3 .
This example tells us that, besides graph-label consistency, we need to study the relation between nodes after aggregation step.
To this end, we ﬁrst deﬁne the post-aggregation node similarity matrix as follows:
[Equation]
where ˆ A ∈ R N × N denotes a general aggregation operator.
S ( ˆ A, X ) is essentially the gram matrix that measures the similarity between each pair of aggregated node features.
Relationship Between S ( ˆ A, X ) and Gradient of SGC SGC [41] is one of the most simple but representative GNN models and its output can be written as:
[Equation]
With the loss function in equation 2, after each gradient descent step, we have ∆ W = γ d L dW , where γ is the learning rate.
The update of Y (cid:48) is (see Appendix E for derivation):
[Equation]
where Z − Y is the prediction error matrix.
The update direction of the prediction for node i is essentially a weighted sum of the prediction error, i.e., ∆( Y (cid:48) )
i, : = (cid:80) j ∈V (cid:2) S ( ˆ A, X ) (cid:3) i,j ( Z − Y ) j, : and (cid:2) S ( ˆ A, X ) (cid:3) i,j can be considered as the weights.
Intuitively, a high similarity value (cid:2) S ( ˆ A, X ) (cid:3) i,j means node i tends to be updated to the same class as node j .
This indicates that S ( ˆ A, X ) is closely related to a single layer GNN model.
Based on the above deﬁnition and observation, we deﬁne the aggregation similarity score as follows.
Deﬁnition 1.
The aggregation similarity score is:
[Equation]
where Mean u ( {·} ) takes the average over u of a given multiset of values or variables. [Equation]
S ( ˆ A, X )) measures the proportion of nodes v ∈ V as which the average weights on the set of nodes in the same class (including v ) is larger than that in other classes.
In practice, we observe that in most datasets, we will have S agg ( S ( ˆ A, X )) ≥ 0 .
To make the metric range in [0,1], like existing metrics, we rescale equation 7 to the following modiﬁed aggregation similarity,
[Equation]
In order to measure the consistency between labels and graph structures without considering node features and to make a fair comparison with the existing homophily metrics in equation 3, we deﬁne the graph ( G ) aggregation ( ˆ A ) homophily and its modiﬁed version 5 as:
[Equation]
As the example shown in Figure 1, when ˆ A = ˆ A rw , it is easy to see that H agg ( G ) =
H M agg ( G ) = and other metrics are 0.
Thus, this new metric reﬂects the fact that nodes in classes 1 and 2 are still highly distinguishable after aggregation, while other metrics mentioned before fail to capture such information and misleadingly give value 0.
This shows the advantage of H agg ( G ) and H M agg ( G ) , which additionally exploit information from aggregation operator ˆ A and the similarity matrix.
To comprehensively compare H M agg ( G ) with the existing metrics on their ability to elucidate the inﬂuence of graph structure on GNN performance, we generate synthetic graphs with different homophily levels and evaluate SGC [41] and GCN [19] on them in the next subsection. [Equation] [Figure] In practice, we will only check H agg ( G )
when H M agg ( G ) = 0 .
[Figure]
3.2 Empirical Evaluation and Comparison on Synthetic Graphs
In this subsection [Figure] we conduct experiments on synthetic graphs generated with different levels of
H M edge ( G ) to assess the output of H M agg ( G ) in comparison with existing metrics.
Data Generation & Experimental Setup We ﬁrst generated 10 graphs for each of 28 edge homophily levels, from 0.005 to 0.95, for a total of 280 graphs.
In every generated graph, we hadclasses, with 400 nodes in each class.
For nodes in each class, we randomly generated 800 intra-class edges and [ 800 H edge ( G ) − 800 ] inter-class edges.
The features of nodes in each class are sampled from node features in the corresponding class of 6 base datasets ( Cora, CiteSeer, PubMed, Chameleon, Squirrel,
Film ).
Nodes were randomly split into train/validation/test sets, in proportion of 60%/20%/20%.
We trained 1-hop SGC ( sgc-1 ) [41] and GCN [19] on the synthetic graphs 6 .
For each value of H edge ( G ) , we take the average test accuracy and standard deviation of runs over the 10 generated graphs with that value.
For each generated graph, we also calculate H node ( G ) , H class ( G ) and H M agg ( G ) .
Model performance with respect to different homophily values is shown in Figure 2.
Comparison of Homophily Metrics
The performance of SGC-1 and GCN is expected to be monotonically increasing if the homophily metric is informative.
However, Figure 2(a)(b)(c) show that the performance curves under H edge ( G ) , H node ( G ) and H class ( G ) are U -shaped 7 , while Figure 2(d) reveals a nearly monotonic curve with a little numerical perturbation around 1.
This indicates that
H M agg ( G ) provides a better indication of the way in which the graph structure affects the performance of SGC-1 and GCN than existing metrics.
(See more discussion on aggregation homophily and theoretical results for regular graphs in Appendix D.)
Adaptive Channel Mixing (ACM)
In prior work [31, 8, 4], it has been shown that high-frequency graph signals, which can be extracted by a high-pass ﬁlter (HP), is empirically useful for addressing heterophily.
In this section, based on the similarity matrix in equation 6, we theoretically prove that a diversiﬁcation operation, i.e., HP ﬁlter, can address some cases of harmful heterophily locally.
Besides, a node-wise analysis shows that different nodes may need different ﬁlters to process their neighborhood information.
Based on the above analysis, in Sec.
4.2 we propose Adaptive Channel Mixing (ACM), a 3-channel architecture which can adaptively exploit local and node-wise information from aggregation, diversiﬁcation and identity channels.
4.1 Diversiﬁcation Helps with Harmful Heterophily
We ﬁrst consider the example shown in Figure 3.
From S ( ˆ A, X ) , we can see that nodes { 1 , 3 } assign relatively large positive weights to nodes in class 2 after aggregation, which will make nodes { 1 , 3 } hard to be distinguished from nodes in class 2.
However, we can still distinguish nodes { 1 , 3 } and { 4 , 5 , 6 , 7 } by considering their neighborhood differences: nodes { 1 , 3 } are different from most of their neighbors while nodes { 4 , 5 , 6 , 7 } are similar to most of their neighbors.
