Original: Node-oriented Spectral Filtering for Graph Neural Networks
Revised: Node-oriented Spectral Filtering for Graph Neural Networks

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Graph neural networks (GNNs) have shown remarkable performance on homophilicgraph data while being far less impressive when handling non-homophilic graphdata due to the inherent low-pass filtering property of GNNs.
Revised: Graph neural networks (GNNs) have shown remarkable performance on homophilicgraph data while being far less impressive when handling non-homophilic graphdata due to the inherent low-pass filtering property of GNNs.

Original: In general, sincethe real-world graphs are often a complex mixture of diverse subgraph patterns,learning a universal spectral filter on the graph from the global perspective asin most current works may still be difficult to adapt to the variation of localpatterns.
Revised: In general, sincethe real-world graphs are often a complex mixture of diverse subgraph patterns,learning a universal spectral filter on the graph from the global perspective asin most current works may still be difficult to adapt to the variation of localpatterns.

Original: On the basis of the theoretical analysis of local patterns, we rethinkthe existing spectral filtering methods and propose the Node-oriented spectral
Revised: On the basis of the theoretical analysis of local patterns, we rethinkthe existing spectral filtering methods and propose the Node-oriented spectral

Original: Filtering for Graph Neural Network (namely NFGNN).
Revised: Filtering for Graph Neural Network (namely NFGNN).

Original: By estimating the nodeoriented spectral filter for each node, NFGNN is provided with the capabilityof precise local node positioning via the generalized translated operator, thusadaptive discriminating the variations of local homophily patterns.
Revised: By estimating the nodeoriented spectral filter for each node, NFGNN is provided with the capabilityof precise local node positioning via the generalized translated operator, thusadaptive discriminating the variations of local homophily patterns.

Original: Furthermore,the utilization of re-parameterization brings a trade-off between global consistencyand local sensibility for learning the node-oriented spectral filters.
Revised: Furthermore,the utilization of re-parameterization brings a trade-off between global consistencyand local sensibility for learning the node-oriented spectral filters.

Original: Meanwhile,we theoretically analyze the localization property of NFGNN, demonstrating thatthe signal after adaptive filtering is still positioned around the corresponding node.
Revised: Meanwhile,we theoretically analyze the localization property of NFGNN, demonstrating thatthe signal after adaptive filtering is still positioned around the corresponding node.

Original: Extensive experimental results demonstrate that the proposed NFGNN achievesmore favorable performance.
Revised: Extensive experimental results demonstrate that the proposed NFGNN achievesmore favorable performance.

Original:  Introduction
Revised:  Introduction

Original: As a powerful tool for analyzing graph data, GNNs are attracting considerable attention from bothacademia and industry.
Revised: As a powerful tool for analyzing graph data, GNNs are attracting considerable attention from bothacademia and industry.

Original: Meanwhile, GNNs have also demonstrated remarkable capabilities in anumber of graph-related applications, including but not limited to recommendation system [14, 38],disease prediction [25, 17], drug discovery [34, 9] and action recognition [40, 31].
Revised: Meanwhile, GNNs have also demonstrated remarkable capabilities in anumber of graph-related applications, including but not limited to recommendation system [14, 40],disease prediction [27, 17], drug discovery [36, 9] and action recognition [42, 33].

Original: In the field of graph machine learning homophily has always remained a common assumption[23, 37] i.e., nodes within the same class tend to connect with each other.
Revised: In the field of graph machine learning homophily has always remained a common assumption[25, 39], i.e., nodes within the same class tend to connect with each other.

Original: However, behind the greatsuccess of the previous efforts, such assumption as a critical limitation doesn’t hold true in manygraph-related scenario which inhibits severely the further extension of GNNs to more general graphdata.
Revised: However, behind the greatsuccess of the previous efforts, such assumption as a critical limitation doesn’t hold true in manygraph-related scenario which inhibits severely the further extension of GNNs to more general graphdata.

Original: As a mater of fact, it is hard to argue that homophily is an inherent characteristic of graphdata [42] and there are also a considerable number of non-homophilic graphs in real-world, where thelinks usually exist between nodes from different classes.
Revised: As a mater of fact, it is hard to argue that homophily is an inherent characteristic of graphdata [45] and there are also a considerable number of non-homophilic graphs in real-world, where thelinks usually exist between nodes from different classes.

Original: For example, in protein structural networks,connections between different types of amino acids are easier to form [8]; in addition, for an airtraffic network, the establishment of the air routes is more for commercial reasons and has little to dowith the activities of airports [27].
Revised: For example, in protein structural networks,connections between different types of amino acids are easier to form [8]; in addition, for an airtraffic network, the establishment of the air routes is more for commercial reasons and has little to dowith the activities of airports [29].

Original: As far as the existing GNNs are concerned, most of them usually adopt message passing architecture inthe spatial domain to aggregate the node feature from neighbors over the given topology structure [36,10, 20].
Revised: As far as the existing GNNs are concerned, most of them usually adopt message passing architecture inthe spatial domain to aggregate the node feature from neighbors over the given topology structure [38,10, 20].

Original: Obviously, the practice that all neighbor nodes are considered to contribute positively to
Revised: Obviously, the practice that all neighbor nodes are considered to contribute positively to

Original: [Figure]
Revised: [Figure]

Original: node aggregation without distinction does not apply to heterogeneous graphs.
Revised: node aggregation without distinction does not apply to heterogeneous graphs.

Original: Besides, the commonlyadopted message passing architectures have been proven to exhibit significant low-pass filteringproperties [2, 39], which is quite contrary to the non-low-pass properties of non-homophilic graphs.
Revised: Besides, the commonlyadopted message passing architectures have been proven to exhibit significant low-pass filteringproperties [2, 41], which is quite contrary to the non-low-pass properties of non-homophilic graphs.

Original: In order to break the homophily limitation, several recent studies have made some exploratory effortsto solve the GNN modeling problem with non-homophily graphs, such as exploring some newaggregation schemes [42, 12, 41] and high-pass spectral filtering [4, 43].
Revised: In order to break the homophily limitation, several recent studies have made some exploratory effortsto solve the GNN modeling problem with non-homophily graphs, such as exploring some newaggregation schemes [45, 12, 44] and high-pass spectral filtering [4, 46].

Original: However, these methodsare still designed for specific heterophilic graph and lack of good scalability to homophilic graph.
Revised: However, these methodsare still designed for specific heterophilic graph and lack of good scalability to homophilic graph.

Original: Actually, whether a graph is homophilic or heterophilic depends on the relatedness of the downstreamtask to the graph construction.
Revised: Actually, whether a graph is homophilic or heterophilic depends on the relatedness of the downstreamtask to the graph construction.

Original: It means that for a fixed topology structure, when combined withdifferent downstream tasks, the identification of its graph property will be very different.
Revised: It means that for a fixed topology structure, when combined withdifferent downstream tasks, the identification of its graph property will be very different.

Original: Taking thedating network as an example, people are more likely to date someone who are opposite sex andabout the same age.
Revised: Taking thedating network as an example, people are more likely to date someone who are opposite sex andabout the same age.

Original: In this case, the graph is likely to be heterophilic if we use the gender of the nodeas the label, while it may be homophilic if we use the age group of the node as the label.
Revised: In this case, the graph is likely to be heterophilic if we use the gender of the nodeas the label, while it may be homophilic if we use the age group of the node as the label.

Original: Therefore,from the perspective of practical application, a simple yet effective GNN model that can be adaptivelyapplied to graphs with mixed structural properties should be more preferred.
Revised: Therefore,from the perspective of practical application, a simple yet effective GNN model that can be adaptivelyapplied to graphs with mixed structural properties should be more preferred.

Original: Moreover, GNN also confronts with another follow-on challenge, i.e., the homophily property isin general not consistent across the whole graph.
Revised: Moreover, GNN also confronts with another follow-on challenge, i.e., the homophily property isin general not consistent across the whole graph.

Original: In a real-world graph, there are always diversesubgraph patterns among different regions [35].
Revised: In a real-world graph, there are always diversesubgraph patterns among different regions [37].

Original: As shown in Fig 1, a network seems like homophilymay also contain a small amount of randomness or heterophily.
Revised: As shown in Fig 1, a network seems like homophilymay also contain a small amount of randomness or heterophily.

