Original: A
Revised: A

Original: N EW
Revised: N EW

Original: P ARADIGM
Revised: P ARADIGM

Original: FOR
Revised: FOR

Original: F EDERATED
Revised: F EDERATED

Original: S TRUCTURE
Revised: S TRUCTURE

Original: N ON -IID S UBGRAPH L EARNING
Revised: N ON -IID S UBGRAPH L EARNING

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Federated graph learning (FGL), a distributed training framework for graph neural networks (GNNs) has attracted much attention for breaking the centralized machine learning assumptions.
Revised: Federated graph learning (FGL), a distributed training framework for graph neural networks (GNNs) has attracted much attention for breaking the centralized machine learning assumptions.

Original: Despite its effectiveness, the differences in data collection perspectives and quality lead to the challenges of heterogeneity, especially the domain-specific graph is partitioned into subgraphs in different institutions.
Revised: Despite its effectiveness, the differences in data collection perspectives and quality lead to the challenges of heterogeneity, especially the domain-specific graph is partitioned into subgraphs in different institutions.

Original: However, existing FGL methods implement graph data augmentation or personalization with community split which follows the cluster homogeneity assumptions.
Revised: However, existing FGL methods implement graph data augmentation or personalization with community split which follows the cluster homogeneity assumptions.

Original: Hence we investigate the above issues and suggest that subgraph heterogeneity is essentially the structure variations.
Revised: Hence we investigate the above issues and suggest that subgraph heterogeneity is essentially the structure variations.

Original: From the observations on FGL, we first define the structure non-independent identical distribution (Non-IID) problem, which presents covariant shift challenges among client-wise subgraphs.
Revised: From the observations on FGL, we first define the structure non-independent identical distribution (NonIID) problem, which presents unique challenges among client-wise subgraphs.

Original: Meanwhile, we propose a new paradigm for general federated data settings called Adaptive Federated Graph Learning (AdaFGL).
Revised: Meanwhile, we propose a new paradigm for general federated data settings called Adaptive Federated Graph Learning (AdaFGL).

Original: The motivation behind it is to implement adaptive propagation mechanisms based on federated global knowledge and non-params label propagation.
Revised: The motivation behind it is to implement adaptive propagation mechanisms based on federated global knowledge and non-params label propagation.

Original: We conduct extensive experiments with community split and structure Non-IID settings, our approach achieves state-of-the-art performance on five benchmark datasets.
Revised: We conduct extensive experiments with community split and structure Non-IID settings, our approach achieves state-of-the-art performance on five benchmark datasets.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: The graph as a relational data structure is widely used to model real-world entity relations such as citation networks Yang et al.
Revised: The graph as a relational data structure is widely used to model real-world entity relations such as citation networks Yang et al.

Original: (2016a), recommended systems Wu et al.
Revised: (2016a), recommended systems Wu et al.

Original: , drug discovery Gaudelet et al.
Revised: , drug discovery Gaudelet et al.

Original: (2021), particle physics Shlomi et al.
Revised: (2021), particle physics Shlomi et al.

Original: (2021), etc.
Revised: (2021), etc.

Original: However, due to the collection agents and privacy concerns, generally, the global domain-specific graph consists of many subgraphs collected by multiple institutions.
Revised: However, due to the collection agents and privacy concerns, generally, the global domain-specific graph consists of many subgraphs collected by multiple institutions.

Original: In order to analyze the local subgraph, each client maintains a powerful graph mining model such as graph neural networks (GNNs), which have achieved stateof-the-art performance in many graph learning tasks Zhang et al.
Revised: In order to analyze the local subgraph, each client maintains a powerful graph mining model such as graph neural networks (GNNs), which have achieved stateof-the-art performance in many graph learning tasks Zhang et al.

Original: (2022b)
Revised: (2022b)

Original: ; Hu et al.
Revised: ; Hu et al.

Original: ; Zhang & Chen (2018).
Revised: ; Zhang & Chen (2018).

Original: Despite its effectiveness, the limited data provide sub-optimal performance in most cases.
Revised: Despite its effectiveness, the limited data provide sub-optimal performance in most cases.

Original: Motivated by the success of federated learning (FL), a natural idea is to combine the GNNs with FL to utilize the distributed subgraphs.
Revised: Motivated by the success of federated learning (FL), a natural idea is to combine the GNNs with FL to utilize the distributed subgraphs.

Original: Recently, federated graph learning (FGL) He et al.
Revised: Recently, federated graph learning (FGL) He et al.

Original: ; Wang et al.
Revised: ; Wang et al.

Original: (2022b) is proposed to achieve collaborative training without directly sharing data, yet an essential concern is the heterogeneity of the distributed subgraphs.
Revised: (2022b) is proposed to achieve collaborative training without directly sharing data, yet an essential concern is the heterogeneity of the distributed subgraphs.

Original: Notably, graph heterogeneity is different from the heterogeneity of labels or features in the fields of computer vision or natural language processing, we suggest that it depends on the graph structure.
Revised: Notably, graph heterogeneity is different from the heterogeneity of labels or features in the fields of computer vision or natural language processing, we suggest that it depends on the graph structure.

Original: However, The existing FGL methods simulate the federated subgraph distributions through community split, which follows the cluster homogeneity assumption as shown in Fig.1(a).
Revised: However, The existing FGL methods simulate the federated subgraph distributions through community split, which follows the cluster homogeneity assumption as shown in Fig.1(a).

Original: Specifically, community split leads to the subgraph structure being consistent and the same as the original graph, e.g., connected nodes are more likely to have the same labels.
Revised: Specifically, community split leads to the subgraph structure being consistent and the same as the original graph, e.g., connected nodes are more likely to have the same labels.

Original: Obviously, it is overly desirable and hard to satisfy in reality, hence we consider a more reasonable setting shown in Fig.1(c).
Revised: Obviously, it is overly desirable and hard to satisfy in reality, hence we consider a more reasonable setting shown in Fig.1(c).

Original: The motivation behind it is due to graph structure directly related to node labels and feature distributions.
Revised: The motivation behind it is due to graph structure directly related to node labels and feature distributions.

Original: Meanwhile, the challenges of structure heterogeneity are ubiquitous in the real world Zheng et al.
Revised: Meanwhile, the challenges of structure heterogeneity are ubiquitous in the real world Zheng et al.

Original: (2022b).
Revised: (2022b).

Original: Specifically, graph structure can be divided into two types: homogeneity means that connected nodes are more likely to have the same label and similar feature distributions and heterogeneity is the opposite.
Revised: Specifically, graph structure can be divided into two types: homogeneity means that connected nodes are more likely to have the same label and similar feature distributions and heterogeneity is the opposite.

Original: Therefore, we first refer to the above problem as structure non-independent identical distribution (Non-IID) in FGL.
Revised: Therefore, we first refer to the above problem as structure non-independent identical distribution (Non-IID) in FGL.

Original: In order to explain it intuitively, we visualize the 3 clients
Revised: In order to explain it intuitively, we visualize the 3 clients

Original: FGL Global Server 1.
Revised: FGL Global Server 1.

Original: Init model 4.
Revised: Init model 4.

Original: Receives message (grads, embs, adj...) 5.
Revised: Receives message (grads, embs, adj...) 5.

Original: Handles messages 6.
Revised: Handles messages 6.

Original: Broadcasts results
Revised: Broadcasts results

Original: FGL Local System 2.
Revised: FGL Local System 2.

Original: Receives model 3.
Revised: Receives model 3.

Original: Local training ...
Revised: Local training ...

Original: Repeat 2.
Revised: Repeat 2.

Original: (b) General FGL Pipeline
Revised: (b) General FGL Pipeline

Original: GNN Client
Revised: GNN Client

Original: Client Client
Revised: Client Client

Original: GNN (a)
Revised: GNN (a)

Original: [Figure] FGLBased on Community Split
Revised: Existing FGLBased on Community Split

Original: Homophily
Revised: Homophily

Original: Heterophily
Revised: Heterophily

Original: Client
Revised: Client

Original: Homophily
Revised: Homophily

Original: Client( [Figure] ) Challenges 
Revised: Client(c) Challenges Based on Structure Non

Original: [Figure]
Revised: [Figure]

Original:  Client partitioning result on Cora in Table.
Revised: Split Client partitioning result on Cora in Table.

Original: 1 and Table.
Revised: 1 and Table.

Original: 2, where Homo represents the homogeneity degree of the local subgraph, and it is computed by a popular metric
Revised: 2, where Homo represents the homogeneity degree of the local subgraph, and it is computed by a popular metric

Original: Pei et al.
Revised: Pei et al.

Original: Obviously, compared to community split, which follows the cluster homogeneity assumption and uniform distribution principle, structure Non-IID brings challenges to the existing FGL methods.
Revised: Obviously, compared to community split, which follows the cluster homogeneity assumption and uniform distribution principle, structure Non-IID brings challenges to the existing FGL methods.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: Based on this, we investigate the above issues through empirical analysis shown in Fig. 2.
Revised: Based on this, we investigate the above issues through empirical analysis shown in Fig. 2.

Original: According to the results, we observe that in case the original graph satisfies the homogeneity assumption then the label distributions satisfy Non-IID.
Revised: According to the results, we observe that in case the original graph satisfies the homogeneity assumption then the label distributions satisfy Non-IID.

Original: It is the opposite when the original graph satisfies the heterogeneity.
Revised: It is the opposite when the original graph satisfies the heterogeneity.

Original: This is due to the fact that the nodes partitioned into the same clients are communities and follow the uniform distribution principle.
Revised: This is due to the fact that the nodes partitioned into the same clients are communities and follow the uniform distribution principle.

Original: Meanwhile, Fig. 2 confirms that our proposed structure Non-IID is a more general approach, which contains two reasonable label distributions and diverse graph structure information.
Revised: Meanwhile, Fig. 2 confirms that our proposed structure Non-IID is a more general approach, which contains two reasonable label distributions and diverse graph structure information.

Original: In addition, the local accuracy indicates that the subgraph structure performs a more important role in FGL compared to the label distributions, which also supports our motivation.
Revised: In addition, the local accuracy indicates that the subgraph structure performs a more important role in FGL compared to the label distributions, which also supports our motivation.

Original: In model performance, we observe that the GGCN improves the structure Non-IID problem, and FedSage+ trains NeighGen to implement local subgraph augmentation by sharing node embeddings, which reduce the covariance shifts among clients.
Revised: In model performance, we observe that the GGCN improves the structure Non-IID problem, and FedSage+ trains NeighGen to implement local subgraph augmentation by sharing node embeddings.

Original: However, the above methods fail to achieve competitive results as SGC on the homogeneous subgraphs while considering heterogeneity.
Revised: However, the above methods fail to achieve competitive results as SGC on the homogeneous subgraphs while considering heterogeneity.