This indicates that although some nodes become similar after aggregation, they are still distinguishable through their local surrounding dissimilarities.
This observation leads us to introduce the diversiﬁcation operation , i.e., HP ﬁlter I − ˆ A [11] to extract information regarding neighborhood differences, thereby addressing harmful heterophily.
As
S ( I − ˆ A, X ) in Fig.
3 shows, nodes { 1 , 3 } will assign negative weights to nodes { 4 , 5 , 6 , 7 } after the diversiﬁcation operation, i.e., nodes 1,3 treat nodes 4,5,6,7 as negative samples and will move away from them during backpropagation.
This example reveals that there are cases in which the diversiﬁcation operation is helpful to handle heterophily, while the aggregation operation is not.
Based on this observation, we ﬁrst deﬁne the diversiﬁcation distinguishability of a node and the graph diversiﬁcation distinguishability value, which measures the proportion of nodes for which the diversiﬁcation operation is potentially helpful.
Deﬁnition 2.
Diversiﬁcation Distinguishability (DD) based on S ( I − ˆ A, X ) .
Given S ( I − ˆ A, X ) , a node v is diversiﬁcation distinguishable if the following two conditions are satisﬁed at the same time,
[Equation]
Then, graph diversiﬁcation distinguishability value is deﬁned as
[Equation]
v is diversiﬁcation distinguishable
[Equation]
We can see that DD ˆ A,X ( G ) ∈ [0 , 1] .
Based on Def. 2 , the effectiveness of diversiﬁcation in addressing heterophily can be theoretically proved under certain conditions:
Theorem 1.
(See Appendix G for proof).
For C = 2 , suppose X
=
Z, ˆ A
= ˆ
A rw .
Then for any
I − ˆ A rw , all nodes are diversiﬁcation distinguishable and DD ˆ A,Z ( G ) = 1 .
With the above results for HP ﬁlters, we will now introduce the concept of ﬁlterbank which combines both LP (aggregation) and HP (diversiﬁcation) ﬁlters and can potentially handle various local heterophily cases.
We then develop ACM framework in the following subsection.
4.2 Filterbank and Adaptive Channel Mixing (ACM) Framework
[Figure] Filterbank For the graph signal x deﬁned on G , a 2-channel linear (analysis) ﬁlterbank [11] 8 includes a pair of ﬁlters H LP , H HP , which retain the low-frequency and high-frequency content of x , respectively.
Most existing GNNs use a uni-channel ﬁltering architecture [19, 40, 15] with either LP or HP channel, which only partially preserves the input information.
Unlike the uni-channel architecture, ﬁlterbanks with H LP + H HP
= I do not lose any information from the input signal, which is called the perfect reconstruction property [11].
Generally, the Laplacian matrices ( L sym , L rw , ˆ L sym , ˆ L rw ) can be regarded as HP ﬁlters [11] and afﬁnity matrices ( A sym , A rw , ˆ A sym , ˆ A rw ) can be treated as LP ﬁlters [33, 14].
Moreover, we extend the concept of ﬁlterbank and view MLPs as using the identity (fullpass) ﬁlterbank with H LP
=
I and H HP = 0 , which also satisﬁes H LP + H HP
= I +
= I .
Node-wise Channel Mixing for Diverse Local Homophily The example in Figure 3 also shows that different nodes may need the local information extracted from different channels, e.g., nodes { 1 , 3 } demand information from the HP channel while node 2 only needs information from the LP channel.
Figure 4 reveals that nodes have diverse distributions of node local homophily H v node across different datasets.
In order to adaptively leverage the LP, HP and identity channels in GNNs to deal with the diverse local heterophily situations, we will now describe our proposed Adaptive Channel
Mixing (ACM) framework.
Adaptive Channel Mixing (ACM)
We will use GCN 9 as an example to introduce the ACM framework in matrix form, but the framework can be combined in a similar manner to many different
GNNs.
The ACM framework includes the following steps:
Step
Feature Extraction for Each Channel:
[Equation]
Learning
[Equation]
Step wise
Channel Mixing:
[Equation]
We will refer to the instantiation which uses option 1 in step 1 as ACM and to the one using option 2 as ACMII.
In step 1, ACM(II)-GCN implement different feature extractions for 3 channels using a set of ﬁlterbanks.
Three ﬁltered components, H lL , H lH , H lI , are obtained.
To adaptively exploit information from each channel, ACM(II)-GCN ﬁrst extract nonlinear information from the ﬁltered signals, then use W l Mix to learn which channel is important for each node, leading to the row-wise weight vectors α lL , α lH , α lI ∈ R N × 1 whose i -th elements are the weights for node i 10 .
These three vectors are then used as weights in deﬁning the updated H l in step 3.
Complexity
The number of learnable parameters in layer l of ACM(II)-GCN is 3 F l − 1 ( F l + 1) + 9 , compared to F l − 1 F l in GCN.
The computation of steps 1-3 takes NF l (8+6 F l − 1 )
+2 F l ( nnz ( H LP )+ nnz
( H HP ))
+ 18 N ﬂops, while the GCN layer takes 2 NF l − 1 F l + 2 F l ( nnz ( H LP )) ﬂops, where nnz ( · ) is the number of non-zero elements.
An ablation study and a detailed comparison on running time are conducted in Sec. 6.1.
Limitations of Diversiﬁcation Like any other method, there exists some cases of harmful heterophily that diversiﬁcation operation cannot work well.
For example, suppose we have an imbalanced dataset where several small clusters with distinctive labels are densely connected to a large cluster.
In this case, the surrounding differences of nodes in small clusters are similar, i.e., the neighborhood differences mainly come from their connections to the same large cluster, and this can lead to the diversiﬁcation operation failing to discriminate them.
See Appendix H for a more detailed discussion.
Related Work
We now discuss relevant work on addressing heterophily in GNNs.
[1] acknowledges the difﬁculty of learning on graphs with weak homophily and propose MixHop to extract features from multi-hop neighborhoods to get more information.
[17] propose measurements based on feature smoothness and label smoothness that are potentially helpful to guide GNNs when dealing with heterophilic graphs.