Original: Although the universality of GNNshas been taken into consideration in [6, 13] through adaptive spectral filter learning, the global filtermodeling without focus on the variation of local structural pattern may still be suboptimal for thegraph that is mixed up of more complex homophilic and heterophilic property.
Revised: Although the universality of GNNshas been taken into consideration in [6, 13] through adaptive spectral filter learning, the global filtermodeling without focus on the variation of local structural pattern may still be suboptimal for thegraph that is mixed up of more complex homophilic and heterophilic property.

Original: Besides, the relationre-estimation based methods [22, 35, 16, 26] have shown some advance in addressing the issue ofthe mixing pattern to a certain extent.
Revised: Besides, the relationre-estimation based methods [24, 37, 16, 28] have shown some advance in addressing the issue ofthe mixing pattern to a certain extent.

Original: In these approaches, different measures of node similarity aredefined to perform potential neighborhood discovery, whereas the design of similarity measure andthe high complexity that comes with it makes them less concise and flexible.
Revised: In these approaches, different measures of node similarity aredefined to perform potential neighborhood discovery, whereas the design of similarity measure andthe high complexity that comes with it makes them less concise and flexible.

Original: In this paper, we first analyze the local mixing patterns in the graph via the label consistencyof the node neighborhoods.
Revised: In this paper, we first analyze the local mixing patterns in the graph via the label consistencyof the node neighborhoods.

Original: A theoretical justification is also given to analyze why the existingnear-neighborhood aggregation mechanisms fail to work for the non-homophily graphs.
Revised: A theoretical justification is also given to analyze why the existingnear-neighborhood aggregation mechanisms fail to work for the non-homophily graphs.

Original: Further,inspired by the generalized translated operator in graph signal processing (GSP), we propose anovel GNN (namely NFGNN) from the perspective of spectral filtering to achieve adaptive localizedgraph spectral filter learning.
Revised: Further,inspired by the generalized translated operator in graph signal processing (GSP), we propose anovel GNN (namely NFGNN) from the perspective of spectral filtering to achieve adaptive localizedgraph spectral filter learning.

Original: The key idea is to estimate the node-oriented filter for each node tosolve the issue of varied local patterns in the graph.
Revised: The key idea is to estimate the node-oriented filter for each node tosolve the issue of varied local patterns in the graph.

Original: To be specific, we first apply the translatedoperator to center the spectral filter at each node, and then the K-order polynomial is used toapproximate the optimal filter to be learned at each node.
Revised: To be specific, we first apply the translatedoperator to center the spectral filter at each node, and then the K-order polynomial is used toapproximate the optimal filter to be learned at each node.

Original: In addition, low-rank approximationbased re-parameterization is used to decompose the filter weight matrix to node-agnostic and nodedependent matrices, improving the flexibility of the model.
Revised: In addition, low-rank approximationbased re-parameterization is used to decompose the filter weight matrix to node-agnostic and nodedependent matrices, improving the flexibility of the model.

Original: It also brings a trade-off between globaland local perspectives.
Revised: It also brings a trade-off between globaland local perspectives.

Original: Meanwhile, we theoretically prove that the filtered signal is localized aroundthe corresponding node, demonstrating that NFGNN achieves the adaptive localized filtering.
Revised: Meanwhile, we theoretically prove that the filtered signal is localized aroundthe corresponding node, demonstrating that NFGNN achieves the adaptive localized filtering.

Original: Finally,an extensive group of experiments on various real-world datasets with different scales verifies thatthe proposed NFGNN achieves more favorable performance.
Revised: Finally,an extensive group of experiments on various real-world datasets with different scales verifies thatthe proposed NFGNN achieves more favorable performance.

Original:  Preliminaries
Revised:  Preliminaries

Original: Notations.
Revised: Notations.

Original: An undirected graph is denoted as G =
Revised: An undirected graph is denoted as G =

Original: ( V , E ) , where V = { v i }
Revised: ( V , E ) , where V = { v i }

Original: |V| i =1 denotes the set ofnodes with |V| = n , and E is the set of edges among nodes.
Revised: |V| i =1 denotes the set ofnodes with |V| = n , and E is the set of edges among nodes.

Original: The topology structure of graph G could be described by the adjacency matrix A ∈ R n × n with A i,j = 1 if ( i, j ) ∈ E or 0 otherwise,and D is the diagonal degree matrix D with its i -th diagonal entry D ii = (cid:80)
Revised: The topology structure of graph G could be described by the adjacency matrix A ∈ R n × n with A i,j = 1 if ( i, j ) ∈ E or 0 otherwise,and D is the diagonal degree matrix D with its i -th diagonal entry D ii = (cid:80)

Original: j A ij .
Revised: j A ij .

Original: Besides, we use
Revised: Besides, we use

Original: L = I − D − 1 / 2 AD − 1 / 2 to denote the symmetric normalized Laplacian matrix of G
Revised: L = I − D − 1 / 2 AD − 1 / 2 to denote the symmetric normalized Laplacian matrix of G

Original: and I is theidentity matrix.
Revised: and I is theidentity matrix.

Original: For each node v ∈ V , we denote its neighborhood by using N ( v ) , and further, the i -hop neighbors
Revised: For each node v ∈ V , we denote its neighborhood by using N ( v ) , and further, the i -hop neighbors

Original: N i ( v ) and the neighbors within i -hops N <
Revised: N i ( v ) and the neighbors within i -hops N <

Original: i ( v ) of node v by N i ( v ) = { m : m ∈ V ∧ d G ( v, m ) =
Revised: i ( v ) of node v by N i ( v ) = { m : m ∈ V ∧ d G ( v, m ) =

Original: i }and N <i ( v ) =
Revised: i }and N <i ( v ) =

Original: { m : m ∈ V ∧ d G ( v, m ) ≤ i } , respectively, where d G ( i, j ) is the shortest pathdistance between two nodes i and j on graph G .
Revised: { m : m ∈ V ∧ d G ( v, m ) ≤ i } , respectively, where d G ( i, j ) is the shortest pathdistance between two nodes i and j on graph G .

Original: Besides, let x =
Revised: Besides, let x =

Original: [ x 1 , · · · , x i , · · · , x n ] T ∈ R ndenote the n -dimensional signal defined on the given graph G , where x i denotes the signal response(feature) at the i -th node v i .
Revised: [ x 1 , · · · , x i , · · · , x n ] T ∈ R ndenote the n -dimensional signal defined on the given graph G , where x i denotes the signal response(feature) at the i -th node v i .

Original: Generally, when each node receives f channels of signals, we will havea feature matrix X
Revised: Generally, when each node receives f channels of signals, we will havea feature matrix X

Original: =
Revised: =

Original: [ X 1 , · · · , X i , · · · , X n ] T ∈ R n × f with each column of it being a graph signalx and its i -th row X i ∈
Revised: [ X 1 , · · · , X i , · · · , X n ] T ∈ R n × f with each column of it being a graph signalx and its i -th row X i ∈

Original: R f representing f - dimensional feature vector associated with node v i 1 .
Revised: R f representing f - dimensional feature vector associated with node v i 1 .

Original: Furthermore, for the node classification task, each node v ∈ V has a class label y v ∈ Y = { 1 , · · · , C } ,where Y is the set of class labels with |Y| =
Revised: Furthermore, for the node classification task, each node v ∈ V has a class label y v ∈ Y = { 1 , · · · , C } ,where Y is the set of class labels with |Y| =

Original: C , and C is the number of classes.
Revised: C , and C is the number of classes.

Original: In addition, we usey v ∈ R C to denote the one-hot vector corresponding to y v .
Revised: In addition, we usey v ∈ R C to denote the one-hot vector corresponding to y v .

Original: Graph Fourier Transform.
Revised: Graph Fourier Transform.

Original: According to graph signal processing theory, the graph Laplacianprovides an effective way of spectral analysis on graphs.
Revised: According to graph signal processing theory, the graph Laplacianprovides an effective way of spectral analysis on graphs.