Original: In order to efficiently analyze distributed subgraphs with both homogeneity and heterogeneity.
Revised: In order to efficiently analyze distributed subgraphs with both homogeneity and heterogeneity.

Original: We propose a simple pipeline called Adaptive Federated Graph Learning (AdaFGL) for more general federated data settings, which consists of three main parts.
Revised: We propose a simple pipeline called Adaptive Federated Graph Learning (AdaFGL) for more general federated data settings, which consists of three main parts.

Original: Specifically, it starts by analyzing the subgraph structure through non-params label propagation and selects the appropriate base model: (i) the federated global knowledge extractor (e.g., MLP, powerful GNNs, or any reasonable embedding models), which does not rely on any learning over the subgraph.
Revised: Specifically, it starts by analyzing the subgraph structure through non-params label propagation and selects the appropriate base model: (i) the federated global knowledge extractor (e.g., MLP, powerful GNNs, or any reasonable embedding models), which does not rely on any learning over the subgraph.

Original: Then, the base predictor is trained based on the global data, which can be done offline or in parallel with local training, benefiting from the flexibility of our approach.
Revised: Then, the base predictor is trained based on the global data, which can be done offline or in parallel with local training, benefiting from the flexibility of our approach.

Original: Finally, the local client implements two adaptive propagation mechanisms: (ii) homogeneity propagation module or (iii) heterogeneity propagation module based on the local subgraph.
Revised: Finally, the local client implements two adaptive propagation mechanisms: (ii) homogeneity propagation module or (iii) heterogeneity propagation module based on the local subgraph.

Original: Notably, with non-params label propagation, the above process is adaptive.
Revised: Notably, with non-params label propagation, the above process is adaptive.

Original: To summarize, the contributions of this paper are as follows: (1) To the best of our knowledge, we are the first to analyze the structure Non-IID problem in FGL, which is a more general federated data setting and brings new challenges.
Revised: To summarize, the contributions of this paper are as follows: (1) To the best of our knowledge, we are the first to analyze the structure Non-IID problem in FGL, which is a more general federated data setting and brings new challenges.

Original: (2) We propose AdaFGL, a new paradigm for structure Non-IID subgraph learning, which shows its flexibility in FGL with impressive performance.
Revised: (2) We propose AdaFGL, a new paradigm for structure Non-IID subgraph learning, which shows its flexibility in FGL with impressive performance.

Original: (3) Extensive experiments demonstrate the effectiveness of AdaFGL.
Revised: (3) Extensive experiments demonstrate the effectiveness of AdaFGL.

Original: Specifically, our approach achieves state-ofthe-art performance in the above two data settings.
Revised: Specifically, our approach achieves state-ofthe-art performance in the above two data settings.

Original: Compared to the best prediction accuracy in the baselines, our method achieves performance gains of 4 .
Revised: Compared to the best prediction accuracy in the baselines, our method achieves performance gains of 4 .

Original: 67% and 2 .
Revised: 67% and 2 .

Original: 65% in structure Non-IID and community split data settings, respectively.
Revised: 65% in structure Non-IID and community split data settings, respectively.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: FedGGCNFedSGCFedSage
Revised: FedGGCNFedSGCFedSage

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Original: FedGGCNFedSGCFedSage+

Original:  P RELIMINARIES
Revised:  P RELIMINARIES

Original: In this section, we first introduce the semi-supervised node classification task.
Revised: In this section, we first introduce the semi-supervised node classification task.

Original: Then, we review the prior diverse GNNs and very recent FGL methods.
Revised: Then, we review the prior diverse GNNs and very recent FGL methods.

Original: Consider a graph G =
Revised: Consider a graph G =

Original: ( V, E ) with | V | = n nodes and | E | = m edges, the adjacency matrix (including self loops) is denoted as ˆ A ∈ R n × n , the feature matrix is denoted as X = { x 1 , x 2 , . . .
Revised: ( V, E ) with | V | = n nodes and | E | = m edges, the adjacency matrix (including self loops) is denoted as ˆ A ∈ R n × n , the feature matrix is denoted as X = { x 1 , x 2 , . . .

Original: , x n } in which x v ∈ R f represents the feature vector of node v , and f represents the dimension of the node attributes.
Revised: , x n } in which x v ∈ R f represents the feature vector of node v , and f represents the dimension of the node attributes.

Original: Besides, Y = { y 1 , y 2 , . . .
Revised: Besides, Y = { y 1 , y 2 , . . .

Original: , y n } is the label matrix, where y v ∈ R | Y | is a one-hot vector and | Y | represents the number of the node classes.
Revised: , y n } is the label matrix, where y v ∈ R | Y | is a one-hot vector and | Y | represents the number of the node classes.

Original: The semi-supervised node classification task is based on the topology of labeled set V L and unlabeled set V U , and the nodes in V U are predicted with the model supervised by V L .
Revised: The semi-supervised node classification task is based on the topology of labeled set V L and unlabeled set V U , and the nodes in V U are predicted with the model supervised by V L .

Original: GNNs.
Revised: GNNs.

Original: As the most popular GNN method, The forward information propagation process of the l -th layer GCN Kipf & Welling (2017) is formulated as
Revised: As the most popular GNN method, The forward information propagation process of the l -th layer GCN Kipf & Welling (2017) is formulated as

Original: [Equation]
Revised: [Equation]

Original: ) where ˆ D represents the degree matrix with ˆ A , r ∈ [0 , 1] denotes the convolution kernel coefficient,
Revised: where ˆ D represents the degree matrix with ˆ A , r ∈ [0 , 1] denotes the convolution kernel coefficient,

Original: W represents the trainable weight matrix, and σ ( · ) represents the non-linear activation function.
Revised: W represents the trainable weight matrix, and σ ( · ) represents the non-linear activation function.

Original: In GCN, we set r = 1 / 2 , and
Revised: In GCN, we set r = 1 / 2 , and

Original: then ˆ D − 1 / 2 ˆ A ˆ D − 1 / 2 is called symmetric normalized adjacency matrix.
Revised: then ˆ D − 1 / 2 ˆ A ˆ D − 1 / 2 is called symmetric normalized adjacency matrix.

Original: Despite their effectiveness, they have limitations in real-world graphs, which have complex heterogeneous relationship patterns.
Revised: Despite their effectiveness, they have limitations in real-world graphs, which have complex heterogeneous relationship patterns.

Original: Some recent researches Liu et al.
Revised: Some recent researches Liu et al.

Original: Chien et al.
Revised: Chien et al.

Original: He et al.
Revised: He et al.

Original: ; Wang et al.
Revised: ; Wang et al.

Original: (2022a)
Revised: (2022a)

Original: ; Yang et al.
Revised: ; Yang et al.

Original: (2022) solve it by higher-order neighborhood discovery or message combination strategies to improve the GNN process via
Revised: (2022) solve it by higher-order neighborhood discovery or message combination strategies to improve the GNN process via

Original: [Equation]
Revised: [Equation]

Original: where h ∗ u denotes the information of multi-hop neighbors N ∗ ( v ) , m ∗ v represents the higher-order messages of node v from the previous layers, Aggregate ( · ) and Update ( · ) denote the message aggregation function and update function, respectively.
Revised: where h ∗ u denotes the information of multi-hop neighbors N ∗ ( v ) , m ∗ v represents the higher-order messages of node v from the previous layers, Aggregate ( · ) and Update ( · ) denote the message aggregation function and update function, respectively.

Original: However, these methods suffer from high computational complexity and fail to achieve competitive performance on the homogeneous graph.
Revised: However, these methods suffer from high computational complexity and fail to achieve competitive performance on the homogeneous graph.

Original: FGL has received growing attention for breaking centralized graph machine learning assumptions.
Revised: FGL has received growing attention for breaking centralized graph machine learning assumptions.

Original: FedGraphNN He et al.
Revised: FedGraphNN He et al.

Original: and
Revised: and

Original: FS-G Wang
Revised: FS-G Wang

Original: et al.
Revised: et al.

Original: (2022b)
Revised: (2022b)

Original: propose general FGL packages, which contain a wide range of graph learning tasks.
Revised: propose general FGL packages, which contain a wide range of graph learning tasks.

Original: GCFL Xie et al.
Revised: GCFL Xie et al.

Original: and FED-PUB Baek et al.
Revised: and FED-PUB Baek et al.

Original: (2022) investigate the personalized technologies in graph-level and node-level, respectively.
Revised: (2022) investigate the personalized technologies in graph-level and node-level, respectively.

Original: Furthermore, some recent researches improve performance with local subgraph augmentation, including FedGNN Wu et al.
Revised: Furthermore, some recent researches improve performance with local subgraph augmentation, including FedGNN Wu et al.

Original: (2021), FedGL Chen et al.
Revised: (2021), FedGL Chen et al.

Original: (2021), and FedSage Zhang et al.
Revised: (2021), and FedSage Zhang et al.

Original: Inspired by FS-G Wang et al.
Revised: Inspired by FS-G Wang et al.

Original: (2022b)
Revised: (2022b)

Original: , we can consider the collaborative training process in FGL as modules.
Revised: , we can consider the collaborative training process in FGL as modules.

Original: Specifically, we model the information such as gradients and node embeddings uploaded bythe clients as messages.
Revised: Specifically, we model the information such as gradients and node embeddings uploaded by

Original: Then we consider the server processes and broadcast results as the various

Original: [Figure]
Revised: [Figure]

Original: message-handling mechanisms.
Revised: the clients as messages.

Revised: Then we consider the server processes and broadcast results as the various message-handling mechanisms.

Original: Here we illustrate the GNNs combined with collaborative training.
Revised: Here we illustrate the GNNs combined with collaborative training.

Original: Its generic form with N clients is defined as
Revised: Its generic form with N clients is defined as

Original: [Equation]
Revised: [Equation]

Original: where f θi and L ce are the i -th local GNN with parameters θ and cross-entropy loss function, respectively.
Revised: where f θi and L ce are the i -th local GNN with parameters θ and cross-entropy loss function, respectively.

Original: The cross-entropy loss can be replaced by any other appropriate loss function depending on the task.
Revised: It can be replaced by any other appropriate loss function depending on the task.

Original: ( A i , X i , Y i ) ∼ D i represents the local subgraph ( A i , X i , Y i ) sampled from the distribution D i .
Revised: ( A i , X i , Y i ) ∼ D i represents the local subgraph ( A i , X i , Y i ) sampled from the distribution D i .

Original: FedAvg McMahan et al.
Revised: FedAvg McMahan et al.