Geom-GCN [35] precomputes unsupervised node embeddings and uses the graph structure deﬁned by geometric relationships in the embedding space to deﬁne the bi-level aggregation process to handle heterophily.
H 2 GCN [45] combines 3 key designs to address heterophily: (1) ego- and neighbor-embedding separation; (2) higher-order neighborhoods; (3) combination of intermediate representations.
CPGNN [44] models label correlations through a compatibility matrix, which is beneﬁcial for heterophilic graphs, and propagates a prior belief estimation into the GNN by using the compatibility matrix.
FAGCN [4] learns edge-level aggregation weights as GAT [40] but allows the weights to be negative, which enables the network to capture high-frequency components in the graph signals.
GPRGNN [8] uses learnable weights that can be both positive and negative for feature propagation.
This allows GPRGNN to adapt to heterophilic graphs and to handle both high- and low-frequency parts of the graph signals (See Appendix J for a more comprehensive comparison between ACM-GNNs, ACMII-GNNs and FAGCN, GPRGNN).
BernNet [16] designs a scheme to learn arbitrary graph spectral ﬁlters with Bernstein polynomial to address heterophily.
[32] points out that homophily is not necessary for GNNs and characterizes conditions that GNNs can perform well on heterophilic graphs.
Empirical Evaluation
In this section, we evaluate the proposed ACM and ACMII framework on real-world datasets (see Appendix D.2 for a performance comparison with basline models on synthetic datasets).
We ﬁrst conduct ablation studies in Sec.
6.1 to validate the effectiveness and efﬁciency of different components of ACM and ACMII.
Then, we compare with state-of-the-art (SOTA) models in Sec. 6.2.
The hyperparameter searching range and computing resources are described in Appendix C.
[Figure]
& Efﬁciency
We will now investigate the effectiveness and efﬁciency of adding HP, identity channels and the adaptive mixing mechanism in the proposed framework by performing an ablation study.
Speciﬁcally, we apply the components of ACM to SGC-1 [41] 11 and the components of ACM and ACMII to
GCN [19] separately.
We run 10 times on each of the 9 benchmark datatsets, Cornell , Wisconsin ,
Texas , Film , Chameleon , Squirrel , Cora , Citeseer and Pubmed used in [36, 35], with the same 60%/20%/20% random splits for train/validation/test used in [8] and report the average test accuracy as well as the standard deviation.
We also record the average running time per epoch (in milliseconds) to compare the computational efﬁciency.
We set the temperature T in equation 4.2 to be 3 , which is the number of channels.
The results in Table 1 show that on most datasets, the additional HP and identity channels are helpful, even for strong homophily datasets such as Cora, CiteSeer and PubMed .
The adaptive mixing mechanism also has an advantage over directly adding the three channels together.
This illustrates the necessity of learning to customize the channel usage adaptively for different nodes.
The t-SNE visualization in Figure 5 demonstrates that the high-pass channel(e) and identity channel(f) can extract meaningful patterns, which the low-pass channel(d) is not able to capture. The output of ACMGCN(c) shows clearer boundaries among classes than GCN(b).
The running time is approximately doubled in the ACM and ACMII framework compared to the original models.
[Table]
6.2 Comparison with Baseline and SOTA Models
Datasets & Experimental Setup
In this section, we evaluate SGC [41] with 1 hop and 2 hops (SGC-1, SGC-2), GCNII [7], GCNII ∗ [7], GCN [19] and snowball networks [29] with 2 and 3 layers (snowball-2, snowball-3) and combine them with the ACM or ACMII framework 12 .
We use ˆ A rw as the LP ﬁlter and the corresponding HP ﬁlter is I − ˆ A rw 13 .
Both ﬁlters are deterministic.
We compare these approaches with several baselines and SOTA GNN models: MLP with 2 layers (MLP-2),
GAT [40], APPNP [20], GPRGNN [8], H 2 GCN [45], MixHop [1], GCN+JK [19, 42, 26], GAT+JK [40, 42, 26], FAGCN [4], GraphSAGE [15], Geom-GCN [35] and BernNet [16].
In addition to thebenchmark datasets used in section 6.1, we further test the above models on a new benchmark dataset, Deezer-Europe [37] 14 .
On each dataset used in [36, 35], we test the models 10 times following the same early stopping strategy, the same 60%/20%/20% random data split 15 and Adam [18] optimizer as used in GPRGNN [8].
For Deezer-Europe , we test the above models 5 times with the same early stopping strategy, the same ﬁxed splits and Adam used in [26].
Structure information channel and residual connection Besides the ﬁltered features, some recent SOTA models additionally use graph structure information, i.e., MLP θ ( A ) , and residual connection to address heterophily problem, e.g., LINKX [25] and GloGNN [24].
MLP θ ( A ) and residual connection can be directly incorporated into ACM and ACMII framework, which leads us to
ACM(II)-GCN+ and ACM(II)-GCN++.
See the details of implementation in Appendix B.
To visualize the performance, in Fig.
6, we plot the bar charts of the test accuracy of SOTA models, three selected baselines (GCN, snowball-2, snowball-3), their ACM(II) augmented models, ACM(II)- See explanation [Figure] ↑ ” shows the range of performance improvement (%) of ACM-GNNs and ACMII-GNNs over baseline GNNs.
See Appendix I for a detailed discussion of the relation between H M agg and GNN performance.
GCN+ and ACM(II)-GCN++ on the 6 most commonly used benchmark heterophily datasets (See
Table 2 in Appendix A.1 for the full results, comparison and ranking).
From Fig.
6, we can see that (1) after being combined with the ACM or ACMII framework, the performance of the three baseline models is signiﬁcantly boosted, by 2.04% ∼ 27.50% on all the 6 tasks.
The ACM and ACMII in fact achieve SOTA performance.
(2) On Cornell, Wisconsin, Texas, Chameleon and Squirrel , the augmented baseline models signiﬁcantly outperform the current SOTA models .
Overall, these results suggest that the proposed approach can help GNNs to generalize better on node classiﬁcation tasks on heterophilic graphs, without adding too much computational cost.