Original: Given the Laplacian matrix L , it can beeigendecomposed into U Λ U T , where U =
Revised: Given the Laplacian matrix L , it can beeigendecomposed into U Λ U T , where U =

Original: [ u 1 , · · · , u l , · · · , u n ] ∈ R n × n is the graph Fourier basisformed by n orthonormal eigenvectors { u l } nl =1 , and Λ =
Revised: [ u 1 , · · · , u l , · · · , u n ] ∈ R n × n is the graph Fourier basisformed by n orthonormal eigenvectors { u l } nl =1 , and Λ =

Original: diag
Revised: diag

Original: [ λ 1 , · · · , λ l , · · · , λ n ] ∈ R n × n is thediagonal matrix of the ordered eigenvalues { λ l } nl
Revised: [ λ 1 , · · · , λ l , · · · , λ n ] ∈ R n × n is thediagonal matrix of the ordered eigenvalues { λ l } nl

Original: =1 , λ l ∈ [0 , 2] .
Revised: =1 , λ l ∈ [0 , 2] .

Original: Notice that { λ l } nl =1 is also identifiedas the frequencies of the graph.
Revised: Notice that { λ l } nl =1 is also identifiedas the frequencies of the graph.

Original: Thus, the graph Fourier transform of the signal x is defined asˆ x = U T x , and ˆ x ( λ l ) indicates the response of x at the frequency λ l .
Revised: Thus, the graph Fourier transform of the signal x is defined asˆ x = U T x , and ˆ x ( λ l ) indicates the response of x at the frequency λ l .

Original: The inverse graph Fouriertransform is defined as x = U ˆ x [32].
Revised: The inverse graph Fouriertransform is defined as x = U ˆ x [34].

Original: Thus, on the basis of the graph Fourier transform, the signal xfiltered by ˆ g can be given as follows:
Revised: Thus, on the basis of the graph Fourier transform, the signal xfiltered by ˆ g can be given as follows:

Original: [Equation]
Revised: [Equation]

Original: where ˆ g = ˆ g (Λ) = diag[ˆ g ( λ 1 ) , · · · , ˆ g ( λ l ) , · · · , ˆ g ( λ n )] is the spectral filter and we have g
Revised: where ˆ g = ˆ g (Λ) = diag[ˆ g ( λ 1 ) , · · · , ˆ g ( λ l ) , · · · , ˆ g ( λ n )] is the spectral filter and we have g

Original: = U ˆ g .
Revised: = U ˆ g .

Original: Since the spectral filtering is equivalent to convolution in the spatial domain [24], Eq. 1 could alsobe defined as the spectral graph convolution z
Revised: Since the spectral filtering is equivalent to convolution in the spatial domain [26], Eq. 1 could alsobe defined as the spectral graph convolution z

Original: = x ∗ G g , where ∗ G denotes the graph convolutionoperator.
Revised: = x ∗ G g , where ∗ G denotes the graph convolutionoperator.

Original:  Motivations
Revised:  Motivations

Original: 3.1 Adaptability to Mixing Local Structural Patterns
Revised: 3.1 Adaptability to Mixing Local Structural Patterns

Original: To measure the homophily of a graph, both edge homophily ratio [42] and node homophily ratio [26]are two widely used metrics.
Revised: To measure the homophily of a graph, both edge homophily ratio [45] and node homophily ratio [28]are two widely used metrics.

Revised: In addition, [23] proposed a more comprehensive homophily metric thatmitigates homogeneity bias from class imbalance.

Revised: It is less sensitive to the number of classes and sizeof each class than edge homophily ratio and node homophily ratio.

Original: Since we aim to analyze the local patterns of the graph via the labelconsistency of the node neighborhoods, the node homophily ratio is adopted in this work.
Revised: Since we aim to analyze the localpatterns of the graph via the label consistency of the node neighborhoods, the node homophily ratiois adopted in this work.

Original: It shouldbe noticed that the edge homophily ratio and the node homophily ratio have similar qualitativebehavior [21].
Revised: It should be noticed that the edge homophily ratio and the node homophilyratio have similar qualitative behavior [21].

Original: In particular, the node homophily ratio H G of the graph G is defined as the average ofthe homophilic 1-hop neighbor ratio of each node v in G and given by:
Revised: In particular, the node homophily ratio H G of the graph G is defined as the average of the homophilic 1-hop neighbor ratio of each node v in G and given by:

Original: [Equation]
Revised: [Equation]

Original: where h N 1 ( v ) denotes the homophilic 1 -hop neighbor ratio of node v centered at node v.
Revised: where h N 1 ( v ) denotes the homophilic 1 -hop neighbor ratio of node v centered at node v.

Original: In essence, H G provides an overall evaluation criterion for homophily of graph.
Revised: In essence, H G provides an overall evaluation criterion for homophily of graph.

Original: Instead, we shouldalso cast more insight into the variation of local structural pattern.
Revised: Instead, we shouldalso cast more insight into the variation of local structural pattern.

Original: Particularly, we first give visualization of the statistical histogram of h N 1 ( v ) and h N < 2 ( v ) .
Revised: Particularly, we first give visualization of the statistical histogram of h N 1 ( v ) and h N < 2 ( v ) .

Original: As shown in Fig 1, even in Coraand Citeseer network, which are usually considered as homophilic graphs, there still exist a smallnumber of completely 1-hop heterophilic subgraphs.
Revised: As shown in Fig 1, even in Cora and Citeseer network, which are usually considered as homophilic graphs, there still exist a smallnumber of completely 1-hop heterophilic subgraphs.

Original: Similarly, there are some subgraphs with a high homophily ratio in Cornell and Actor network.
Revised: Similarly, there are some subgraphs with a highhomophily ratio in Cornell and Actor network.

Original: Obviously, these observations mean that, for a graphwith complex topological structure, it is definitely a mixture of homophilic and heterophilic localsubgraphs.
Revised: Obviously, these observations mean that, for a graphwith complex topological structure, it is definitely a mixture of homophilic and heterophilic localsubgraphs.

Original: Furthermore, for the two heterophily graphs, we can find that the statistical histogramof h N < 2 ( v ) is much different from that of h N 1 ( v ) , demonstrating that the associated local subgraphpatterns for each node varied generally with the change of neighborhood range.
Revised: Furthermore, for the two heterophily graphs, we can find that the statistical histogramof h N < 2 ( v ) is much different from that of h N 1 ( v ) , demonstrating that the associated local subgraphpatterns for each node varied generally with the change of neighborhood range.

Original: Based on the above analysis on the variation of local structural pattern, it motivates us to considerthat it is conducting adaptive modeling for the graph nodes with different degree of homophily is anecessity, and further, we should improve the effectiveness of GNNs for the nodes with various localstructural patterns.
Revised: Based on the above analysis on the variation of local structural pattern, it motivates us to considerthat it is conducting adaptive modeling for the graph nodes with different degree of homophily is anecessity, and further, we should improve the effectiveness of GNNs for the nodes with various localstructural patterns.

Original: 3.2 Aggregatability of Near-neighbors
Revised: 3.2 Aggregatability of Near-neighbors

Original: To facilitate the discussion of the aggregatability of near-neighbors, we first give two definitionsabout the neighborhood, i.e., heterophily-preferred and homophily-preferred :
Revised: To facilitate the discussion of the aggregatability of near-neighbors, we first give two definitionsabout the neighborhood, i.e., heterophily-preferred and homophily-preferred :

Original: Definition 3.1.
Revised: Definition 3.1.

Original: For a node v with label y v , N ( v ) is expected to be heterophily-preferred if P ( y m =
Revised: For a node v with label y v , N ( v ) is expected to be heterophily-preferred if P ( y m =

Original: y v | y v ) ≤ P ( y m ̸
Revised: y v | y v ) ≤ P ( y m ̸

Original: = y v | y v ) , ∀ m ∈ N ( v ) .
Revised: = y v | y v ) , ∀ m ∈ N ( v ) .

Original: Conversely, N ( v ) is expected to be homophily-preferred.
Revised: Conversely, N ( v ) is expected to be homophily-preferred.

Original: Intuitively, the near-neighbor aggregation is definitely effective when the near-neighbor subgraphis completely homophilic, while it may not capture adequate homophilic information when theneighborhood is expectedly heterophily-preferred.
Revised: Intuitively, the near-neighbor aggregation is definitely effective when the near-neighbor subgraphis completely homophilic, while it may not capture adequate homophilic information when theneighborhood is expectedly heterophily-preferred.