Original: (2017) is an efficient FL algorithm for collaborative training of the global model in the server, which can be defined as
Revised: (2017) is an efficient FL algorithm, which can be defined as

Original: [Equation]
Revised: [Equation]

Original: where t represents the round number of the FL process, W represents the model weights, η represents the learning rate, g represents the gradient calculated from the Eq. 3, n i and n represent the i -th local client data size and the global data size, respectively.
Revised: where t represents the round number of the FL process, W represents the model weights, η represents the learning rate, g represents the gradient calculated from the Eq. 3, n i and n represent the i -th local client data size and the global data size, respectively.

Original:  A DA FGL P IPELINE
Revised:  A DA FGL P IPELINE

Original: The basic idea of AdaFGL is to perform adaptive propagation mechanisms based on federated global knowledge and non-params label propagation for general federated data settings.
Revised: The basic idea of AdaFGL is to perform adaptive propagation mechanisms based on federated global knowledge and non-params label propagation.

Original: Therefore, we propose a simple pipeline with three main parts as shown in Fig. 3, which combine the global knowledge embeddings and local structure properties.
Revised: The pipeline with three main parts as shown in Fig. 3, which combine the global knowledge embeddings and local structure properties.

Original: Note this pipeline is not trained in an endto-end way, the above decoupling process utilizes the computational capacity of the local system while minimizing communication costs and the risk of privacy leakage.
Revised: The above decoupling process utilizes the computational capacity of the local system while minimizing communication costs and the risk of privacy leakage.

Original: AdaFGLis a new paradigm for FGL, which can tightly combine FL and graph machine learning through the base predictor and adaptive propagation mechanisms. This design allows our approach to instantly benefit from the evolution of FL and GNNs.
Revised: AdaFGL can benefit from the evolution of FL and GNN through the base predictor and adaptive propagation. Notably, the base predictor obtained by federated training and personalized propagation are viewed as two decoupled modules that are executed sequentially.

Original: We will see the massive potential through the details of  the AdaFGL.
Revised: Meanwhile, both of them accomplish the training without sharing local private data.

Original: 3. F EDERATED G LOBAL K NOWLEDGE E XTRACTOR
Revised: 3. F EDERATED G LOBAL K NOWLEDGE E XTRACTOR

Original: In FGL, limited data yields sub-optimal performance in most cases.
Revised: In FGL, limited data yields sub-optimal performance in most cases.

Original: Therefore, we aim to achieve collaborative training by exploiting global data.

Original:  AdaFGL starts to perform non-params label propagation to select the appropriate federated global knowledge extractor (e.g. MLP, diverse GNNs, or any reasonable graph machine learning models).
Revised: Therefore, AdaFGL starts to perform non-params label propagation to select the appropriate federated global knowledge extractor (e.g. MLP, diverse GNNs).

Original: Note this process does not rely on any learning over the subgraph.
Revised: Note this process does not rely on any learning over the subgraph.

Original: Specifically, the labeled nodes are initialized as y 0 v = y v , ∀ v ∈ V L , and the unlabeled nodes are denoted as y 0 u
Revised: Specifically, the labeled nodes are initialized as y 0 v = y v , ∀ v ∈ V L , and the unlabeled nodes are denoted as y 0 u

Original: = ( 1 | Y | , . .
Revised: = ( 1 | Y | , . .

Original: , 1 | Y | ) , ∀ u ∈ V U .
Revised: , 1 | Y | ) , ∀ u ∈ V U .

Original: Then, the non-params label propagation of the k -step is expressed as
Revised: Then, the non-params label propagation of the k -step is expressed as

Original: [Equation]
Revised: [Equation]

Original: We follow the approximate calculation of the personalized PageRank matrix Klicpera et al.
Revised: We follow the approximate calculation of the personalized PageRank matrix Klicpera et al.

Original: (2019) to implement label propagation, where N v represents the one-hop neighbors of v , and we default set α = 0 .
Revised: (2019), where N v represents the one-hop neighbors of v , and we default set α = 0 .

Original: Then, we design the homogeneity confidence score (HCS) computed by the number of correct predictions, and the default ratio of the boolean mask is 0 .
Revised: Then, we design the homogeneity confidence score (HCS) computed by the number of correct predictions, and the default ratio of the boolean mask is 0 .

Revised: Finally, we set thresholds λ for the adaptive binary selection of the homogeneity propagation module and heterogeneity propagation module in each client.

Revised: In experiments, we default set λ = 0 .

Original:  To demonstrate that AdaFGL is a simple yet effective framework, we choose simple models (e.g. MLP or SGC) and FedAvg to achieve federated training.
Revised: 6 To demonstrate that AdaFGL is a simple yet effective framework, we choose simple models (e.g. MLP or SGC) and FedAvg to achieve federated training.

Original: Due to the flexibility of AdaFGL, they can be replaced by any other powerful GNNs and federated training methods to improve predictive accuracy.
Revised: Due to the flexibility of AdaFGL, they can be replaced by any other powerful GNNs and federated methods.

Original: From the perspective of FL in Non-IID data, we default choose MLP as the base predictor, which is independent of the graph structure.
Revised: From the perspective of FL in Non-IID data, we default choose MLP as the base predictor, which is independent of the graph structure.

Original: Then we quote the convergence theorem Li et al.
Revised: Then we quote the convergence theorem Li et al.

Original: (2020) in T rounds and E epochs, the federated global knowledge extractor error bound ϵ fed is expressed as
Revised: (2020) in T rounds and E epochs, the federated global knowledge extractor error bound ϵ fed is expressed as

Original: [Equation]
Revised: [Equation]

Original: It assumes that the mapping function satisfies L -smooth and µ -strongly convex, where φ and ϕ represent the local random gradient and the degree of model heterogeneity, respectively, γ = max { 8 L/µ, E } , ω denotes the divergence of local model, and W ∗ represents the global optima.
Revised: It assumes that the mapping function satisfies L -smooth and µ -strongly convex, where φ and ϕ represent the local random gradient and the degree of model heterogeneity, respectively, γ = max { 8 L/µ, E } , ω denotes the divergence of local model, and W ∗ represents the global optima.

Original: We observe that the base predictor error bound is mainly determined by the differences in the node feature distributions, and the model performance will be further hurt if the graph structure is considered.
Revised: We observe that the base predictor error bound is mainly determined by the differences in the node feature distributions, and the model performance will be further hurt if the graph structure is considered.

Original: Based on this, we are motivated to propose adaptive propagation mechanisms, which improve this problem through personalized techniques.
Revised: Therefore, we are motivated to propose adaptive propagation mechanisms. Specifically, we implement the binary selection of the homogeneity propagation module or heterogeneity propagation module in each client by comparing the HCS value and the threshold λ . We will describe the technical details of personalized propagation strategies.

Original: 3. A DAPTIVE H OMOGENEITY P ROPAGATION
Revised: 3. A DAPTIVE H OMOGENEITY P ROPAGATION

Original: After that, we use the base predictor to embed local subgraph nodes into the global knowledge space  and improve the accuracy with  local homogeneous structure.
Revised: After that, we use the base predictor to embed local subgraph nodes into the global knowledge space X global and improve the accuracy with the local homogeneous structure.

Original: The motivation behind it is that the feature propagation satisfying homogeneity has a significant positive impact on prediction performance, which has also been confirmed in many recent research works Zhang et al.
Revised: The motivation behind it is that the feature propagation satisfying homogeneity has a significant positive impact on prediction performance, which has also been confirmed in many recent research works Zhang et al.

Original: (2022a)
Revised: (2022a)

Original: ; Wang & Leskovec (2020).
Revised: ; Wang & Leskovec (2020).

Original: Hence we expect to utilize local smoothing features to correct the predictions.
Revised: Hence we expect to utilize local smoothing features to correct the predictions.

Original: Then, we first define the homogeneous feature propagation
Revised: Then, we first define the homogeneous feature propagation

Original: [Equation]
Revised: [Equation]

Original: where graph − operator ( · ) represents the graph operator in feature propagation, we default to use symmetric normalized adjacency as shown in Eq. 1.
Revised: where graph − operator ( · ) represents the graph operator in feature propagation, we default to use symmetric normalized adjacency as shown in Eq. 1.

Original: X ( k ) smooth represents the local smoothing features after K -steps propagation, message − updater ( · ) denotes the model training process, and we use f θ to represent the linear regression or MLP with parameters of θ .
Revised: X ( k ) smooth represents the local smoothing features after K -steps propagation, message − updater ( · ) denotes the model training process, and we use f θ to represent the linear regression or MLP with parameters of θ .

Original: In order to correct the global embedding and local information, we use the local message update mechanism and online distillation to achieve an effective combination of the local smooth structure prior and the global embeddings, which can be written as
Revised: In order to correct the global embedding and local information, we use the local message update mechanism and online distillation to achieve an effective combination of the local smooth structure prior and the global embeddings, which can be written as

Original: [Equation]
Revised: [Equation]

Original: Based on this, we can make local smoothing information and global embeddings to achieve mutual supervision and end-to-end training by gradient updating.
Revised: Based on this, we can make local smoothing information and global embeddings to achieve mutual supervision and end-to-end training by gradient updating.

Original: This exploits the local structure information to reduce the error bound.
Revised: This exploits the local structure information to reduce the error bound.

Original: Notably, the above adaptive process is accomplished in the local client and has no additional communication costs and privacy concerns.
Revised: Notably, the above adaptive process is accomplished in the local client and has no additional communication costs and privacy concerns.

Original: 3. A DAPTIVE H ETEROGENEITY P ROPAGATION
Revised: 3. A DAPTIVE H ETEROGENEITY P ROPAGATION

Original: In contrast, in order to break the heterogeneous structure limitations, we optimize the messagepassing framework byglobal embeddings  to detect subgraph heterogeneous patterns.
Revised: In contrast, in order to break the heterogeneous structure limitations, we optimize the messagepassing framework by embeddings X global to detect subgraph heterogeneous patterns.

Original: Specifically, we propose an adaptive propagation mechanism by discovering the global dependency of the current node and modeling the positive or negative impact of the messages.
Revised: Specifically, we propose an adaptive propagation mechanism by discovering the global dependency of the current node and modeling the positive or negative impact of the messages.

Original: Intuitively, we first expect to optimize the propagation probability matrix and align the local structure by global embeddings
Revised: Intuitively, we first expect to optimize the propagation probability matrix and align the local structure by global embeddings

Original: [Equation]
Revised: [Equation]

Original: Obviously, the original propagation probability matrix introduces high error, we improve it by scaling the aggregated messages and making it trainable.
Revised: Obviously, the original propagation probability matrix introduces high error, we improve it by scaling the aggregated messages and making it trainable.