Conclusions and Limitations
We have presented an analysis of existing homophily metrics and proposed new metrics which are more informative in terms of correlating with GNN performance.
To our knowledge, this is the ﬁrst work analyzing heterophily from the perspective of post-aggregation node similarity.
The similarity matrix and the new metrics we deﬁned mainly capture linear feature-independent relationships of each node.
This might be insufﬁcient when nonlinearity and feature-dependent information is important for classiﬁcation.
In the future, it would be useful to investigate if a similarity matrix could be deﬁned which is capable of capturing nonlinear and feature-dependent relations between aggregated node.
We have also proposed a multi-channel mixing mechanism which leverages the intuitions gained in the ﬁrst part of the paper and can be combined with different GNN architectures, enabling adaptive ﬁltering (high-pass, low-pass or identity) at different nodes.
Empirically, this approach shows very promising results, improving the performance of the base GNNs with which it is combined and achieving SOTA results at the cost of a reasonable increase in computation time.
As discussed in
Sec. 4.2, however, the ﬁlterbank method cannot properly handle all cases of harmful heterophily, and alternative ideas should be explored as well in the future.
Acknowledge
The authors would like to very special thanks to William L. Hamilton for valuable discussion and advice.
The project was partially supported by DeepMind and NSERC.
A More Experimental Results A.1 Comparison with SOTA Models on 60%/20%/20% Random Splits
The main results of the full sets of experiments 16 with statistics of datasets are summarized in Table 2, where we report the mean accuracy (%) and standard deviation.
We can see that after applied in
ACM or ACMII framework, the performance of baseline models are boosted on almost all tasks and achieve SOTA performance on 9 out of 10 datasets.
Especially, ACMII-GCN+ performs the best in terms of average rank (4.40) across all datasets.
Overall, It suggests that ACM or ACMII framework can signiﬁcantly increase the performance of GNNs on node classiﬁcation tasks on heterophilic graphs and maintain highly competitive performance on homophilic datasets.
[Table]
A.
Comparison with SOTA Models on
See table 3 for the results and table 13 14 the optimal searched hyperparameters. The results and comparison give us the same conclusion as in Appendix A.1.
[Table] , 5, 28]; results on the rest models are run by ourselves and the hyperparameter searching range is the same as table 9. A.3 Discussion of Random Walk and Symmetric Renormalized Filters [Table]
The deﬁnitions of the similarity matrix, (modiﬁed) aggregation similarity score and diversiﬁcation distinguishability value can be extended to symmetric normalized Laplacian or other aggregation operations.
Yet unfortunately, we cannot extend Theorem 1 at this moment, because we need a condition that the row sum of ˆ A is not greater than 1 in the proof.
This condition is guaranteed for random walk normalized Laplacian but not for symmetric normalized Laplacian.
While in practice, we evaluate our models with symmetric ﬁlters and compare them with random walk ﬁlters.
From table 4 we can see that, there are no big differences between these two ﬁlters. Datasets/Models [Table] A.4 Ablation Study
W
From table 5 we can see that ACM(II) with W mix shows superiority in most datasets, although it is not statistically signiﬁcant on some of them.
One possible explanation of the function of W mix is that it could help alleviate the dominance and bias to majority: Suppose in a dataset, most of the nodes need more information from LP channel than HP and identity channels, then W L , W H , W I tend to learn larger α L than α H and α I .
For the minority nodes that need more information from HP or identity channels, they are hard to get large α H or α I values because W L , W H , W I are biased to the majority.
And W mix can help us to learn more diverse alpha values when W L , W H , W I are biased.
Attention with more complicated design can be found for the node-wise adaptive channel mixing mechanism, but we do not explore this direction deeper in this paper because investigating attention function is not the main contribution of our paper.
A.5 Learn Weights with Raw Features v.s. Combined Features
[Table]
Construct the combined feature H l Comb =
[ H lL , H lH , H lI ] , Replace the ﬁrst line in Step 2 by the following lines:
[Equation]
The performance comparison can be found in table 6.
From the results, we do not ﬁnd signiﬁcant difference between the frameworks with combined features and raw features.
The reason is that the necessary nonlinear information from each channel is combined in (cid:2)
˜ α lL , ˜ α lH , ˜ α lI (cid:3) and W l Mix is enough to learn to mix the combined weights from different channels.
The learning of redundant information in the feature extraction step for each channel will not improve the performance.
Meanwhile, A disadvantage of the combined feature is that it increases the computational cost.
Thus, we decide to use the raw features.
A.6 H v node Distributions of Different Datasets
See Figure 7 for H v node distributions.
We can see that Wisconsin and Texas have high density in low homophily area, Cornell, Chameleon, Squirrel and Film have high density in low and middle homophily area, Cora, CiteSeer and PubMed have high density in high homophily area.
[Figure]
A.7 Distributions of Learned α L , α H , α I in the Hidden and Output Layers of ACN-GCN
See Figure 8 for the distributions of weights in hidden layers and Figure 9 for the distributions of weights in output layers.
Alpha values
D e n s i t y
Alpha values
L
H
I (a) Cornell 0.250.500. Alpha values D e n s i t y Alpha values
L
H
I (b) Wisconsin 0.25 0.
Alpha values
D e n s i t y
Alpha values
L
H
I (c) Texas 0.25 0.
Alpha values
D e n s i t y
Alpha values
L
H
I (d) Chameleon 0.
Alpha values
D e n s i t y
Alpha values
L
H
I (e) Squirrel 0.
Alpha values
D e n s i t y
Alpha values
L
H
I (f) Film 0.250.500.
Alpha values
D e n s i t y
Alpha values
L
H
I (g) Cora 0.250.500.
Alpha values
D e n s i t y
Alpha values
L
H
I (h) Citeseer 0.250.500.
Alpha values
D e n s i t y
Alpha values
L
H I (i) Pubmed [Figure] L
H
I
[Figure]
n [Figure] i [Figure]
L
H
I ( [Figure] e [Figure]
s i t y
Alpha values
L
H
I (
[Figure]
s i t y
[Figure]
L
[Figure]
I
[Figure]
L
H
I ( [Figure] L [Figure]
L H I
[Figure]
B Details of the Implementation B.1 Implementation of ACM-GCMII
Unlike other baseline GNN models, GCNII and GCNII* are not able to be applied under ACMII framework and we will make an explanation as follows.