Original: According to Definition 3.1, it can be inferredthat the aggregation of expectedly homophily-preferred neighborhoods is also beneficial to the noderepresentation.
Revised: According to Definition 3.1, it can be inferredthat the aggregation of expectedly homophily-preferred neighborhoods is also beneficial to the noderepresentation.

Original: Classical GNNs [19, 36, 10] commonly suffer from the over-smoothing problem andare thus limited to shallow networks, which means that each node just aggregates the informationabout its neighbors within 2 or 3-hops.
Revised: Classical GNNs [19, 38, 10] commonly suffer from the over-smoothing problem andare thus limited to shallow networks, which means that each node just aggregates the informationabout its neighbors within 2 or 3-hops.

Original: Thus, whether the near-neighborhood is homophily-preferredor heterophily-preferred will be of great importance for them.
Revised: Thus, whether the near-neighborhood is homophily-preferredor heterophily-preferred will be of great importance for them.

Original: To empirically analyze the preference ofnear-neighborhood, we first propose a label entropy S N i ( v ) to measure the neighbor label distributionof node v , which is defined as :
Revised: To empirically analyze the preference ofnear-neighborhood, we first propose a label entropy S N i ( v ) to measure the neighbor label distributionof node v , which is defined as :

Original: [Equation]
Revised: [Equation]

Original: where N ( y ) i ( v ) = { m : m ∈ N i ( v )
Revised: where N ( y ) i ( v ) = { m : m ∈ N i ( v )

Original: ∧
Revised: ∧

Original: y m = y } and ε = 1e-10 is a constant to avoid overflow.
Revised: y m = y } and ε = 1e-10 is a constant to avoid overflow.

Original: Clearly,the larger the label entropy S N i ( v ) is, the more random the neighbor label distribution of v willbe.
Revised: Clearly,the larger the label entropy S N i ( v ) is, the more random the neighbor label distribution of v willbe.

Original: As shown in Fig 2, most nodes in the homophily graphs have low S N 1 ( v ) and most nodes inthe heterophily graphs have high S
Revised: As shown in Fig 2, most nodes in the homophily graphs have low S N 1 ( v ) and most nodes inthe heterophily graphs have high S

Original: N 1 ( v ) .
Revised: N 1 ( v ) .

Original: Besides, for all four graphs, the statistical histogram of
Revised: Besides, for all four graphs, the statistical histogram of

Original: S N < 2 ( v ) is shifted to the right overall compared to S N 1 ( v ) .
Revised: S N < 2 ( v ) is shifted to the right overall compared to S N 1 ( v ) .

Original: These observations suggest that theneighbor label distribution of each node tends to be uniform as the neighborhood range increases.
Revised: These observations suggest that theneighbor label distribution of each node tends to be uniform as the neighborhood range increases.

Original: Combined with the definition 3.1, we can conclude that the near-neighbor based aggregation is notthe optimal solution for heterophily graphs.
Revised: Combined with the definition 3.1, we can conclude that the near-neighbor based aggregation is notthe optimal solution for heterophily graphs.

Original: [Figure]
Revised: [Figure]

Original: Theoretically, we explore the preference of the 2-hop neighborhood N 2 ( v ) for multi-class nodeclassification and have the following proposition:
Revised: Theoretically, we explore the preference of the 2-hop neighborhood N 2 ( v ) for multi-class nodeclassification and have the following proposition:

Original: Proposition 3.1.
Revised: Proposition 3.1.

Original: For each node v in a graph G , let’s assume the class labels of its neighbors{ y m : m ∈ N ( v ) } are conditionally independent when given y v , and P ( y m =
Revised: For each node v in a graph G , let’s assume the class labels of its neighbors{ y m : m ∈ N ( v ) } are conditionally independent when given y v , and P ( y m =

Original: y v | y v )
Revised: y v | y v )

Original: =
Revised: =

Original: α ,
Revised: α ,

Original: P ( y m =
Revised: P ( y m =

Original: y | y v ) = 1 − α |Y|− 1 , ∀ y ̸
Revised: y | y v ) = 1 − α |Y|− 1 , ∀ y ̸

Original: = y v .
Revised: = y v .

Original: Then, the 2-hop neighborhood N 2 ( v ) of a node v will always beexpectedly heterophily
Revised: Then, the 2-hop neighborhood N 2 ( v ) of a node v will always beexpectedly heterophily-preferred if α ≤ 2 |Y| .

Original: For the proof of the above proposition 3.1, please refer to the supplemental materials.
Revised: For the proof of the above proposition 3.1, please refer to the supplemental materials.

Original: Different from the statement that 2-hop neighbor aggregation could help GNNs to learn moreeffective node representation [42, 1], proposition 3.1 shows that the aggregation of N 2 ( v ) is hard tobe beneficial for GNNs when the neighborhood label distribution tends to be uniform.
Revised: Different from the statement that 2-hop neighbor aggregation could help GNNs to learn moreeffective node representation [45, 1], proposition 3.1 shows that the aggregation of N 2 ( v ) is hard tobe beneficial for GNNs when the neighborhood label distribution tends to be uniform.

Original: This finding isalso consistent with our empirical analysis mentioned above.
Revised: This finding isalso consistent with our empirical analysis mentioned above.

Original:  The Proposed NFGNN
Revised:  The Proposed NFGNN

Original: The observations in Sect.
Revised: The observations in Sect.

Original: 3 points out that the real-world graph is often made up of a mixture of variouslocal patterns, while the near-neighbor aggregation mechanism does not handle the heterophilic localpatterns well.
Revised: 3 points out that the real-world graph is often made up of a mixture of variouslocal patterns, while the near-neighbor aggregation mechanism does not handle the heterophilic localpatterns well.

Original: Various elaborately designed aggregation mechanisms in the spatial domain have beenproposed to tackle these issues.
Revised: Various elaborately designed aggregation mechanisms in the spatial domain have beenproposed to tackle these issues.

Original: Different from these methods based on spatial domain, spectral graphconvolution aims to learn a specific spectral filter for a given graph structure and node labels, thuspreserving the appropriate frequency components for the downsteam tasks.
Revised: Different from these methods based on spatial domain, spectral graphconvolution aims to learn a specific spectral filter for a given graph structure and node labels, thuspreserving the appropriate frequency components for the downsteam tasks.

Original: Therefore, spectral graphconvolution also possesses a strong expressive power [2] and can work well for both heterophily andhomophily graphs.
Revised: Therefore, spectral graphconvolution also possesses a strong expressive power [2] and can work well for both heterophily andhomophily graphs.

Original: However, existing spectral-based methods are still not flexible enough.
Revised: However, existing spectral-based methods are still not flexible enough.

Original: Theyusually estimate a globally consistent filter from the perspective of the whole graph [20, 13, 7, 3],which may be inappropriate for some local patterns.
Revised: Theyusually estimate a globally consistent filter from the perspective of the whole graph [20, 13, 7, 3],which may be inappropriate for some local patterns.

Original: In this section, we rethink the globally consistentspectral graph convolutions, and propose a localized spectral filter learning method to break thelimitation.
Revised: In this section, we rethink the globally consistentspectral graph convolutions, and propose a localized spectral filter learning method to break thelimitation.

Original: 4.1 Polynomial Filter
Revised: 4.1 Polynomial Filter

Original: Based on Eq 1, early spectral GNNs [5, 15] directly eigendecompose the normalized Laplacianmatrix L to obtain the Fourier basis U and treat the ˆ g as the trainable parameters.
Revised: Based on Eq 1, early spectral GNNs [5, 15] directly eigendecompose the normalized Laplacianmatrix L to obtain the Fourier basis U and treat the ˆ g as the trainable parameters.

Original: However, theexpensive eigenvalue decomposition restricts the availability of these methods greatly.
Revised: However, theexpensive eigenvalue decomposition restricts the availability of these methods greatly.

Original: To circumventthe eigendecomposition, K -order polynomial approximation is adopted to parameterize the spectralfilter:
Revised: To circumventthe eigendecomposition, K -order polynomial approximation is adopted to parameterize the spectralfilter:

Original: [Equation]
Revised: [Equation]

Original: where γ k denotes the learnable fitting coefficient.
Revised: where γ k denotes the learnable fitting coefficient.