Original: Formally, let p
Revised: Formally, let p

Original: ij ∈
Revised: ij ∈

Original: A prop correspond to the i -th row and j -th col of A prop , we define the scaling operator d ij = dis ( P ii , P ij ) for j ̸
Revised: A prop correspond to the i -th row and j -th col of A prop , we define the scaling operator d ij = dis ( P ii , P ij ) for j ̸

Original: = i , where dis ( · ) is a distance function or a function positively relative with the difference, which can be implemented using identity distance.
Revised: = i , where dis ( · ) is a distance function or a function positively relative with the difference, which can be implemented using identity distance.

Original: Thus the corrected propagation matrix is expressed as
Revised: Thus the corrected propagation matrix is expressed as

Original: [Equation]
Revised: [Equation]

Original: The purpose of it is to measure the global dependency of the current node through the probability difference.
Revised: The purpose of it is to measure the global dependency of the current node through the probability difference.

Original: Then, we further model the positive and negative impacts of the messages to implement effective aggregation, which is formally represented as follows
Revised: Then, we further model the positive and negative impacts of the messages to implement effective aggregation, which is formally represented as follows

Original: [Equation]
Revised: [Equation]

Original: where H (0) =
Revised: where H (0) =

Original: X align , PoSign ( · ) and NeSign ( · ) represent the trainable adaptive propagation probabilities, it can be replaced by any reasonable nonlinear activation function.
Revised: X align , PoSign ( · ) and NeSign ( · ) represent the trainable adaptive propagation probabilities, it can be replaced by any reasonable nonlinear activation function.

Original: Here we analyze the error bound for the above adaptive heterogeneous propagation mechanism.
Revised: Here we analyze the error bound for the above adaptive heterogeneous propagation mechanism.

Original: The proof of the following theorem and reasonable assumptions are given in Appendix.
Revised: The proof of the following theorem and reasonable assumptions are given in Appendix.

Original: A.
Revised: A.

Original: Theorem 3.1 Suppose that the latent ground-truth mapping Φ : x → y from node features to node labels is differentiable and satisfies L -Lipschitz constraint, the following approximation error is
Revised: Theorem 3.1 Suppose that the latent ground-truth mapping Φ : x → y from node features to node labels is differentiable and satisfies L -Lipschitz constraint, the following approximation error is

Original: [Equation]
Revised: [Equation]

Original: where ⋆ represents the global optimal, ϵ denotes immediate neighbors error, O ( · ) denotes a higher order infinitesimal, ϕ and κ represent propagation matrix and model differences, respectively.
Revised: where ⋆ represents the global optimal, ϵ denotes immediate neighbors error, O ( · ) denotes a higher order infinitesimal, ϕ and κ represent propagation matrix and model differences, respectively.

Original: The core of the above propagation mechanisms is to generate embeddings based on other nodes in the embedding space.
Revised: The core of the above propagation mechanisms is to generate embeddings based on other nodes in the embedding space.

Original: In other words, it means that any node representation can be mapped to a linear combination of existing node representations, which has been applied in many studies
Revised: In other words, it means that any node representation can be mapped to a linear combination of existing node representations, which has been applied in many studies

Original: Zheng et al.
Revised: Zheng et al.

Original: (2022a)
Revised: (2022a)

Original: ; Yang et al.
Revised: ; Yang et al.

Original: However, most of the methods use ranking mechanisms for representation and fail to consider modeling propagation processes, which has limitations.
Revised: However, most of the methods use ranking mechanisms for representation and fail to consider modeling propagation processes, which has limitations.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: In this section, we conduct experimental analysis on five benchmark datasets with community split and structure Non-IID settings to validate the effectiveness of AdaFGL.
Revised: In this section, we conduct experimental analysis on five benchmark datasets with community split and structure Non-IID settings to validate the effectiveness of AdaFGL.

Original: We aim to answer the following five questions.
Revised: We aim to answer the following five questions.

Original: Q1 :
Revised: Q1 :

Original: Compared with other state-of-the-art FGL baselines, can AdaFGL achieve better predictive accuracy in the community split setting?
Revised: Compared with other state-of-the-art FGL baselines, can AdaFGL achieve better predictive accuracy in the community split setting?

Original: Q2 :
Revised: Q2 :

Original: How does structure NonIID influence existing methods and can AdaFGL improve it?
Revised: How does structure NonIID influence existing methods and can AdaFGL improve it?

Original: Q3 :
Revised: Q3 :

Original: Are knowledge distillation and message detection working in adaptive propagation mechanisms?
Revised: Are knowledge distillation and message detection working in adaptive propagation mechanisms?

Original: Q4 : Why AdaFGL can achieve desirable predictions utilizing the interactions between decoupled modules?
Revised: Q4 : Why AdaFGL can achieve desirable predictions utilizing the interactions between decoupled modules?

Original: Q5 :
Revised: Q5 :

Original: Compared with existing FGL methods and heterogeneous GNNs, what are the advantages of AdaFGL?
Revised: Compared with existing FGL methods and heterogeneous GNNs, what are the advantages of AdaFGL?

Original: [Table] [Table]
Revised: [Table] [Table]

Original: 4. E XPERIMENTAL S ETUP AND B ASELINES
Revised: 4. E XPERIMENTAL S ETUP AND B ASELINES

Original: Existing FGL methods implement data partitioning by community split Wang et al.
Revised: Existing FGL methods implement data partitioning by community split Wang et al.

Original: (2022b)
Revised: (2022b)

Original: based on the cluster homogeneity assumptions.
Revised: based on the cluster homogeneity assumptions.

Original: We follow it while proposing a more convincing strategy structure Non-IID.
Revised: We follow it while proposing a more convincing strategy structure Non-IID.

Original: Due to space limitations, the data partitioning results and the implementation details of the structure Non-IID can be found in Appendix.
Revised: Due to space limitations, the data partitioning results and the implementation details of the structure Non-IID can be found in Appendix.

Original: A.3 and Appendix.
Revised: A.3 and Appendix.

Original: A.6.
Revised: A.6.

Original: To demonstrate the effectiveness of AdaFGL, we combine powerful GNNs with FedAvg as the representative methods such as FedMLP, FedSGC, FedNLGNN, and FedGGCN.
Revised: To demonstrate the effectiveness of AdaFGL, we combine powerful GNNs with FedAvg as the representative methods.

Original: Meanwhile, we compare the recently proposed FGL methods such as FedGL, FedSage+, and GCFL+.
Revised: Meanwhile, we compare the recently proposed FGL methods such as FedGL, FedSage+, and GCFL+.

Original: FedSGC efficiently exploits the local structure prior by performing feature propagation multiple times.
Revised: FedSGC efficiently exploits the local structure prior by performing feature propagation.

Original: FedNLGNN implements node embeddings by MLP or GCN and then utilizes the ranking mechanism to discover potential neighbors.
Revised: FedNLGNN implements node embeddings by MLP or GCN and then utilizes the ranking mechanism to discover potential neighbors.

Original: FedGGCN further exploits the relationship between over-smoothing and heterogeneity to achievenode degree weighted propagation.
Revised: FedGGCN further exploits the relationship between oversmoothing and heterogeneity to achieve weighted propagation.

Original: FedGL aims to optimize the local model performance using globalsupervised information, it is essentially graph structure learning in the subgraph partitioning setting without overlapping nodes.
Revised: FedGL aims to optimize the local model performance using global information, it is essentially graph structure learning in the subgraph partitioning setting without overlapping nodes.

Original: FedSage+ performs local graph augmentation to improve prediction performance by sharing global node embeddings and topological relationships of missing subgraphs.
Revised: FedSage+ performs local graph augmentation to improve prediction performance by sharing global node embeddings and topological relationships of missing subgraphs.

Original: GCFL+ implements the clustering process to perform the personalized update mechanisms.
Revised: GCFL+ implements the clustering process to perform the personalized update mechanisms.

Original: More details about baseline methods can be referred to Appendix.
Revised: More details about baseline methods can be referred to Appendix.

Original: A.2.
Revised: A.2.

Original: 4. O VERALL P ERFORMANCE
Revised: 4. O VERALL P ERFORMANCE

Original: We first present the complete results on Cora and Chameleon in Table.
Revised: We first present the complete results on Cora and Chameleon in Table.

Original: 4 and Table.
Revised: 4 and Table.

Original: 3, which are two representative homogeneous and heterogeneous datasets.
Revised: 3, which are two representative homogeneous and heterogeneous datasets.

Original: Due to the space limitation, the details about the experiment environment and results in other datasets can be found in AppendixA.6.
Revised: Due to the space limitation, the details about the experiment environment and results in other datasets can be found in AppendixA.6.

Original: Notably, since we randomly inject homogeneous or heterogeneous information into the structure
Revised: Notably, since we randomly inject homogeneous or heterogeneous information into the structure

Original: Non-IID data partitioning process, the model performance does not directly relate to the number of clients.
Revised: Non-IID data partitioning process, the model performance does not directly relate to the number of clients.

Original: Meanwhile, in the community split setting, the process of model aggregation by multiple clients to achieve federated learning can be considered as ensemble learning.
Revised: Meanwhile, in the community split setting, the process of model aggregation by multiple clients to achieve federated learning can be considered as ensemble learning.

Original: Therefore, the prediction performance gets better with the increasing number of clients in some cases.
Revised: Therefore, the prediction performance gets better with the increasing number of clients in some cases.

Original: [Figure]
Revised: [Figure]

Original: Squirrel
Revised: Squirrel

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Revised: 16 Nodes

Original: Federated Extractor   
Revised: Federated Extractor

Original: 16 Nodes
Revised: 16 Nodes

Original: Adapative Propagation
Revised: Adapative Propagation

Original: To answer Q1 , Table.
Revised: To answer Q1 , Table.

Original: 3 shows the comparison results with the baseline methods in community split setting.
Revised: 3 shows the comparison results with the baseline methods in community split setting.

Original: For the homogeneous dataset Cora, compared with the most competitive FGL methods,
Revised: For the homogeneous dataset Cora, compared with the most competitive FGL methods,

Original: AdaFGL achieved accuracy gains of 1 .
Revised: AdaFGL achieved accuracy gains of 1 .

Original: 18% , 2 . 25% , and 1 .
Revised: 18% , 2 . 25% , and 1 .

Original: 64% in multiple client settings, respectively.
Revised: 64% in multiple client settings, respectively.

Original: Meanwhile, AdaFGL exceeds the best methods among all considered baselines on the heterogeneous datasets by a margin of 6 . 37% to 8 .
Revised: Meanwhile, AdaFGL exceeds the best methods among all considered baselines on the heterogeneous datasets by a margin of 6 . 37% to 8 .