[Equation]
W ( (cid:96) )
(cid:17)(cid:17)
GCNII*: H ( (cid:96) +1) =
σ (cid:16) (1 − α (cid:96) )
ˆ AH ( (cid:96) )
(cid:16) (1 − β (cid:96) )
I n + β (cid:96) W ( (cid:96) ) 1 (cid:17)
+ + α (cid:96) H (0) (cid:16) (1 − β (cid:96) )
I n + β (cid:96) W ( (cid:96) ) 2 (cid:17)(cid:17)
From the above formulas of GCNII and GCNII ∗ we cam see that, without major modiﬁcation, GCNII and GCNII* are hard to be put into ACMII framework.
In ACMII framework, before apply ˆ A , we ﬁrst implement a nonlinear feature extractor σ ( H (cid:96) W ( (cid:96) ) ) .
But in GCNII and GCNII*, before multiplying W (cid:96) ( or W (cid:96) 1 , W (cid:96) 2 ) to extract features, we need to add another term including H (0) , which are not ﬁltered by ˆ A .
This makes the order of aggregator ˆ A and nonlinear extractor unexchangable and thus, incompatible with ACMII framework.
So we did not implement GCNII and GCNII* in ACMII framework.
B.2 Implementation of ACM(II)-GCN+ and ACM(II)-GCN++
Besides the features extracted by different ﬁlters, some recent SOTA models use additional graph structure information explicitly, i.e., MLP θ ( A ) , to address heterophily problem, e.g., LINKX [25] and GloGNN [24] and is found effective on some datasets, e.g., Chameleon, Squirrel .
The explicit structure information can be directly incorporated into ACM and ACMII framework, and we have
ACM(II)-GCN+ and ACM(II)-GCN++ as follows.
• ACM-GCN+ and ACMII-GCN+ have an option to include structure information channel (the 4-th channel) in each layer and their differences from ACM-GCN and ACMII-GCN are highlighted in red) as follows, Step 1.
Feature Extraction for LP, HP, Identity and Structure Information Channel:
H lA = ReLU (cid:0) AW lA
(cid:1)
, W lA ∈ R N × F l , get H lL , H lH , H lI with the same step as ACM-GCN and ACMII-GCN.
Step 2.
Row-wise Feature-based Weight Learning with Layer Normalization (LN)
˜ H lL =
LN ( H lL ) , ˜ H
lH = LN ( H lH ) ,
˜ H lI =
LN ( H lI ) , ˜ H lA = LN ( H lA ) , ˜
α lL =
Sigmoid (cid:16) ˜ H lL ˜ W lL (cid:17) , ˜ α lH =
Sigmoid (cid:16) ˜ H lH ˜ W lH (cid:17) , ˜ α lI =
Sigmoid (cid:16) ˜
H lI ˜ W lI (cid:17) , ˜ α lA =
Sigmoid (cid:16) ˜
H lA ˜ W lA (cid:17) , ˜ W l − 1 L , ˜ W l − 1 H , ˜ W l − 1 I , ˜ W lA ∈ R F l ×
Step 3.
Node-wise Adaptive Channel Mixing: Option
without structure information
[Equation]
Option 2: with structure information
[Equation]
• ACM-GCN++ and ACMII-GCN++ have an option to include structure information channel (the 4-th channel) in each layer and residual connection and their differences from ACMGCN+ and ACMII-GCN+ are highlighted in red) as follows, Step 1.
Feature Extraction for LP, HP, Identity and Structure Information Channel, Get H X : H X = ReLU ( XW X ) ∈ R F × F (cid:48) ,
H lA = ReLU (cid:0) AW lA (cid:1) , W lA ∈ R N × F
[Equation]
, get H lL , H lH , H lI with the same step as ACM-GCN and ACMII-GCN.
Step 2.
Row-wise Feature-based Weight Learning with Layer Normalization (LN)
[Equation]
˜ H lA = LN ( H lA ) , ˜
α lL =
Sigmoid (cid:16) ˜ H lL ˜ W lL (cid:17) , ˜ α lH =
Sigmoid (cid:16) ˜ H lH ˜ W lH (cid:17) , ˜ α lI =
Sigmoid (cid:16) ˜ H lI ˜ W lI (cid:17) , ˜ α lA = Sigmoid (cid:16) ˜ H lA ˜ W lA (cid:17) , ˜ W l − 1 L , ˜ W l − 1 H , ˜ W l − 1 I , ˜ W lA ∈ R F (cid:48) ×
Step 3.
Node-wise Adaptive Channel Mixing:
Option 1: without structure information
[Equation]
Option 2: with structure information
[Equation]
H
[Equation]
lL
[Equation]
lL [Equation] lI [Equation]
+ H X
The results of ACM-GCN+, ACMII-GCN+, ACM-GCN++ and ACMII-GCN++ trained on random 60%/20%/20% splits are reported in table 2 in Computing Resources For all experiments on synthetic datasets and real-world datasets, we use NVIDIA V100 GPUs with 16/32GB GPU memory, 8-core CPU, 16G Memory. The software implementation is based on PyTorch and PyTorch Geometric [12].
C Hyperparameter Searching Range & Optimal Hyperparameters C.1 Hyperparameter Searching Range for Synthetic Experiments
Hyperparameter Searching Range for Synthetic Experiments [Table] [Table]
[Table]
C.2 Hyperparameter Searching Range for Ablation Study
[Table]
C.3 Hyperparameter Searching Range for GNNs on Real-world Datasets
See table 9 for the hyperparameter seaching range of baseline GNNs, ACM-GNNs, ACMII-GNNs and several SOTA models.
C.4 Searched Optimal Hyperparameters for Baselines and ACM(II)-GNNs on Real-world Tasks
See the reported optimal hyperparameters on random 60%/20%/20% splits for baseline GNNs in table 10, for ACM-GNNs and ACMII-GNNs in table 11 and for ACM(II)-GCN+ and ACM(II)-GCN++ in table 12.