Original: By plugging Eq. 4 into Eq.
Revised: By plugging Eq. 6 into Eq.

Original: 1, the spectral filteringcan be rewritten as:
Revised: 1, the spectral filteringcan be rewritten as:

Original: [Equation]
Revised: [Equation]

Original: Except for the reduced complexity, another advantage of the polynomial-parameterized filter is thelocalization property.
Revised: Except for the reduced complexity, another advantage of the polynomial-parameterized filter is thelocalization property.

Original: When the filter ˆ g centers at node v i , the value at node v j after filtering by ˆ g isequal to (cid:80) Kk =0 γ k ( L k ) i,j .
Revised: When the filter ˆ g centers at node v i , the value at node v j after filtering by ˆ g isequal to (cid:80) Kk =0 γ k ( L k ) i,j .

Original: Meanwhile, ( L k ) i,j will be 0 if d G ( i, j ) > K [11].
Revised: Meanwhile, ( L k ) i,j will be 0 if d G ( i, j ) > K [11].

Original: The above facts showthat the K -order polynomial spectral filter is exactly localized in N <K ( i ) .
Revised: The above facts showthat the K -order polynomial spectral filter is exactly localized in N <K ( i ) .

Original: Due to the high efficiency, various polynomial kernels are used for spectral filter parameterization,such as Chebyshev basis [7] and Bernstein basis [13].
Revised: Due to the high efficiency, various polynomial kernels are used for spectral filter parameterization,such as Chebyshev basis [7] and Bernstein basis [13].

Original: Interestingly, many spatial aggregationmethods can also be essentially attributed to polynomial-parameterized spectral convolution [20, 6].
Revised: Interestingly, many spatial aggregationmethods can also be essentially attributed to polynomial-parameterized spectral convolution [20, 3].

Original: Nevertheless, such polynomial filters are still globally consistent, or node independent.
Revised: Nevertheless, such polynomial filters are still globally consistent, or node independent.

Original: In otherwords, the filter ˆ g is applied for all nodes with the fixed fitting coefficients { γ k } Kk =0 that are trainedon the whole graph, and makes no specific discrimination for each node when performing filtering.
Revised: In otherwords, the filter ˆ g is applied for all nodes with the fixed fitting coefficients { γ k } Kk =0 that are trainedon the whole graph, and makes no specific discrimination for each node when performing filtering.

Original: Thus, even it is localized, the polynomial-parameterized spectral filters are still unable to effectivelymodel the complex local structural patterns with a mixture of homophily and heterophily.
Revised: Thus, even it is localized, the polynomial-parameterized spectral filters are still unable to effectivelymodel the complex local structural patterns with a mixture of homophily and heterophily.

Revised: Intuitively,compared to learning a globally shared filter ˆ g ( λ l ) as a trade-off solution for different local patternsacross the whole graph, learning an appropriate node-specific filter ˆ g i ( λ l ) for node i to fit the localpattern where it is located seems to be a better choice.

Revised: [Equation]

Revised: For such practice, what needs to be figured out is how to learn ˆ g i and ensure it is still positionedaround node i .

Revised: To this end, we introduce the node-oriented filtering.

Original: 4.2 Translated Filter for Node-oriented Filtering
Revised: 4.2 Translated Filter for Node-oriented Filtering

Original: Inspired by the generalized translation operator, we develop an adaptive localized spectral filteringon graph G using the polynomial-parameterized spectral convolution.
Revised: Inspired by the generalized translation operator, we develop an adaptive localized spectral filteringon graph G using the polynomial-parameterized spectral convolution.

Original: It takes full into account thespecific effect of the node where the filter is positioned.
Revised: It takes full into account thespecific effect of the node where the filter is positioned.

Original: Definition 4.1.
Revised: Definition 4.1.

Original: ( Generalized translation operator )
Revised: ( Generalized translation operator )

Original: [32] For any signal g ∈ R n defined on a givengraph G and any i ∈ { 0 , 1 , · · · , n − 1 } , we define a generalized translation operator T i : R n → R n via generalized convolution with a Kronecker delta function δ i centered at the i -th node v i :
Revised: [34] For any signal g ∈ R n defined on a givengraph G and any i ∈ { 0 , 1 , · · · , n − 1 } , we define a generalized translation operator T i : R n → R nvia generalized convolution with a Kronecker delta function δ i centered at the i -th node v i :

Original: [Equation]
Revised: [Equation]

Original: where u Tl ( i ) denotes the i -th element of u Tl .
Revised: where u Tl ( i ) denotes the i -th element of u Tl .

Original: Definition 4.1 shows that a signal could be centered at a specific node through a kernelized operatoracting on ˆ g [32].
Revised: Definition 4.1 shows that a signal could be centered at a specific node through a kernelized operatoracting on ˆ g [34].

Original: Then, we can perform the spectral convolution of signal x with the filter signal gwhen g is centered at a specific node v i :
Revised: Then, we can perform the spectral convolution of signal x with the filter signal gwhen g is centered at a specific node v i :

Original: [Equation]
Revised: [Equation]

Original: Let ˆ g i ( λ l ) =
Revised: Let ˆ g i ( λ l ) =

Original: √ Nu Tl ( i )ˆ g ( λ l ) , Eq. 7 becomes:
Revised: √ Nu Tl ( i )ˆ g ( λ l ) , Eq. 8 becomes:

Original: [Equation]
Revised: [Equation]

Original: Recall the inverse graph Fourier transform, it can be derived that x
Revised: Recall the inverse graph Fourier transform, it can be derived that x

Original: i =
Revised: i =

Original: U i : ˆ x , where U i : indicatesthe i -th row of U .
Revised: U i : ˆ x , where U i : indicatesthe i -th row of U .

Original: Hence, U i : could also approximated from x i according to U i :
Revised: Hence, U i : could also approximated from x i according to U i :

Original: ≈ x i q , whereq =
Revised: ≈ x i q , whereq =

Original: pinv(ˆ x ) ∈ R n is the pseudoinverse of ˆ x .
Revised: pinv(ˆ x ) ∈ R n is the pseudoinverse of ˆ x .

Original: Then, note that u l ( i ) =
Revised: Then, note that u l ( i ) =

Original: u Tl ( i ) , it can be derived thatˆ
Revised: u Tl ( i ) , it can be derived thatˆ

Original: g i ( λ l ) ≈ ˜
Revised: g i ( λ l ) ≈ ˜

Original: g
Revised: g

Original: i ( λ ) = √ N ( x i q l )ˆ g ( λ l ) , where q l denotes the l -th element of q .
Revised: i ( λ ) = √ N ( x i q l )ˆ g ( λ l ) , where q l denotes the l -th element of q .

Original: Therefore, ˆ g i can beapproximated by a specific filter ˜
Revised: Therefore, ˆ g i can beapproximated by a specific filter ˜

Original: g
Revised: g

Original: i corresponding to node v i , which also considers the impact fromthe feature x i associated with node v i in estimating the filter weight.
Revised: i corresponding to node v i , which also considers the impact fromthe feature x i associated with node v i in estimating the filter weight.

Original: Without loss of generality, the K -order polynomial approximation can be used to directly parameterize˜ g i .
Revised: Without loss of generality, the K -order polynomial approximation can be used to directly parameterize˜ g i .

Original: As a traditionally used approximate kernel in GSP [7], the Chebyshev polynomial T k ( · ) areadopted to parameterize ˜
Revised: As a traditionally used approximate kernel in GSP [7], the Chebyshev polynomial T k ( · ) areadopted to parameterize ˜

Original: g
Revised: g

Original: i , that is, ˜ g
Revised: i , that is, ˜ g

Original: i = (cid:80) Kk
Revised: i = (cid:80) Kk

Original: =

Revised: =

Original: η
Revised: η

Original: i,k T k (˜Λ) , where ˜Λ = 2Λ /λ max − I .
Revised: i,k T k (˜Λ) , where ˜Λ = 2Λ /λ max − I .