Original: In the community split setting, we improve the prediction accuracy by utilizing the local smoothing prior and adaptive propagation mechanisms.
Revised: In the community split setting, we improve the prediction accuracy by utilizing the local smoothing prior and adaptive propagation mechanisms.

Original: To answer Q2 , We demonstrate the performance of existing methods in the face of structure Non-IID challenges in Table.
Revised: To answer Q2 , We demonstrate the performance of existing methods in the face of structure Non-IID challenges in Table.

Original: Although FedGGCN performs well in general, it cannot obtain competitive performance.
Revised: Although FedGGCN performs well in general, it cannot obtain competitive performance.

Original: Despite FedSage+ achieving effective local graph augmentation by sharing global data, structure Non-IID is a natural challenge, and this weakness is amplified when heterogeneity is high.
Revised: Despite FedSage+ achieving effective local graph augmentation by sharing global data, structure Non-IID is a natural challenge, and this weakness is amplified when heterogeneity is high.

Original: In addition, FedSage+ needs more training nodes to have better prediction performance, and this is the reason why it cannot achieve its original experimental results.

Original: In contrast, our method achieves performance gains of 1 . 45% , 3 .
Revised: In contrast, our method achieves performance gains of 1 . 45% , 3 .

Original: 77% , and 1 . 87% compared to the highest prediction accuracy.
Revised: 77% , and 1 . 87% compared to the highest prediction accuracy.

Original: Impressively, AdaFGL improves performance by 9 .
Revised: Impressively, AdaFGL improves performance by 9 .

Original: 82% , 13 . 06% , and 13 .
Revised: 82% , 13 . 06% , and 13 .

Original: 29% in the structure Non-IID setting for the heterogeneous dataset Chameleon.
Revised: 29% in the structure Non-IID setting for the heterogeneous dataset Chameleon.

Original: from the observation of the comparison results with the baselines, our method has significant advantages, especially in terms of robustness and impressive performance when dealing with the structure Non-IID problem.
Revised: From the observation of the comparison results with the baselines, our method has significant advantages, especially in terms of robustness and impressive performance.

Original: 4. A BLATION E XPERIMENTS
Revised: 4. A BLATION E XPERIMENTS

Original: To answer Q3 , we present the ablation experiment results in Table.
Revised: To answer Q3 , we present the ablation experiment results in Table.

Original: 3 and Table.
Revised: 3 and Table.

Original: 4, where HomoKD represents the online distillation in the homogeneous propagation module and HeteTA represents the trainable probability propagation matrix in the heterogeneous propagation module.
Revised: 4, where HomoKD represents the online distillation in the homogeneous propagation module and HeteTA represents the trainable probability propagation matrix in the heterogeneous propagation module.

Original: We observe that the online distillation enhances Homogeneous propagation by combining local smoothing features and local embeddings, it can effectively improve model performance without adding additional computation costs.
Revised: We observe that the online distillation enhances Homogeneous propagation by combining local smoothing features and local embeddings, it can effectively improve model performance without adding additional computation costs.

Original: In essence, it achieves mutually supervised end-to-end learning of global and local information.
Revised: In essence, it achieves mutually supervised end-to-end learning of global and local information.

Original: Furthermore, the trainable probability propagation matrix optimizes the heterogeneity propagation module.
Revised: Furthermore, the trainable probability propagation matrix optimizes the heterogeneity propagation module.

Original: It learns the global optimal propagation mechanism and detects positive and negative messages to generate embeddings.
Revised: It learns the global optimal propagation mechanism and detects positive and negative messages to generate embeddings.

Original: HeteTA can discover the global dependence of the current node and achieve effective message aggregation, which is proved by Theorem. 3.1.
Revised: HeteTA can discover the global dependence of the current node and achieve effective message aggregation, which is proved by Theorem. 3.1.

Original: 4. V ISUALIZATION AND E XPLAINABILITY A NALYSIS
Revised: 4. V ISUALIZATION AND E XPLAINABILITY A NALYSIS

Original: To answer Q4 , we present the local prediction accuracy trends with the competitive baseline methods in Fig. 4.
Revised: To answer Q4 , we present the local prediction accuracy trends with the competitive baseline methods in Fig. 4.

Original: According to it, we can notice that our method achieves the best performance in most 
Revised: According to it, we can notice that our method achieves the best performance in most cases under both community split and structure Non-IID data settings, while the overall trend is

Original: [Table] [Table]
Revised: [Table] [Table]

Original: cases under both community split and structure Non-IID data settings, while the overall trend is optimized.Meanwhile, we minimize the high variance of homogeneous and heterogeneous subgraphs, which is a significant improvement compared with the other methods.
Revised: optimized.

Original: Meanwhile, we perform hyperparameter sensitivity analysis experiments on AdaFGL. Due to space limitations, the relevant experimental results  and conclusions can be found in Appendix.
Revised: Due to space limitations, the relevant experimental results about the hyperparameter sensitivity analysis experiments on AdaFGL and conclusions can be found in Appendix.

Original: A.5.
Revised: A.

Revised: [Table]

Original: In order to illustrate the effectiveness of the federated global knowledge extractor and the adaptive propagation mechanisms
Revised: In order to illustrate the effectiveness of the federated global knowledge extractor and the adaptive propagation mechanisms

Original: [Table]
Revised: [Table]

Original: we also analyze the explainability by presenting the heat maps shown in 
Revised: we also analyze the explainability by presenting the heat maps shown in Fig.

Original: Fig.
Revised: We perform Structure Non-IID partitioning for 10 clients on PubMed, then select the client with the highest number of nodes with homogeneity and heterogeneity. Based on this, we randomly sampled 20 nodes to obtain the similarity score by computing the embedding transpose.

Original: From the observation of the results
Revised: From the observation of the results

Original: [Table]
Revised: [Table]

Original: we notice that the federated global knowledge extractor only obtains fuzzy results and cannot be optimized for  local subgraph.
Revised: we notice that the federated global knowledge extractor only obtains fuzzy results and cannot be optimized for the local subgraphs.

Original: Fortunately
Revised: Fortunately

Original: [Table]
Revised: [Table]

Original: we achieve an effective combination of global knowledge and local subgraph structure prior to obtain explicit node embeddings, which is also demonstrated through the final output in Fig.[Table]
Revised: we achieve an effective combination of global knowledge and local subgraph structure prior to obtaining explicit node embeddings, which is also demonstrated through the final output in Fig.

Original: 4. M ETHODS C OMPARISON
Revised: 4. M ETHODS C OMPARISON

Original: To answer Q
Revised: To answer Q

Original: [Table]
Revised: [Table]

Original: we review three recent FGL methods and analyze our approach to them in terms of three aspects: method type, exchange messages, and the ability to solve structure Non-IID problems as shown in Table.5.
Revised: we review three recent FGL methods and analyze our approach to them in terms of three aspects: method type, exchange messages, and the ability to solve structure Non-IID problems as shown in Table.5.

Original: Obviously, although FedSage+ can achieve competitive results, it introduces significant communication costs and privacy concerns.
Revised: Obviously, although FedSage+ can achieve competitive results, it introduces significant communication costs and privacy concerns.

Original: GCFL+ has limitations in model selection leads to its failure to handle the structure Non-IID problem in subgraph learning.
Revised: GCFL+ has limitations in model selection leads to its failure to handle the structure Non-IID problem in subgraph learning.

Original: In our experiments,
Revised: In our experiments,

Original: FedGL is essentially a local graph structure learning process.
Revised: FedGL is essentially a local graph structure learning process.

Original: In contrast, our approach can utilize the computational capabilities of the local system while minimizing communication costs and privacy concerns.
Revised: In contrast, our approach can utilize the computational capabilities of the local system while minimizing communication costs and privacy concerns.

Original: More experimental details can be found in Appendix.
Revised: More experimental details can be found in Appendix.

Original: A.4.
Revised: A.4.

Original: Then, we compare the effectiveness of existing GNNs and our approach to handling heterogeneous graph, which focuses on two points: Neighbor Discovery and Message Combination, which is shown in Fig.
Revised: Then, we compare the effectiveness of existing GNNs and our approach to handling heterogeneous graph, which focuses on two points: Neighbor Discovery and Message Combination, which is shown in Fig.

Original: We observe that MLP ignores graph structure prior which leads to the failure to handle heterogeneous graphs.
Revised: We observe that MLP ignores graph structure prior which leads to the failure to handle heterogeneous graphs.

Original: Although FedGL and FedSage+ can improve this problem by utilizing global information for local graph augmentation, the limitations of propagation lead to the fact that they are still not the best solutions.
Revised: Although FedGL and FedSage+ can improve this problem by utilizing global information for local graph augmentation, the limitations of propagation lead to the fact that they are still not the best solutions.

Original: Therefore, we are motivated by these methods and propose adaptive propagation mechanisms to improve the performance, which has been validated to be effective.
Revised: Therefore, we are motivated by these methods and propose adaptive propagation mechanisms to improve the performance, which has been validated to be effective.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we discover and define the structure Non-IID problem in FGL, which is a new challenge for existing methods.
Revised: In this paper, we discover and define the structure Non-IID problem in FGL, which is a new challenge for existing methods.

Original: Based on this, we propose a new paradigm AdaFGL for more general federated data settings.
Revised: Based on this, we propose a new paradigm AdaFGL for more general federated data settings.

Original: Specifically, we investigate the structure Non-IID problem in FGL for supplementing the existing community split data partitioning approach, which is a more practical federated data setting.
Revised: Specifically, we investigate the structure Non-IID problem in FGL for supplementing the existing community split data partitioning approach, which is a more practical federated data setting.

Original: To implement effective FGL on heterogeneous distributed subgraphs, we propose
Revised: To implement effective FGL on heterogeneous distributed subgraphs, we propose

Original: AdaFGL which consists of the federated global knowledge extractor and adaptive propagation modules.
Revised: AdaFGL which consists of the federated global knowledge extractor and adaptive propagation modules.

Original: It combines FL and GNNs tightly and benefits from their evolution.
Revised: It combines FL and GNNs tightly and benefits from their evolution.

Original: Extensive experiments based on the community split and structure Non-IID data settings demonstrate the effectiveness of
Revised: Extensive experiments based on the community split and structure Non-IID data settings demonstrate the effectiveness of

Original: AdaGFL.
Revised: AdaGFL.

Original: We believe that the ability to fully utilize the graph structure information is the key to achieving efficient FGL, thus the research on graph structure in FGL is a promising direction.
Revised: We believe that the ability to fully utilize the graph structure information is the key to achieving efficient FGL, thus the research on graph structure in FGL is a promising direction.

Original: A A PPENDIX O UTLINE
Revised: A A PPENDIX O UTLINE

Original: The appendix is organized as follows:
Revised: The appendix is organized as follows:

Original: A.1 Theory error bounds for adaptive heterogeneous propagation modules.
Revised: A.1 Theory error bounds for adaptive heterogeneous propagation modules.