See the reported optimal hyperparameters on ﬁxed 48%/32%/20% splits for ACM(II)-GNNs and
FAGCN in table 13 and for ACM(II)-GCN+ and ACM(II)-GCN++ in table 14.
[Table]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
Squirrel
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
[Table]
Chameleon
[Table]
Deezer
[Table]
[Table]
[Table]
[Table]
[Table]
25s ACMII-
[Table]
++
[Table]
[Table]
[Table]
Film
ACM-
[Table]
+ [Table] [Table] [Table]
ACMII-GCN+
[Table]
[Table] Chameleon
ACM-
[Table]
+
[Table] [Table]
+
[Table]
[Table]
58s ACMII-GCN++
[Table] [Table]
[Table] [Table]
[Table]
Squirrel
[Table] [Table]
+
[Table]
[Table]
[Table]
[Table]
[Table] [Table]
+
[Table]
[Table]
[Table]
Y 65.
[Table]
D Experimental Setup and Further Discussion on Synthetic Graphs D.1 Detailed Description of Data Generation Process
• For each node v , we ﬁrst randomly generate its degree d v .
• Given d v , for any h , we sample hd v intra-class edges and (1 − h )
d v inter-class edges.
More speciﬁcally in our synthetic experiments, for a given h , • we generate node degree d v for nodes in each class from multinomial distribution with numpy.random.multinomial(800/h, numpy.ones(400)/400, size=1)[0] .
• For a sampled d v , we generate intra-class edges from (does not include self-loop) numpy.random.multinomial(h d v , numpy.ones(399)/399, size=1)[0] and interclass edges from numpy.random.multinomial((1-h) d v , numpy.ones(1600)/1600, size=1)[0] .
For each generated graph, we calculate their H node , H class , H M agg .
Then, we reorder the value of the metrics in ascend order for x-axis and plot the corresponding test accuracy.
Here is a simpliﬁed example of how we draw Figure 2.
Suppose we generate 3 graphs with H edge = 0 . 1 , 0 .
9 , the test accuracy of GCN on these 3 synthetic graphs are 0 . 8 , 0 .
For the generated graphs, we calculate their H M agg , and suppose we get H M agg = 0 .
Then we will draw the performance of GCN under H M agg with ascend x-axis order [0 . 4 , 0 . 7 , 0 . 8] and the corresponding reordered y-axis is [0 . 5 , 0 . 8 , 0 . 9] . Other ﬁgures are drawn with the same process. D.2 Model Comparison on Synthetic Graphs
[Figure] [Figure] In order to separate the effects of nonlinearity and graph structure, we compare SGC with 1 hop (sgc-1) with MLP-1 (linear model).
For GCN which includes nonlinearity, we use MLP-2 as its corresponding graph-agnostic baseline model. We train the above GNN models, graph-agnostic baseline models and ACM-GNN models on all synthetic datasets and plot the mean test accuracy with standard deviation on each dataset. From Figure 10 and Figure 11, we can see that on each H M agg ( G ) level, ACM-GNNs will not underperform baseline GNNs and the graph-agnostic models.
But when H M agg ( G ) is small, baseline GNNs will be outperformed by graph-agnostic models by a large margin.
This demonstrate that the ACM framework can help GNNs to perform well on harmful graphs while keep competitive on less harmful graphs.
D.3 Further Discussion of Aggregation Homophily on Regular Graphs
We notice that in Figure 2(a), the performance of SGC-1 and GCN both have a turning point, i.e., when H edge ( G ) is smaller than a certain value, the performance will get better instead of getting worse.
With some extra restriction on node degree in data generation process, we ﬁnd that this interesting phenomenon can be theoretically explained by the following proposition 1 based on our proposed similarity matrix which can verify the usefulness of H M agg ( G ) .
We ﬁrst generate regular graphs , i.e., each node has the same degree, as follows,
Generate Synthetic Regular Graphs We ﬁrst generate 180 graphs in total with 18 edge homophily levels varied from 0.05 to 0.9, each corresponding to 10 graphs.
For every generated graph, we have 5 classes with 400 nodes in each class.
For each node, we randomly generate 10 intra-class edges and [ 10 H edge ( G ) − 10 ] inter-class edges.
The features of nodes in each class are sampled from node features in the corresponding class of the base dataset.
Nodes are randomly split into 60%/20%/20% for train/validation/test.
We train 1-hop SGC ( sgc-1 ) [41] and GCN [19] on synthetic data (see
Appendix C.1 for hyperparameter searching range).
For each value of H edge ( G ) , we take the average test accuracy and standard deviation of runs over 10 generated graphs.
We plot the performance curves in Figure 12.
From Figure 12 we can see that the turning point is a bit less than 0.2.
We derive the following proposition for d -regular graph to explain and predict it.
Proposition 1.
(See Appendix F for proof).
Suppose there are C classes in the graph G and G is a d -regular graph (each node has d neighbors).
Given d , edges for each node are i.i.d.
generated, such that each edge of any node has probability h to connect with nodes in the same class and probability 1 − h to connect with nodes in different classes.
Let the aggregation operator ˆ
A
=
ˆ
A rw .
Then, for nodes v , u 1 and u 2 , where Z u 1 , : =
Z v, : and Z u 2 , : (cid:54) =
Z v, : , we have
[Equation]
and the minimum of g ( h ) is reached at
[Equation]
where d intra = dh , which is the expected number of neighbors of a node that have the same label as the node.
The value of g ( h ) in equation 12 is the expected differences of the similarity values between nodes in the same class as v and nodes in other classes.
g ( h ) is strongly related to the deﬁnition of aggregation homophily and its minimum potentially implies the turning point of performance curves.
In the synthetic experiments, we have d intra = 10 , C = 5 and the minimum of g ( h ) is reached at h = 5 / 27 ≈ 0 .
1852 , which corresponds to the lowest point in the performance curve in Figure 12.
In other words, the H edge ( G ) where SGC-1 and GCN perform worst is where g ( h ) gets the smallest value, instead of the point with the smallest edge homophily value, i.e., H edge ( G ) = 0 .