Original: Meanwhile,for each node, we focus only on the convolution result of the filter positioned at that node, i.e.,z i =
Revised: Meanwhile,for each node, we focus only on the convolution result of the filter positioned at that node, i.e.,z i =

Original: δ i ( U ˜ g
Revised: δ i ( U ˜ g

Original: i U T x ) , here δ i =
Revised: i U T x ) , here δ i =

Original: [0 , · · · , 1 i , · · · , 0] ∈ R n denotes a row vector with only the i -thelement being 1 and the remains being zeros.
Revised: [0 , · · · , 1 i , · · · , 0] ∈ R n denotes a row vector with only the i -thelement being 1 and the remains being zeros.

Original: Thus, the node-oriented localized filtering can be asfollows:
Revised: Thus, the node-oriented localized filtering can be as:

Original: [Equation]
Revised: [Equation]

Original: where Ψ =
Revised: where Ψ =

Original: [ η i,k ] ik ∈ R n × ( K +1) is the trainable coefficient matrix, ˜ L = U ˜Λ U T .
Revised: [ η i,k ] ik ∈ R n × ( K +1) is the trainable coefficient matrix, ˜ L = U ˜Λ U T .

Original: Meanwhile, similarto the above discussion on localized filter, we have the following Proposition4.1 to claim that theadaptively filtered signal z is also approximately positioned around the node i .
Revised: Meanwhile, similarto the above discussion on localized filter, we have the following Proposition4.1 to claim that theadaptively filtered signal z is also approximately positioned around the node i .

Original: Proposition 4.1.
Revised: Proposition 4.1.

Original: Given a signal x defined on a graph G and a filter T i ( g ) that translated to a givencenter node v i in G , the filtered signal z = x ∗ T i ( g ) is approximately localized around the node i .
Revised: Given a signal x defined on a graph G and a filter T i ( g ) that translated to a givencenter node v i in G , the filtered signal z = x ∗ T i ( g ) is approximately localized around the node i .

Original: To prove the proposition 4.1, let’s first introduce the following Lemma 4.1.
Revised: To prove the proposition 4.1, let’s first introduce the following Lemma 4.1.

Original: Lemma 4.1 ( [33]) .
Revised: Lemma 4.1 ( [35]) .

Original: Let ˆ p K be the polynomial approximation with degree K to the spectrum of a graphsignal φ , i.e., ˆ φ ( λ l )
Revised: Let ˆ p K be the polynomial approximation with degree K to the spectrum of a graphsignal φ , i.e., ˆ φ ( λ l )

Original: ≈ ˆ p K ( λ l ) =
Revised: ≈ ˆ p K ( λ l ) =

Original: (cid:80) Kk =

Revised: (cid:80) Kk =

Original: γ k λ kl .
Revised: γ k λ kl .

Original: If d G ( i, n ) > K , then T i ( φ ) n ≈ T i ( p K ) n = 0 ,where p K denotes the signal corresponding to ˆ p K .
Revised: If d G ( i, n ) > K , then T i ( φ ) n ≈ T i ( p K ) n = 0 ,where p K denotes the signal corresponding to ˆ p K .

Original: Proof.
Revised: Proof.

Original: According to Definition 4.1 and the properties of convolution, we notice that z can be rewrittenas:
Revised: According to Definition 4.1 and the properties of convolution, we notice that z can be rewrittenas:

Original: [Equation]
Revised: [Equation]

Original: Furthermore, let φ = x ∗ g and ˆ p K be the polynomial approximation with degree K to ˆ φ .
Revised: Furthermore, let φ = x ∗ g and ˆ p K be the polynomial approximation with degree K to ˆ φ .

Original: From
Revised: From

Original: Lemma 4.1, T i ( φ ) n ≈ T i ( p K ) n = 0 will hold if d G ( i, n ) > K .
Revised: Lemma 4.1, T i ( φ ) n ≈ T i ( p K ) n = 0 will hold if d G ( i, n ) > K .

Original: Then, we have z n =
Revised: Then, we have z n =

Original: T i ( φ ) n ≈ 0if d G ( i, n ) > K .
Revised: T i ( φ ) n ≈ 0if d G ( i, n ) > K .

Original: This completes the proof.
Revised: This completes the proof.

Original: Input: X ∈ R n × f , ˜ L ∈ R n × n , K
Revised: Input: X ∈ R n × f , ˜ L ∈ R n × n , K

Original: Output: Z
Revised: Output: Z

Original: Learnable Parameters: W , Γ , Θ .
Revised: Learnable Parameters: W , Γ , Θ .

Original: 1: X (0)  MLP Θ(
Revised: 1: X (0) ← MLP Θ

Original:  )  ← ˜ LX .
Revised: ( X ) , X (1) ← ˜ LX .

Original: /* Feature Transformation */ 2:  do do
Revised: /* Feature Transformation */ 2: = 1 do X

Revised: ) ←

Original:  Z i end for end for
Revised: > do Z i ← η X + Z i end for end for

Original: 4.3 The Implementation of NFGNN
Revised: 4.3 The Implementation of NFGNN

Original: According to the proposed adaptive localized filtering in Eq 9, we will further formalize the architecture of the proposed NFGNN.
Revised: According to the proposed adaptive localized filtering in Eq 10, we will further formalize thearchitecture of the proposed NFGNN.

Original: As pointed out in [39, 20], the entanglement of feature transformationand filtering may be harmful to the performance and robustness of the GNN model.
Revised: As pointed out in [41, 20], the entanglement of featuretransformation and filtering may be harmful to the performance and robustness of the GNN model.

Original: Hence, we adoptthe similar way by first applying a MLP to perform the non-linear transformation for the raw featurematrix X .
Revised: Hence, we adopt the similar way by first applying a MLP to perform the non-linear transformationfor the raw feature matrix X .

Original: Then, the spectral filtering operation can be implemented by a recursive way due to thestable recurrence relation of T k ( · ) : 
Revised: Then, the spectral filtering operation can be implemented by a recursiveway due to the stable recurrence relation of T k ( · ) : [Equation]

Original: [Equation]
Revised: [Equation]

Original: Accordingly, given the input X , we will have X ( k ) =
Revised: Accordingly, given the input X , we will have X ( k ) =

Original: T k (˜ L ) X .
Revised: T k (˜ L ) X .

Original: Notice that, the scale of the trainable coefficient matrix Ψ is positive proportional to the number n ofnodes.
Revised: Notice that, the scale of the trainable coefficient matrix Ψ is positive proportional to the number n ofnodes.

Original: With the increase of n , this will inevitably involve learning a large number of parameters.
Revised: With the increase of n , this will inevitably involve learning a large number of parameters.

Original: Atthe same time, the model also needs to learn K parameters for a single node, which is intractable tobe optimized.
Revised: Atthe same time, the model also needs to learn K parameters for a single node, which is intractable tobe optimized.

Original: In addition, learning such a large number of parameters can also lead to overfitting,especially in the case of small number of labels.
Revised: In addition, learning such a large number of parameters can also lead to overfitting,especially in the case of small number of labels.

Original: To achieve parameter lightweight for Ψ , weuse a separable low-rank approximation to re-parameterize it.
Revised: To achieve parameter lightweight for Ψ , weuse a separable low-rank approximation to re-parameterize it.

Original: Specifically, Ψ is assumed to bedecomposed into two trainable parameter matrices Ψ
Revised: Specifically, Ψ is assumed to bedecomposed into two trainable parameter matrices Ψ

Original: = H Γ with η i,k
Revised: = H Γ with η i,k

Original: = H i : Γ : k ,
Revised: = H i : Γ : k ,

Original: where H ∈ R n × dand
Revised: where H ∈ R n × dand

Original: Γ ∈ R
Revised: Γ ∈ R

Original: d × ( K +1) are the node-dependent matrix and node-agnostic matrix, respectively.
Revised: d × ( K +1) are the node-dependent matrix and node-agnostic matrix, respectively.

Original: As Γ is seen as node-agnostic, it can be directly trained as general parameters, which is very similarto the learning of the polynomial coefficients in [7].
Revised: As Γ is seen as node-agnostic, it can be directly trained as general parameters, which is very similarto the learning of the polynomial coefficients in [7].

Original: But for H , since we treat it as node dependent, asimple yet effective nonlinear transformation (MLP) is applied, i.e., H = σ ( XW ) , where W ∈ R f × dand σ ( · ) are the learnable weight matrix and activation function, respectively.
Revised: But for H , since we treat it as node dependent, asimple yet effective nonlinear transformation (MLP) is applied, i.e., H = σ ( XW ) , where W ∈ R f × dand σ ( · ) are the learnable weight matrix and activation function, respectively.