Original: A.2 More details about the compared baselines.
Revised: A.2 More details about the compared baselines.

Original: A.3 Datasets description and structure Non-IID data setting.
Revised: A.3 Datasets description and structure Non-IID data setting.

Original: A.4 Communication costs analysis.
Revised: A.4 Communication costs analysis.

Original: A.5 Hyperparameter sensitivity analysis.
Revised: A.5 Hyperparameter sensitivity analysis.

Original: A.6 Experimental environment and additional base results.
Revised: A.6 Experimental environment and additional base results.

Original: A. T HEORY ERROR BOUNDS FOR ADAPTIVE HETEROGENEOUS PROPAGATION
Revised: A. T HEORY ERROR BOUNDS FOR ADAPTIVE HETEROGENEOUS PROPAGATION

Original: To demonstrate the effectiveness of the adaptive heterogeneous propagation module, we prove its error bound.
Revised: To demonstrate the effectiveness of the adaptive heterogeneous propagation module, we prove its error bound.

Original: We first make the reasonable following assumption and definitions.
Revised: We first make the reasonable following assumption and definitions.

Original: Assumption A.1 Φ is L -smooth, ∀ x
Revised: Assumption A.1 Φ is L -smooth, ∀ x

Original: [Equation]
Revised: [Equation]

Original: , x
Revised: , x

Original: [Equation]
Revised: [Equation]

Original: (Φ
Revised: (Φ

Original: [Equation]
Revised: [Equation]

Original: Then we quote the embedding method theorem Linial et al.
Revised: Then we quote the embedding method theorem Linial et al.

Original: Definition A.1 Given two metric spaces ( V , d ) and ( Z, d ′ ) and mapping function Φ : V → Z , the distortion ϵ distor is definied as ∀ u, v ∈ V , 1 /ϵ distor d ( u, v ) ≤ d ′ (Φ( u ) , Φ( v )) ≤ d ( u, v ) .
Revised: Definition A.1 Given two metric spaces ( V , d ) and ( Z, d ′ ) and mapping function Φ : V → Z , the distortion ϵ distor is definied as ∀ u, v ∈ V , 1 /ϵ distor d ( u, v ) ≤ d ′ (Φ( u ) , Φ( v )) ≤ d ( u, v ) .

Original: Theorem A.1 (Bourgain theorem)
Revised: Theorem A.1 (Bourgain theorem)

Original: Given any finite metric space ( V , d ) with V = n , there exists an embedding of ( V , d ) into R k under any l p metric, where k
Revised: Given any finite metric space ( V , d ) with V = n , there exists an embedding of ( V , d ) into R k under any l p metric, where k

Original: = O (log 2 n ) , and the distortion of the embedding is O (log n ) .
Revised: = O (log 2 n ) , and the distortion of the embedding is O (log n ) .

Original: It defines the distortion O (log n ) in the embedding space ( V , d ) for mapping methods satisfying the above conditions.
Revised: It defines the distortion O (log n ) in the embedding space ( V , d ) for mapping methods satisfying the above conditions.

Original: Based on this, we consider a graph G with fixed structure represented by ˜
Revised: Based on this, we consider a graph G with fixed structure represented by ˜

Original: A
Revised: A

Original: =
Revised: =

Original: ˆ D − 1 / 2 ˆ
Revised: ˆ D − 1 / 2 ˆ

Original: A ˆ D − 1 / 2 , embeddings represented with H in the forward propagation, and nodes mapping function Φ( H ) , which satisfies the Theorem.
Revised: A ˆ D − 1 / 2 , embeddings represented with H in the forward propagation, and nodes mapping function Φ( H ) , which satisfies the Theorem.

Original: A.1, it can be expressed as
Revised: A.1, it can be expressed as

Original: [Equation]
Revised: [Equation]

Original: ) k , d ( H , S 1 , 2 ) k , . . .
Revised: ) k , d ( H , S 1 , 2 ) k , . . .

Original: , d ( H , S log n,c log n ) k (cid:19) , where d ( H , S i,j ) = min u ∈ S i,j d ( H , u ) .
Revised: , d ( H , S log n,c log n ) k (cid:19) , where d ( H , S i,j ) = min u ∈ S i,j d ( H , u ) .

Original: S i,j ⊂ V , i = 1 , 2 , . . .
Revised: S i,j ⊂ V , i = 1 , 2 , . . .

Original: , log n, j = 1 , 2 , . .
Revised: , log n, j = 1 , 2 , . .

Original: . , c log n represents c log 2 n random sets, where c is a constant.
Revised: . , c log n represents c log 2 n random sets, where c is a constant.

Original: It is chosen by including each point in V independently with probability 1 / 2 i .
Revised: It is chosen by including each point in V independently with probability 1 / 2 i .

Original: Then motivated by Xie et al.
Revised: Then motivated by Xie et al.

Original: (2021) and the above conclusions, we have the following model weights difference proposition.
Revised: (2021) and the above conclusions, we have the following model weights difference proposition.

Original: Proposition A.1 Assume the propagation probability matrix, hidden embeddings, and label difference with global optima f ⋆θ and local model f θ are bounded with
Revised: Proposition A.1 Assume the propagation probability matrix, hidden embeddings, and label difference with global optima f ⋆θ and local model f θ are bounded with

Original: [Equation]
Revised: [Equation]

Original: Based on this, given that ∥ H · H ⋆ ∥ 22 = ∥
Revised: Based on this, given that ∥ H · H ⋆ ∥ 22 = ∥

Original: H · ( H + E H )
Revised: H · ( H + E H )

Original: ∥ 22 ≥ ∥ H E H ∥ 22 .
Revised: ∥ 22 ≥ ∥ H E H ∥ 22 .

Original: Let ∥ X E H ∥
Revised: Let ∥ X E H ∥

Original: 22 = δ H , then we have (cid:13)(cid:13)
Revised: 22 = δ H , then we have (cid:13)(cid:13)

Original: H ⋆ − 1 − H − 1 (cid:13)(cid:13) 2 

Revised: H ⋆ − 1 − H − 1 (cid:13)(cid:13) 2

Original: =
Revised: =

Original: ∥
Revised: ∥

Original: E H − 1 ∥ ≤ ϵ H /δ H .
Revised: E H − 1 ∥ ≤ ϵ H /δ H .

Original: If we choose SGC Wu et al.
Revised: If we choose SGC Wu et al.

Original: for the forward propagation, the model weights difference with the influence of feature difference is represented as
Revised: for the forward propagation, the model weights difference with the influence of feature difference is represented as

Original: [Equation]
Revised: [Equation]

Original: Similarly
Revised: Similarly

Original: [Equation]
Revised: [Equation]

Original: there exists ∥ P · P ⋆ ∥ 22 = ∥
Revised: there exists ∥ P · P ⋆ ∥ 22 = ∥

Original: P ·
Revised: P ·

Original: [Equation]
Revised: [Equation]

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: E P
Revised: E P

Original: [Equation]
Revised: [Equation]

Original: ∥ 22 ≥ ∥ P E P ∥ 22 , ∥ P E P ∥ 22 = δ P
Revised: ∥ 22 ≥ ∥ P E P ∥ 22 , ∥ P E P ∥ 22 = δ P

Original: [Equation]
Revised: [Equation]

Original: and (cid:13)(cid:13) P ⋆
Revised: and (cid:13)(cid:13) P ⋆

Original: [Equation]
Revised: [Equation]

Original: we can obtain the model weight differences with the influence of structure difference.
Revised: we can obtain the model weight differences with the influence of structure difference.

Original: [Equation]
Revised: [Equation]

Original: Proof A.1 Here, based on the Eq.
Revised: Proof A.1 Here, based on the Eq.

Original: [Equation]
Revised: [Equation]

Original: we consider the adaptive heterogeneous propagation process
Revised: we consider the adaptive heterogeneous propagation process

Original: [Equation]
Revised: [Equation]

Original: Take node i as an example, given that Φ( · ) is differentiable, where contains the gradient update of the model difference ϕ .
Revised: Take node i as an example, given that Φ( · ) is differentiable, where contains the gradient update of the model difference ϕ .

Original: Meanwhile, in order to quantify the difference between our trainable propagation probability matrix and the global optimum, we define
Revised: Meanwhile, in order to quantify the difference between our trainable propagation probability matrix and the global optimum, we define

Original: [Equation]
Revised: [Equation]

Original: where P ⋆ represents the optimal propagation probability matrix.
Revised: where P ⋆ represents the optimal propagation probability matrix.

Original: Then, we use Pos , Neg to denote the positive and negative message propagation weights P =
Revised: Then, we use Pos , Neg to denote the positive and negative message propagation weights P =

Original: ˆ A ( l ) prop , there exist
Revised: ˆ A ( l ) prop , there exist

Original: [Equation]
Revised: [Equation]

Original: where Pos ( l ) ij + Neg ( l ) ij = ˆ A ( l ) prop
Revised: where Pos ( l ) ij + Neg ( l ) ij = ˆ A ( l ) prop

Original: [ i :] .
Revised: [ i :] .

Original: Then, we perform a first-order Taylor expansion with Peano’s form of remainder at H ( l ) i and consider the model differences (cid:88)
Revised: Then, we perform a first-order Taylor expansion with Peano’s form of remainder at H ( l ) i and consider the model differences (cid:88)

Original: j ̸
Revised: j ̸

Original: = i
Revised: = i

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: Φ( H ( l ) ) =
Revised: Φ( H ( l ) ) =

Original: (cid:88)
Revised: (cid:88)

Original: [Equation]
Revised: [Equation]

Original: ̸
Revised: ̸

Original: = i
Revised: = i

Original: [Equation]
Revised: [Equation]

Original: l
Revised: l

Original: [Equation]
Revised: [Equation]

Original: l )
Revised: l )

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ( H
Revised: ( H

Original: [Equation]
Revised: [Equation]

Original: l )
Revised: l )

Original: [Equation]
Revised: [Equation]

Original: H ( l )
Revised: H ( l )

Original: [Equation]
Revised: [Equation]

Original: − H ( l )
Revised: − H ( l )

Original: [Equation]
Revised: [Equation]

Original: ( |
Revised: ( |

Original: [Equation]
Revised: [Equation]

Original: l )
Revised: l )

Original: [Equation]
Revised: [Equation]

Original: ( l )
Revised: ( l )

Original: [Equation]
Revised: [Equation]

Original: ||
Revised: ||

Original: [Equation]
Revised: [Equation]

Original: ) (cid:33) =
Revised: ) (cid:33) =

Original: (cid:88)
Revised: (cid:88)

Original: j ̸
Revised: j ̸

Original: = i
Revised: = i

Original: [Equation]
Revised: [Equation]

Original: ( H ( l ) )
Revised: ( H ( l ) )

Original: + (cid:88)
Revised: + (cid:88)

Original: j ̸
Revised: j ̸

Original: = i
Revised: = i

Original: P
Revised: P

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ( l ) j ) ∂
Revised: ( l ) j ) ∂

Original: ( H ( l ) )
Revised: ( H ( l ) )

Original: [Equation]
Revised: [Equation]

Original: ( H ( l )
Revised: ( H ( l )

Original: [Equation]
Revised: [Equation]

Original: − H ( l ) i ) + (cid:88)
Revised: − H ( l ) i ) + (cid:88)

Original: j ̸
Revised: j ̸

Original: = i
Revised: = i

Original: P ⋆ij O
Revised: P ⋆ij O

Original: ( || H ( l ) j − H ( l ) i || 2 ) .
Revised: ( || H ( l ) j − H ( l ) i || 2 ) .