This reveals the advantage of H agg ( G ) over H edge ( G ) by taking use of the similarity matrix.
E Details of Gradient Calculation in equationE.1 Derivation in Matrix Form
This derivation is similar to [30].
In output layer, we have
[Equation]
where 1 C ∈ R C × 1 , ( · ) − 1 is point-wise inverse function and each element of Y is positive.
Then
[Equation]
Note that
[Equation]
Then,
[Equation]
where the 4-th equation holds due to
[Equation]
exp( Y (cid:48) ) 1 C 1 TC (cid:1) − 1 .
Thus, we have
[Equation]
For Y (cid:48) and W , we have
[Equation]
and d L = trace
[Equation]
To get d L dW we have,
[Equation]
E.2 Component-wise Derivation
Denote ˜ X
=
XW .
We rewrite L as follows:
[Equation]
[Equation]
Expand L component-wisely, we have
[Equation]
Note that C (cid:80) c
=1 Z j,c = 1 for any j .
Consider the derivation of L over ˜ X j (cid:48) ,c (cid:48) :
[Equation]
[Table]
[Equation]

[Equation]

[Equation]
Writing the above in matrix form
[Equation]
we have
[Equation]
F Proof of Proposition
Proof.
According to the given assumptions, for node v , we have ˆ A v,k =
d
+1 , the expected number of intra-class edges is dh (here the self-loop edge introduced by ˆ A is not counted based on the deﬁnition of edge homophily and data generation process) and inter-class edges is (1 − h ) d .
Suppose there are
C ≥ 2 classes.
Consider matrix ˆ AZ ,
Then, we have E
[Equation]
d
[Equation]
, where 1 is the indicator function.
When v is in class c , we have (cid:80)
[Equation]
introduced by ˆ A ).
When v is not in class c , we have (cid:80)
[Equation]
d inter-class edges uniformly distributed in the other C − 1 classes).
For nodes v, u , we have ( ˆ AZ ) v, : , ( ˆ AZ ) u, : ∈ R C and since elements in ˆ A v,k and ˆ A u,k (cid:48) are independently generated for all k, k (cid:48) ∈ V , we have
[Equation]
Thus,
[Equation]
u, v are in the same class
[Equation]
v are in different classes
For nodes u 1 , u 2 , and v , where Z u 1 , : = Z v, : and Z u 2 , : (cid:54) = Z v, : [Equation]
{+(16) = ( C − 1) 2 ( hd + 1) 
 + ( C − 1) [(1 − h ) d ]+}
[Equation]
− ( C − 1) (
[Equation]
( hd + 1)(1 − h ) d ) − ( C − 2)
[(1 − h ) d ]
[Equation]
C
[Equation]
Setting g
( h )
= 0 , we obtain the optimal h :
[Equation]
For the data generation process in the synthetic experiments, we ﬁx d intra , then d = d intra /h , which is a function of h .
We change d in equation 17 to d intra /h , leading to
[Equation]
It is easy to observe that h satisfying equation 18 still makes g ( h ) = 0 , when d in g ( h ) is replaced by d intra /h .
From equation 18 we obtain the optimal h in terms of d intra :
[Equation]
F .1 An Extension of Proposition
Base on the deﬁnition of aggregation similarity, we have
[Equation]
|V|
Then,
[Equation]
Consider the random variable
[Equation]
Since RV is symmetrically distributed and under the conditions in proposition 1, its expectation is E [ RV ] = g ( h ) as showed in equation 16.
Since the minimum of g ( h ) is 0 and RV is symmetrically distributed, we have P ( RV ≥ 0) ≥ 0 .
5 and this can explain why H agg ( G ) is always greater than 0. in many real-world tasks.
G Proof of Theorem
Proof.
Deﬁne W cv =
( ˆ AZ ) v,c .
Then,
[Equation]
Note that
[Equation]
For any node v , let the class v belongs to be denoted by c v .
For two nodes v, u , if Z v, : (cid:54) =
Z u, : , we have
[Equation]
Then, from equation 19 it follows that
[Equation]
When C = 2 ,
[Equation]
If Z v, : = Z u, : , i.e., c v = c u , we have
[Equation]
[Equation]
Then, from equation 19 it follows that
[Equation]
Thus, if C = 2 , for any v ∈ V , if Z u, : (cid:54) =
Z v, : , we have S ( I − ˆ A, Z ) v,u ≤ 0 ; if Z u, : =
Z v, : , we have S ( I − ˆ A, Z ) v,u ≥ 0 .
Apparently, the two conditions in equation 10 are satisﬁed.
Thus v is diversiﬁcation distinguishable and DD ˆ A,X ( G ) = 1 .
The theorem is proved.
H Discussion of the Limitations of Diversiﬁcation Operation
Class
Class
Class 3 𝑋 = 𝑍 1: [1,0,0,0] 2: [1,0,0,0] 3: [1,0,0,0] 4: [0,1,0,0] 5: [0,1,0,0] 6: [0,0,1,0] 7: [0,0,1,0] 8: [0,0,0,1] 9:
[Figure]
From the black box area of S ( I − ˆ A, X ) in the example in Figure 13 we can see that nodes in class 1 and 4 assign non-negative weights to each other although there is no edge between them; nodes in class 2 and 3 assign non-negative weights to each other as well.
This is because the surrounding differences of class 1 are similar as class 4, so are class 2 and 3.
In real-world applications, when nodes in several small clusters connect to a large cluster, the surrounding differences of the nodes in the small clusters will become similar.
In such case, HP ﬁlter are not able to distinguish the nodes from different small clusters [Figure] I
The Similarity, Homophily and DD ˆ A,X ( G ) Metrics and Their Estimations
Firstly, we would like to clarify that, for each curve in the synthetic experiments, the node features are ﬁxed and we only change the homophily values.
But in real-world tasks, different datasets have different features and aggregated features.
Thus, to get more instructive information for different datasets and compare them, we need to consider more metrics, e.g. feature-label consistency and aggregated-feature-label consistency.
With the similarity score of the features S agg ( S ( I, X )) and aggregated features S agg (cid:16) S ( ˆ A, X ) (cid:17) listed in Table 15, our methods open up a new perspective on analyzing and comparing the performance of graph-agnostic models and graph-aware models in real-world tasks.