Original: It is worthy of notethat, through low-rank approximation based re-parameterization, the parameter complexity of Ψ isreduced from O ( n × ( K + 1)) to O
Revised: It is worthy of notethat, through low-rank approximation based re-parameterization, the parameter complexity of Ψ isreduced from O ( n × ( K + 1)) to O

Original: (( K + 1) ×
Revised: (( K + 1) ×

Original: d + f × d ) , and we can flexibly adjust the modelcapacity by changing d .
Revised: d + f × d ) , and we can flexibly adjust the modelcapacity by changing d .

Original: Particularly, we set d = 1 in this work, and thus the node-agnostic matrixΓ ∈ R 1 × ( K +1) is closely related to { γ k } K k =0 in Eq.4.
Revised: Particularly, we set d = 1 in this work, and thus the node-agnostic matrixΓ ∈ R 1 × ( K +1) is closely related to { γ k } K k =0 in Eq.

Original: We summarize the proposed node-orientedfiltering in Algorithm1.
Revised: We summarize the proposed node-orientedfiltering in Algorithm1.

Original:  Experimental Results and Analysis5.1 Experimental Settings
Revised:  Experimental Results and Analysis

Revised: 5.1 Experimental Settings

Original: Datasets.
Revised: Datasets.

Original: To provide a comprehensive evaluation of our method, several graphs from variousdomains with different homophily ratios are used, including 5 homophilic graphs: citation graphs
Revised: To provide a comprehensive evaluation of our method, several graphs from variousdomains with different homophily ratios are used, including 5 homophilic graphs: citation graphs

Original: Cora, CiteSeer, PubMed [29], and co-purchase graphs Computers and Photo [30]; 5 heterophilicgraphs:
Revised: Cora, CiteSeer, PubMed [31], and co-purchase graphs Computers and Photo [32]; 5 heterophilicgraphs:

Original: Wikipedia graphs Chameleon and Squirrel [28], the Actor cooccurrence graph, and webpagegraphs Texas and Cornell from WebKB [26].
Revised: Wikipedia graphs Chameleon and Squirrel [30], the Actor cooccurrence graph, and webpagegraphs Texas and Cornell from WebKB [28].

Original: The statistics of these datasets are summarized insupplemental materials.
Revised: The statistics of these datasets are summarized insupplemental materials.

Original: Baselines.
Revised: Baselines.

Original: Several baselines have been selected for comparison, including 6 methods that can beseen as spectral filtering based, GCN [19], SGC [39], ChebNet [7], APPNP [20], GPRGNN [6],
Revised: Several baselines have been selected for comparison, including 6 methods that can beseen as spectral filtering based, GCN [19], SGC [41], ChebNet [7], APPNP [20], GPRGNN [6],

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: BernNet [13], and 3 spatial aggregation based methods, GAT [36], Geom-GCN [26], BMGCN [10],and MLP.
Revised: BernNet [13], and 3 spatial aggregation based methods, GAT [38], Geom-GCN [28], BMGCN [10],and 3 non-GNN baselines, MLP, LINK [43], and LINKX [22].

Original: For GPRGNN, BernNet, and BMGCN, we directly use the open-source codes released bythe original paper.
Revised: For GPRGNN, BernNet, and BMGCN,we directly use the open-source codes released by the original paper.

Original: For the others, we use the models that are provided by [6].
Revised: For the others, we use themodels that are provided by [6].

Original: Experimental Setup.
Revised: Experimental Setup.

Original: For the node classification task, we follow the random split ratio in [6] to splitthe dataset into training/validation/test sets.
Revised: For the node classification task, we follow the random split ratio in [6] to splitthe dataset into training/validation/test sets.

Original: Specifically, The sparse splitting ratio (2.5%/2.5%95/%)is used for homophilic graphs, and the dense splitting ratio (60%/20%/20%) is used for heterophilicgraphs.
Revised: Specifically, The sparse splitting ratio (2.5%/2.5%95/%)is used for homophilic graphs, and the dense splitting ratio (60%/20%/20%) is used for heterophilicgraphs.

Original: We run each experiment 50 times with random initialization, and random data splits.
Revised: We run each experiment 50 times with random initialization, and random data splits.

Original: Finally,we report the average results with a 95% confidence interval.
Revised: Finally,we report the average results with a 95% confidence interval.

Original: We set the degree of the polynomial
Revised: We set the degree of the polynomial

Original: K = 10 for all datasets.
Revised: K = 10 for all datasets.

Original: The Adam [18] is employed as the optimizer for the NFGNN training.
Revised: The Adam [18] is employed as the optimizer for the NFGNN training.

Original: For
Revised: For

Original: GPRGNN, BernNet, and BMGCN, we use the best combination of hyperparameters provided in theoriginal paper to report the results for each dataset.
Revised: GPRGNN, BernNet, and BMGCN, we use the best combination of hyperparameters provided in theoriginal paper to report the results for each dataset.

Original: 5.2 Performance Comparison
Revised: 5.2 Performance Comparison

Original: The average results of running 50 times on the node classification task are reported in Table 1, whereaccuracy is used as the evaluation metric with a 95% confidence interval.
Revised: The average results of running 50 times on the node classification task are reported in Table 1, whereaccuracy is used as the evaluation metric with a 95% confidence interval.

Original: NFGNN outperforms allthe baselines on 7 datasets and achieves comparable results on the other 3 datasets.
Revised: NFGNN outperforms allthe baselines on 6 datasets and achieves comparable results on the other 4 datasets.

Original: In particular, on
Revised: In particular, on

Original: Chameleon and Squirrel graphs, NFGNN outperforms the SOTA method BMGCN by a large margin,i.e., 2.83% and 5.03%, demonstrating the superiority of our method.
Revised: Chameleon and Squirrel graphs, NFGNN outperforms the SOTA method BMGCN by a large margin,i.e., 2.83% and 5.03%, demonstrating the superiority of our method.

Original: Meanwhile, it can be observed several interesting phenomena in Table 1.
Revised: Meanwhile, it can be observed several interesting phenomena in Table 1.

Original: i ) GCN and GAT are eveninferior to MLP on some heterophilic graphs, which shows that positive near-neighbor aggregation isindeed out of power in some cases.
Revised: i ) GCN and GAT are eveninferior to MLP on some heterophilic graphs, which shows that positive near-neighbor aggregation isindeed out of power in some cases.

Original: Besides, the performance of MLP also shows that the utilizationof node features is also very important for GNNs.
Revised: Besides, the performance of MLP also shows that the utilizationof node features is also very important for GNNs.

Original: ii )
Revised: ii )

Original: The filter-learning based methods generallyhave good a performance on both the homophilic and heterophilic graphs, indicating that adaptivefilter learning has better transferability than filter pre-designing.
Revised: The filter-learning based methods generallyhave good a performance on both the homophilic and heterophilic graphs, indicating that adaptivefilter learning has better transferability than filter pre-designing.

Original: 5.3 Node-level Analysis
Revised: 5.3 Node-level Analysis

Original: A motivation of the proposed NFGNN is to solve the mixed local patterns discussed in Sect.
Revised: A motivation of the proposed NFGNN is to solve the mixed local patterns discussed in Sect.

Original: Therefore, we divide the test nodes into 5 different intervals according to the homophilic 1-hopneighbor ratio h N 1 ( v ) and report the mean accuracy of each interval.
Revised: Therefore, we divide the test nodes into 5 different intervals according to the homophilic 1-hopneighbor ratio h N 1 ( v ) and report the mean accuracy of each interval.

Original: The results of GCN, NFGNNwith only γ (marked as NFGNN w/o NF) and NFGNN are shown in Fig 3.
Revised: The results of GCN, NFGNN [Table] with only γ (marked as NFGNN w/o NF) and NFGNN are shown in Fig 3.

Original: It should be noticed that the
Revised: It should be noticed that the

Original: NFGNN w/o NF is equivalent to learning a globally consistent filter using the Chebyshev polynomial.
Revised: NFGNN w/o NF is equivalent to learning a globally consistent filter using the Chebyshev polynomial.