Original: Now, we let (cid:80)
Revised: Now, we let (cid:80)

Original: j ̸ = i P ⋆ij
Revised: j ̸ = i P ⋆ij

Original: ( H ( l ) j − H ( l ) i ) =
Revised: ( H ( l ) j − H ( l ) i ) =

Original: − ϵ i , there exist (cid:88)
Revised: − ϵ i , there exist (cid:88)

Original: j ̸
Revised: j ̸

Original: = i
Revised: = i

Original: P ⋆ij Φ( H ( l ) )
Revised: P ⋆ij Φ( H ( l ) )

Original: =   κ i + P ii + (cid:88)
Revised: =   κ i + P ii + (cid:88)

Original: j ̸ = i (Pos ( l ) ij + Neg ( l ) ij ) 
Revised: j ̸ = i (Pos ( l ) ij + Neg ( l ) ij ) 

Original:  ϕ H ( l ) , = (cid:88)
Revised:  ϕ H ( l ) , = (cid:88)

Original: [Equation]
Revised: [Equation]

Original: ̸
Revised: ̸

Original: = i
Revised: = i

Original: [Equation]
Revised: [Equation]

Original: l
Revised: l

Original: [Equation]
Revised: [Equation]

Original: l
Revised: l

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ( l
Revised: ( l

Original: [Equation]
Revised: [Equation]

Original: ϵ
Revised: ϵ

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: (cid:88)
Revised: (cid:88)

Original: j ̸
Revised: j ̸

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: l
Revised: l

Original: [Equation]
Revised: [Equation]

Original: l )
Revised: l )

Original: [Equation]
Revised: [Equation]

Original: ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)
Revised: ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)

Original: j ̸
Revised: j ̸

Original: = i
Revised: = i

Original: P ⋆ij Φ( H ( l
Revised: P ⋆ij Φ( H ( l

Original: [Equation]
Revised: [Equation]

Original: −
Revised: −

Original: [Equation]
Revised: [Equation]

Original: According to Cauchy-Schwarz inequality and L-Lipschitz property, we have
Revised: According to Cauchy-Schwarz inequality and L-Lipschitz property, we have

Original: [Equation]
Revised: [Equation]

Original: Therefore, the approximation of H ( l ) i + (cid:80)
Revised: Therefore, the approximation of H ( l ) i + (cid:80)

Original: j ̸
Revised: j ̸

Original: = i (Pos ( l ) ij + Neg ( l ) ij ) H ( l ) j is bounded by
Revised: = i (Pos ( l ) ij + Neg ( l ) ij ) H ( l ) j is bounded by

Original: [Equation]
Revised: [Equation]

Original: j ̸ = i
Revised: j ̸ = i

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: j ̸ = i
Revised: j ̸ = i

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: O (cid:16)(cid:13)(cid:13)(cid:13)
Revised: O (cid:16)(cid:13)(cid:13)(cid:13)

Original: [Equation]
Revised: [Equation]

Original: ( l ) i (cid:13)(cid:13)(cid:13) 

Revised: ( l ) i (cid:13)(cid:13)(cid:13)

Original: (cid:17) + (cid:16)(cid:13)(cid:13)(cid:13) H
Revised: (cid:17) + (cid:16)(cid:13)(cid:13)(cid:13) H

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: H ( l ) (cid:13)(cid:13)(cid:13) 2 (cid:17) , where H ⋆ represents the global optimal embeddings.
Revised: H ( l ) (cid:13)(cid:13)(cid:13) 2 (cid:17) , where H ⋆ represents the global optimal embeddings.

Original: Based on this, we obtain the theory error bound for heterogeneous propagation.
Revised: Based on this, we obtain the theory error bound for heterogeneous propagation.

Original: From the observation of error bounds, we reveal that in theory, the adaptive heterogeneous propagation process can minimize the immediate neighbors error
Revised: From the observation of error bounds, we reveal that in theory, the adaptive heterogeneous propagation process can minimize the immediate neighbors error

Original: ϵ i , the model difference ϕ , and the propagation probability matrix difference κ to scale the error to improve the predictive performance.
Revised: ϵ i , the model difference ϕ , and the propagation probability matrix difference κ to scale the error to improve the predictive performance.

Original: A. C OMPARED B ASELINES
Revised: A. C OMPARED B ASELINES

Original: The main characteristic of all baselines are listed below:
Revised: The main characteristic of all baselines are listed below:

Original: FedMLP: The combination of FedAvg and MLP, we employ a two-layer MLP with the hidden dimension of 64 .
Revised: FedMLP: The combination of FedAvg and MLP, we employ a two-layer MLP with the hidden dimension of 64 .

Original: It generates node embeddings based on the original features while ignoring graph structure information in the forward propagation process.
Revised: It generates node embeddings based on the original features while ignoring graph structure information in the forward propagation process.

Original: [Table]
Revised: [Table]

Original: FedSGC:
Revised: FedSGC:

Original: We combine FedAvg and SGC Wu et al.
Revised: We combine FedAvg and SGC Wu et al.

Original: (2019), we default to use the 3 -layer feature propagation process, which follows the homogeneous assumption and thus fails to deal with heterogeneous graph.
Revised: (2019), we default to use the 3 -layer feature propagation process, which follows the homogeneous assumption and thus fails to deal with heterogeneous graph.

Original: FedNLGNN: Implementation of NLGNN (NLMLP or NLGCN)
Revised: FedNLGNN: Implementation of NLGNN (NLMLP or NLGCN)

Original: Liu et al.
Revised: Liu et al.

Original: based on FedAvg, we select the more effective version to present the model performance.
Revised: based on FedAvg, we select the more effective version to present the model performance.

Original: It depends on the embedding model and suffers from representational limitations.
Revised: It depends on the embedding model and suffers from representational limitations.

Original: FedGGCN: The combination of FedAvg and GGCN Yan et al.
Revised: FedGGCN: The combination of FedAvg and GGCN Yan et al.

Original: (2021), we follow the experimental setup of the original paper as much as possible, which can handle heterogeneous graphs effectively, but cannot achieve competitive results on homogeneous graphs.
Revised: (2021), we follow the experimental setup of the original paper as much as possible, which can handle heterogeneous graphs effectively, but cannot achieve competitive results on homogeneous graphs.

Original: FedGL Chen et al.
Revised: FedGL Chen et al.

Original: As a FGL training framework, it strongly relies on the overlapping nodes assumption, which in our data setting is essentially local graph structure learning.
Revised: As a FGL training framework, it strongly relies on the overlapping nodes assumption, which in our data setting is essentially local graph structure learning.

Original: GCFL+ Xie et al.
Revised: GCFL+ Xie et al.

Original: (2021): due to the limitations of personalized techniques in model selection, they cannot fundamentally solve the structure Non-IID challenges.
Revised: (2021): due to the limitations of personalized techniques in model selection, they cannot fundamentally solve the structure Non-IID challenges.

Original: FedSage+ Zhang et al.
Revised: FedSage+ Zhang et al.

Original: It trains NeighGen to achieve local subgraph augmentation by sharing global missing subgraph feature and topology information for the most powerful results, but suffers from privacy leakage risk and additional computational costs.
Revised: It trains NeighGen to achieve local subgraph augmentation by sharing global missing subgraph feature and topology information for the most powerful results, but suffers from privacy leakage risk and additional computational costs.

Original: For fairness, we follow the experimental setup of the baseline methods paper as much as possible, and in other cases, we show the best prediction accuracy.
Revised: For fairness, we follow the experimental setup of the baseline methods paper as much as possible, and in other cases, we show the best prediction accuracy.

Original: In addition, the number of rounds for the above baseline methods is 50 , and the local epoch is 20 .
Revised: In addition, the number of rounds for the above baseline methods is 50 , and the local epoch is 20 .

Original: A. D ATASETS DESCRIPTION AND STRUCTURE N ON -IID DATA SETTING
Revised: A. D ATASETS DESCRIPTION AND STRUCTURE N ON -IID DATA SETTING

Original: The statistics of datasets are summarized in Table.
Revised: The statistics of datasets are summarized in Table.

Original: 7, which contains both homogeneity and heterogeneity.
Revised: 7, which contains both homogeneity and heterogeneity.

Original: In our experiments, we use five benchmark datasets containing homogeneity and heterogeneity, for which details are given below.
Revised: In our experiments, we use five benchmark datasets containing homogeneity and heterogeneity, for which details are given below.

Original: Cora , Citeseer , and Pubmed Yang et al.
Revised: Cora , Citeseer , and Pubmed Yang et al.

Original: (2016b) are three popular citation network datasets.
Revised: (2016b) are three popular citation network datasets.

Original: In these three networks, papers from different topics are considered as nodes, and the edges are citations among the papers.
Revised: In these three networks, papers from different topics are considered as nodes, and the edges are citations among the papers.

Original: The node attributes are binary word vectors, and class labels are the topics papers belong to.
Revised: The node attributes are binary word vectors, and class labels are the topics papers belong to.

Original: Chameleon and Squirrel are two web page datasets collected from Wikipedia Rozemberczki et al.
Revised: Chameleon and Squirrel are two web page datasets collected from Wikipedia Rozemberczki et al.

Original: (2021), where nodes are web pages on specific topics and edges are hyperlinks between them.
Revised: (2021), where nodes are web pages on specific topics and edges are hyperlinks between them.

Original: Based on this, we illustrate the structure Non-IID data partitioning process in detail.
Revised: Based on this, we illustrate the structure Non-IID data partitioning process in detail.

Original: The core of it is the Dirichlet process He et al.
Revised: The core of it is the Dirichlet process He et al.

Original: (2020), Its basic analysis is as follows.
Revised: (2020), Its basic analysis is as follows.