Here are 2 examples.
Example 1: People observe that GCN (graph-aware model) underperforms MLP-2 (graph-agnostic model) on Cornell, Wisconsin, Texas, Film and people commonly believe that the bad graph structure (low H edge , H node , H class values) is the reason for performance degradation.
But based on the high aggregation homophily values, the graph structure inconsistency is not the main cause of the performance degradation.
And from Table 15 we can see that the S agg (cid:16) S ( ˆ A, X ) (cid:17) for those 4 datasets are lower than their corresponding S agg ( S ( I, X )) , which implies that it is the aggregated-feature-label inconsistency that causes the performance degradation, i.e. the aggregation step actually decrease the quality of node features rather than making them more distinguishable.
For the rest 5 datasets Chameleon, Squirrel, Cora, Citeseer, PubMed , we all have S agg (cid:16) S ( ˆ A, X ) (cid:17) larger than S agg ( S ( I, X )) except PubMed , which means the aggregated features have higher quality than raw features.
We can see that the proposed metrics are much more instructive than the existing ones.
Example 2:
According to H edge , H node , H class , the value for Chameleon , and Squirrel are extremely low indicating graph structure are bad for GNNs.
But on contrary, GCN outperforms MLP-2 on those 2 datasets.
Traditional homophily metrics fail to explain such phenomenon but our method can give an explanation from different angles: For Chameleon, its modiﬁed aggregation homophily is not low and its S agg (cid:16) S ( ˆ A, X ) (cid:17) is higher than its S agg ( S ( I, X )) , which means its graph-label consistency together with aggregated-feature-label consistency help the graph-aware model obtain the performance gain; for Squirrel, its modiﬁed aggregation homophily is low but its S agg (cid:16) S ( ˆ A, X ) (cid:17) is higher than its S agg ( S ( I, X )) , which means although its graph-label consistency is bad, the aggregated-feature-label consistency is the key factor to help the graph-aware model perform better.
We also need to point out that (modiﬁed) aggregation similarity score, S agg (cid:16) S ( ˆ A, X ) (cid:17) and
S agg ( S ( I, X )) are not deciding values because they do not consider the nonlinear structure in the features.
In practice, a low score does not tell us the GNN models will deﬁnitely perform bad.
[Table]
Furthermore, in most real-world applications, not all labels are available to calculate the dataset statistics.
Thus, we randomly split the data into 60%/20%/20% for training/validation/test, and only use the training labels for the estimation of the statistics.
We repeat each estimation for 10 times and report the mean with standard deviation.
The results are shown in table 15.
Analysis From the reported results we can see that the estimations are accurate and the errors are within the acceptable range, which means the proposed metrics and similarity scores can be accurately estimated with a subset of labels and this is important for real-world applications.
[-See table 
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J
A Detailed Explanation of the Differences Between ACM(II)-GNNs and GPRGNN, FAGCN
Differences with GPRGNN [8]:
• GPRGNN does not feed distinct node-wise feature transformation to different "multi-scale channels"
We ﬁrst rewrite GPRGNN as
[Equation]
[Equation] γ
[Equation]
, where H ( k ) [Equation]
ˆ
[Equation] k [Equation] H
[Equation] X i : ) .
From the above equation we can see that Z
[Equation]
the node-wise feature transformation in GPRGNN is only learned by the same θ for all the "multi-scale channels".
But in the ACM framework, different channels extract distinct information with different parameters separately.
• GPRGNN does not have node-wise mixing mechanism.
There is no node-wise mixing in GPRGNN.
The mixing mechanism in GPRGNN is Z = K (cid:80) k
=0 diag ( γ k , γ k , . . .
, γ k ) H ( k ) , i.e. for each "multi-scale channel k ", all nodes share the same mixing parameter γ k .
But in the ACM framework, the node-wise channel mixing can be written as Z = K (cid:80) k
=0 diag ( γ 1 k , γ 2 k , . . .
, γ Nk ) H ( k ) where K is the number of channels, N is the number of nodes and γ ik , i = 1 , . .
, N are the mixing weights that are learned by node i to mix channel k . ACM and ACMII allow GNNs to learn more diverse mixing parameters in diagonal than GPRGNN and thus, have stronger expressive power than GPRGNN.
Differences with FAGCN [4]:
• The targets of node-wise operations in ACM (channel mixing) and FAGCN (negative message passing) are different.
Instead of using a ﬁxed low-pass ﬁlter ˆ A , FAGCN tries to learn a more powerful aggregator ˆ A (cid:48) based on ˆ A by allowing negative message passing.
The node-wise operation in FAGCN is similar to GAT [3] which is trying to modify the node-wise ﬁltering (message passing) process , i.e. for each node i , it assigns different weights α ij ∈ [ − 1 , 1] to different neighborhood nodes (equation 7 in FAGCN paper).
The goal of this node-wise operation in FAGCN is to learn a new ﬁlter during the ﬁltering process node-wisely .
But in ACM, the nodewise operation is to mix the ﬁltered information from each channel which is processed by different ﬁxed ﬁlters.
The targets of two the node-wise operations are actually different things.
• FAGCN does not learn distinct information from different "channels".
FAGCN only uses simple addition to mix information instead of node-wise channel mixing mechanism The learned ﬁlter ˆ A (cid:48) can be decomposed as ˆ A (cid:48)
=
ˆ A (cid:48)
+( − ˆ A (cid:48) 2 ) , where ˆ A (cid:48) 1 and − ˆ A (cid:48) 2 represent positive and negative edge (propagation) information, respectively.
But FAGCN does not feed distinct information to ˆ A (cid:48) 1 and − ˆ A (cid:48) 2 .
Moreover, the aggregated ˆ A (cid:48) 1 X and "diversiﬁed" information ( − ˆ A (cid:48) 2 ) X are simply added together instead of using any node-wise mixing mechanism.
In ACM, we learn distinct information separately in each channel with different parameters and add them adaptively and node-wisely instead of just adding them together.
In section 6.1, the ablation study has empirically shown that node-wise adaptive channel mixing is better than simple addition.