Original: It can be seen from Fig. 3(c) and (d) that NFGNN has a promising and similar performance on allfive intervals, which shows that NFGNN can effectively capture the various local patterns under the [Table] condition as long as the amount of trainable data is sufficient.
Revised: It can be seen from Fig. 3(c) and (d) that NFGNN has a promising and similar performance on allfive intervals, which shows that NFGNN can effectively capture the various local patterns under thecondition as long as the amount of trainable data is sufficient.

Original: Besides, both NFGNN and NFGNN w/o
Revised: Besides, both NFGNN and NFGNN w/o

Original: NF perform better than GCN on the semi-supervised node classification task, as shown in Fig.
Revised: NF perform better than GCN on the semi-supervised node classification task, as shown in Fig.

Original: 3(a)and (b).
Revised: 3(a)and (b).

Original: It suggests that adaptive learning filters are no less expressive than pre-designed filters, evenin the semi-supervised case.
Revised: It suggests that adaptive learning filters are no less expressive than pre-designed filters, evenin the semi-supervised case.

Original: 5.4 Effectiveness of the Node-oriented Filtering
Revised: 5.4 Effectiveness of the Node-oriented Filtering

Original: To evaluate the effectiveness of the proposed node-oriented filtering more comprehensively, we firstcompare the performance of NFGNN and NFGNN w/o NF.
Revised: To evaluate the effectiveness of the proposed node-oriented filtering more comprehensively, we firstcompare the performance of NFGNN and NFGNN w/o NF.

Original: Further, since the node-oriented filteringis independent of the polynomial basis, the Chebyshev basis is replaced by the Monomial basis and
Revised: Further, since the node-oriented filteringis independent of the polynomial basis, the Chebyshev basis is replaced by the Monomial basis and

Original: Bernstein basis, respectively, and we check the improvement bring by the node-oriented filteringmechanism for them.
Revised: Bernstein basis, respectively, and we check the improvement bring by the node-oriented filteringmechanism for them.

Original: For the Bernstein basis, we refer to the implementation form given in [13].
Revised: For the Bernstein basis, we refer to the implementation form given in [13].

Original: The results on six graphs are summarized in 5.4.
Revised: The results on six graphs are summarized in 5.4.

Original: Firstly, it can be seen that the globally consistentfilters learned using three different bases have leading performance on different datasets, respectively,illustrating the effectiveness of using a polynomial approximation to learn filters.
Revised: Firstly, it can be seen that the globally consistentfilters learned using three different bases have leading performance on different datasets, respectively,illustrating the effectiveness of using a polynomial approximation to learn filters.

Original: Furthermore,except for the Monomial basis on Texas graph, the node-oriented filtering mechanism has differentenhancements for each basis.
Revised: Furthermore,except for the Monomial basis on Texas graph, the node-oriented filtering mechanism has differentenhancements for each basis.

Original: The improvements not only validate the effectiveness of the proposednode-oriented filtering, but also demonstrate that the polynomial filter and the node-oriented filteringcan each other to some extent.
Revised: The improvements not only validate the effectiveness of the proposednode-oriented filtering, but also demonstrate that the polynomial filter and the node-oriented filteringcan each other to some extent.

Original: Discussion on the Connection of NFGNN with other Spectral-based GNNs

Original: Studying spectral-based GNNs in accordance with the idea of graph signal processing theory is oneof the origins of GNNs.

Original: With different starting points, the spatial-designed GNNs aim to design theneighborhood aggregation mechanism based on the topological characteristics of the graph, focusingmore on the local relationship between nodes and their neighbors.

Original: In contrast, spectral-based GNNsare dedicated to the design of the filtering of the graph signal in the spectral domain, analyzing thegraph more from a global perspective.

Original: The proposed NFGNN in this paper provides a new form oftrade-off between global and local perspectives in the spectral domain. Particularly, NFGNN can beseen as a extension of the existing methods for estimating global filters. For spectral-based GNNs,the scalability of spectral convolution and inductive learning setting is still a key issue to be solved atpresent, and it is still one of the directions of graph neural networks that can be expected because ofthe great transferability it exhibits.  Conclusion 
Revised:  Conclusion and Discussion

Original: In this paper, we first analyze in depth the local patterns in graph data and the aggregatability of
Revised: In this paper, we first analyze in depth the local patterns in graph data and the aggregatability of

Original: Near-neighbors.
Revised: Near-neighbors.

Original: Motivated by these observations, we rethink the spectral-based GNNs and propose
Revised: Motivated by these observations, we rethink the spectral-based GNNs and propose

Original: NFGNN for node-oriented spectral filtering via the generalized translated operator.
Revised: NFGNN for node-oriented spectral filtering via the generalized translated operator.

Original: Comparedto previous methods that learn a global filter, NFGNN performs spectral filtering through filterstranslated on specific nodes to address the issue of local patterns.
Revised: Comparedto previous methods that learn a global filter, NFGNN performs spectral filtering through filterstranslated on specific nodes to address the issue of local patterns.

Original: Through recursive form and reparameterization trick, the oriented-filtering is implemented in a simple way.
Revised: Through recursive form and reparameterization trick, the oriented-filtering is implemented in a simple way.

Original: The experimental resultson several real-world graph datasets verify that our NFGNN achieves more remarkable performanceover currently available alternatives.
Revised: The experimental resultson several real-world graph datasets verify that our NFGNN achieves more remarkable performanceover currently available alternatives.

Revised: Studying spectral-based GNNs in accordance with the idea of graph signal processing theory is oneof the origins of GNNs.

Original: Checklist
Revised: With different starting points, the spatial-designed GNNs aim to design theneighborhood aggregation mechanism based on the topological characteristics of the graph, focusingmore on the local relationship between nodes and their neighbors. In contrast, spectral-based GNNsare dedicated to the design of the filtering of the graph signal in the spectral domain, analyzing thegraph more from a global perspective. The proposed NFGNN in this paper provides a new form oftrade-off between global and local perspectives in the spectral domain. Particularly, NFGNN can beseen as a extension of the existing methods for estimating global filters.

Revised: For spectral-based GNNs,the scalability of spectral convolution and inductive learning setting are still key issues to be solved atpresent, and it is still one of the directions of graph neural networks that can be expected because ofthe great transferability it exhibits.

Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Original: [Yes] See Sect.
Revised: [Yes] See Sect.

Original: 1, our contribution is in the last paragraph.
Revised: 1, our contribution is in the last paragraph.

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] See in Sect. 
Revised: [Yes] See in Sect. ?? .

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [Yes]
Revised: [Yes]

Original: The
Revised: The

Original: broader impact is discussed in supplemental materials.
Revised: broader impact is discussed in supplemental materials.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [Yes] The code and instructions related to the experiment are in the supplementary materials.
Revised: [Yes] The code andinstructions related to the experiment are in the supplementary materials.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they werechosen)?

Original: [Yes] See the experimental setup in Sect.
Revised: [Yes] See the experimental setup in Sect.

Original: 5 and more detail in supplemental materials.
Revised: 5 and more detail in supplementalmaterials.

Original: (c) Did you report error bars (e
Revised: (c) Did you report error bars (e

Original: g., with respect to the random seed after running experiments multiple times)?
Revised: g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] The confidence interval is reported.
Revised: [Yes] The confidence interval is reported.

Original: (d) Did you include the total amount of compute and the type of resources used (e
Revised: (d) Did you include the total amount of compute and the type of resources used (e

Original: g., typeof GPUs, internal cluster, or cloud provider)?
Revised: g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See in the supplemental materials.
Revised: [Yes] See in the supplemental materials.

Original: If you are using existing assets (e
Revised: If you are using existing assets (e

Original: g., code, data, models) or curating/releasing new assets...
Revised: g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes]
Revised: [Yes]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A]
Revised: [N/A]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: [Table]

Original: using/curating?
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: [Table]

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: [Table] ) Did you include the full text of instructions given to participants and screenshots, if[Table]
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
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Original: [Table]

Original: ) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

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Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: ) Did you include the estimated hourly wage paid to participants and the total amount
Revised: c) Did you include the estimated hourly wage paid to participants and the total amount

Original: [Table]

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]

Original: [Table]