Original: The pdf of the Dirichlet distribution is defined as
Revised: The pdf of the Dirichlet distribution is defined as

Original: [Equation]
Revised: [Equation]

Original: where α i ∈ { α 1 , . .
Revised: where α i ∈ { α 1 , . .

Original: . , α k } > 0 is the dimensionless distribution parameter, the scale (or concentration)
Revised: . , α k } > 0 is the dimensionless distribution parameter, the scale (or concentration)

Original: ϑ
Revised: ϑ

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

Original: α
Revised: α

Original: i , the base measure ( α ⋆ 1 , . . .
Revised: i , the base measure ( α ⋆ 1 , . . .

Original: , α ⋆k ) , α ⋆i =
Revised: , α ⋆k ) , α ⋆i =

Original: α
Revised: α

Original: i /ϑ , and Γ( n ) =
Revised: i /ϑ , and Γ( n ) =

Original: ( n − 1)! .
Revised: ( n − 1)! .

Original: Dirichlet is a distribution over Multinomials, thus there is (cid:80) i p i
Revised: Dirichlet is a distribution over Multinomials, thus there is (cid:80) i p i

Original: = 1 , p i ≥ 0 , where p i represents the probability.
Revised: = 1 , p i ≥ 0 , where p i represents the probability.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: It determines the mean distribution [Table] and the scale affects the variance, then we obtain
Revised: It determines the mean distribution, and the scale affects the variance, then we obtain

Original: [Equation]
Revised: [Equation]

Original: which means that a Dirichlet with small scale ϑ favors extreme distributions, but this prior belief is very weak and is easily overwritten by data.
Revised: which means that a Dirichlet with small scale ϑ favors extreme distributions, but this prior belief is very weak and is easily overwritten by data.

Original: As ϑ → ∞ , the covariance → 0 , the samples → base measure.
Revised: As ϑ → ∞ , the covariance → 0 , the samples → base measure.

Original: Based on this, we start sampling the edges to determine the attribution of a pair of nodes.
Revised: Based on this, we start sampling the edges to determine the attribution of a pair of nodes.

Original: If a conflicting set of nodes exists it is sampled again and finally generates induced subgraphs.
Revised: If a conflicting set of nodes exists it is sampled again and finally generates induced subgraphs.

Original: Then, we randomly inject homogeneous or heterogeneous information based on the label prior, which can solve unreal structure loss and enhance structure identity.
Revised: Then, we randomly inject homogeneous or heterogeneous information based on the label prior, which can solve unreal structure loss and enhance structure identity.

Revised: We propose to set three probabilities p iso , p homo , and p hete for each client individually to represent the probability of avoiding isolated nodes, increasing homogeneous edges, and increasing heterogeneous edges in the subgraph, respectively.

Revised: Specifically, p iso represents the probability of isolated nodes generating edges with other nodes, which can effectively be used to prevent the generation of isolated nodes.

Revised: p homo applies to the subgraph of clients that are selected to enhance homogeneity, and it represents the probability of connection between two nodes with the same label based on the label prior information.

Revised: Correspondingly, p hete represents the probability used to perform the structure information injection for the client subgraph that performs the heterogeneity enhancement.

Original: A. C OMMUNICATION COSTS ANALYSIS
Revised: A. C OMMUNICATION COSTS ANALYSIS

Original: The advantage of our approach is to exploit the local structure prior while making full use of the global information, which considers the characteristics of GNNs.
Revised: The advantage of our approach is to exploit the local structure prior while making full use of the global information, which considers the characteristics of GNNs.

Original: It has the benefit of reducing the communication costs and privacy concerns during the federated training process.
Revised: It has the benefit of reducing the communication costs and privacy concerns during the federated training process.

Original: Meanwhile, thanks to the utilization of the local structure information, we can obtain models with better representational power to improve the performance.
Revised: Meanwhile, thanks to the utilization of the local structure information, we can obtain models with better representational power to improve the performance.

Original: To demonstrate the effectiveness of our method, we provide the experimental results of AdaFGL with the two most competitive methods, FedSage and FedGGCN, as shown in Tables.
Revised: To demonstrate the effectiveness of our method, we provide the experimental results of AdaFGL with the two most competitive methods, FedSage and FedGGCN, as shown in Tables.

Original: A.4 and Table.
Revised: A.4 and Table.

Original: A.4.
Revised: A.4.

Original: According to the experimental results, we observe that AdaFGL maintains the low communication costs and achieves a satisfying result, which mainly benefits from the utilization of local structure information by the adaptive propagation modules.
Revised: According to the experimental results, we observe that AdaFGL maintains the low communication costs and achieves a satisfying result, which mainly benefits from the utilization of local structure information by the adaptive propagation modules.

Original: Compared to FedSage, which is the current most competitive FGL approach, suffers from the performance improvement and communication costs dilemma, which also brings more privacy concerns.
Revised: Compared to FedSage, which is the current most competitive FGL approach, suffers from the performance improvement and communication costs dilemma, which also brings more privacy concerns.

Original: A. H YPERPARAMETER SENSITIVITY ANALYSIS
Revised: P r o p a g a t i o n S m oo t h

Original: Here we conduct the hyperparameter sensitivity of AdaFGL, and the experimental results are shown in Fig.
Revised: [Figure] [Figure] K D L o ss R a t i o 0. Cora

Original: In our experiments, we analyze the ratio of online distillation loss in the homogeneous propagation module and the smoothing coefficient of the trainable propagation matrix in the heterogeneous propagation module.
Revised: P r o p a g a t i o n S m oo t h 0.0 0.2 0. K D L o ss R a t i o 0. Chameleon

Original: According to the experimental results, we observe that AdaFGL performs robustness except for the extreme cases, and it is suitable for more generalized federated data settings and easy to obtain optimal settings.
Revised: [Table]

Original: Furthermore, we obtain the conclusion from the [Figure] Cora [Figure]
Revised: A. H YPERPARAMETER SENSITIVITY ANALYSIS

Original: Chameleon

Original: [Table]

Original: 57±1.

Original: 72.28±0.

Original: 74.40±0.

Original: 32.45±

Original: [Table]

Original: 35.32±

Original: [Table] 29.96± [Table] 56 GCFL+ 73.25±0.
Revised: Here we conduct the hyperparameter sensitivity of AdaFGL, and the experimental results are shown in Fig.

Original: 73.30±0. 76.28±0. 27.17±0. 28.03±0.
Revised: In our experiments

Original: 27.33±0.39 FedSage+ 69.15±0.

Original: 71.88±0.

Original: 75.49±

Original: [Table]

Original: 36.72±

Original: [Table]

Original: 43.65±

Original: [Table]

Original: 31.27±

Original: [Table]

Original: 93 w/o HomoKD 71.46±

Original: [Table]
Revised: [Table]

Original: 74.26±

Original: [Table]

Original: [Table]

Original: 12±

Original: [Table]

Original: [Table]

Original: 02±

Original: [Table]

Original: [Table]

Original: 38±

Original: [Table]

Original: [Table]

Original: 43±

Original: [Table]

Original: 53 w/o HeteTA [Table] 57± [Table]
Revised: we analyze the ratio of online distillation loss in the homogeneous propagation module and the smoothing coefficient of the trainable propagation matrix in the heterogeneous propagation module.

Original: [Table]

Original: 57±

Original: [Table]

Original: [Table]

Original: 81±

Original: [Table]

Original: [Table]

Original: 94±

Original: [Table]

Original: [Table]

Original: 79±

Original: [Table]

Original: 33.16±

Original: [Table] 40 AdaFGL 73.69± [Table] 75.32±
Revised: According to the experimental results, we observe that AdaFGL performs robustness except for extreme cases.

Original: [Table]

Original: 77.05±

Original: [Table]

Original: [Table]

Original: 02±

Original: [Table]

Original: 49.38±

Original: [Table]

Original: [Table] 43± [Table] results generated by the extreme knowledge distillation loss ratios, where the low confidence base predictor results instead affect the homogeneous propagation module.
Revised: Furthermore, we obtain the conclusion from the results generated by the extreme knowledge distillation loss ratios, where the low confidence base predictor results instead affect the homogeneous propagation module.

Original: Motivated by this, in order to avoid global embeddings with low confidence from influencing the propagation module, we measure the confidence of the global model according to the characteristics of the base predictor to achieve more effective local training.
Revised: Motivated by this, in order to avoid global embeddings with low confidence from influencing the propagation module, we measure the confidence of the global model according to the characteristics of the base predictor.

Original: A. E XPERIMENTAL ENVIRONMENT AND ADDITIONAL BASE RESULTS
Revised: A. E XPERIMENTAL ENVIRONMENT AND ADDITIONAL BASE RESULTS

Original: The experiments are conducted on a machine with Intel(R)
Revised: The experiments are conducted on a machine with Intel(R)

Original: Xeon(R)
Revised: Xeon(R)

Original: Gold 6230R CPU @ 2.10GHz, and a single NVIDIA GeForce RTX 3090 with 24GB memory.
Revised: Gold 6230R CPU @ 2.10GHz, and a single NVIDIA GeForce RTX 3090 with 24GB memory.

Original: The operating system of the machine is Ubuntu 18.04.6.
Revised: The operating system of the machine is Ubuntu 18.04.6.

Original: As for software versions, we use Python 3.8, Pytorch 1.11.0, and CUDA 11.4.
Revised: As for software versions, we use Python 3.8, Pytorch 1.11.0, and CUDA 11.4.

Original: To alleviate the influence of randomness, we repeat each method 10 times and report the statistical characteristics.
Revised: To alleviate the influence of randomness, we repeat each method 10 times and report the statistical characteristics.

Original: The hyper-parameters of baselines are set according to the original paper if available.
Revised: The hyper-parameters of baselines are set according to the original paper if available.

Original: We use Optuna Akiba et al.
Revised: We use Optuna Akiba et al.

Original: (2019) to implement hyperparameters search.
Revised: (2019) to implement hyperparameters search.

Original: Following the above principles, we present the results of two data partitioning as follows.[Table] [Table] [Table]
Revised: Following the above principles, we present the results of two data partitioning as follows.

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: Client   0.

Original: Client   0.

Original: Community

Original: [Table]

Original: Nodes

Original: [Table]
Revised: [Table]

Original: Edges

Original: Homo

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client   0.

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: Nodes

Original: [Table]
Revised: [Table]

Original: Edges

Original: Homo

Original: Client   0.

Original: Client

Original: [Table]

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client   0.

Original: Client

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: Homo

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: Client

Original: [Table]

Original: Community

Original: [Table]

Original: Homo

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: Edges

Original: Homo

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: Client

Original: [Table]

Original: [Table]

Original: [Table]