HGOT: Self-supervised Heterogeneous Graph Neural Network with Optimal Transport
Authors: Yanbei Liu
Chongxu Wang
Zhitao Xiao
Lei Geng
Yanwei Pang
Xiao Wang
Venue: ICML 2025 poster (2025)
Keywords: Heterogeneous graph
Self-supervised learning
Optimal transport
Primary area: deep_learning->graph_neural_networks

Abstract: Heterogeneous Graph Neural Networks (HGNNs), have demonstrated excellent capabilities in processing heterogeneous information networks. Self-supervised learning on heterogeneous graphs, especially contrastive self-supervised strategy, shows great potential when there are no labels. However, this approach requires the use of carefully designed graph augmentation strategies and the selection of positive and negative samples. Determining the exact level of similarity between sample pairs is non-trivial.To solve this problem, we propose a novel self-supervised Heterogeneous graph neural network with Optimal Transport (HGOT) method which is designed to facilitate self-supervised learning for heterogeneous graphs without graph augmentation strategies. Different from traditional contrastive self-supervised learning, HGOT employs the optimal transport mechanism to relieve the laborious sampling process of positive and negative samples. Specifically, we design an aggregating view (central view) to integrate the semantic information contained in the views represented by different meta-paths (branch views). Then, we introduce an optimal transport plan to identify the transport relationship between the semantics contained in the branch view and the central view. This allows the optimal transport plan between graphs to align with the representations, forcing the encoder to learn node representations that are more similar to the graph space and of higher quality.
Extensive experiments on four real-world datasets demonstrate that our proposed HGOT model can achieve state-of-the-art performance on various downstream tasks. In particular, in the node classification task, HGOT achieves an average of more than 6\% improvement in accuracy compared with state-of-the-art methods.

HGOT: Self-supervised Heterogeneous Graph Neural Network with Optimal Transport
Yanbei Liu
Chongxu Wang'
Zhitao Xiao
Lei Geng Yanwei Pang?
Xiao Wang?
Abstract
model can achieve state-of seefthe-ar performance on various downstream tasks. In particular, in the node classification task, HGOT achieves an average of more than 6% improvement in accuracy compared with state-of-the-art methods.
Heterogeneous Graph Neural Networks (HGNNs), have demonstrated excellent capabil- ities in processing heterogeneous information networks. Self-supervised learning on het- crogencous graphs. especially contrastive self-supervised strategy, shows great potential when there no labels. However, this approach requires the use of carefully designed graph augmentation strategies and the selection of positive and negative samples. Determining the exact level of similarity between sample pairs non-trivial. To solve this problem, we propose a novel self-supervised Heterogeneous graph neural network with Optimal Transport (HGOT) method which is designed to facilitate self-supervised learning for heterogeneous graphs without graph augmentation strategies. Different from traditional contrastive self-supervised learning, HGOT employs the optimal transport mechanism to relieve the laborious sampling process of positive and negative samples. Specif- ically, we design aggregating view (central view) integrate the semantic information contained in the views represented by different meta-paths (branch views). Then, we introduce an optimal transport plan to identify the transport relationship between the semantics contained in the branch view and the central view. This allows the optimal transport plan between graphs to align with the representations, forcing the encoder to learn node representations that are more similar to the graph space and of higher quality. Extensive experiments on four real-world datasets demonstrate that our proposed HGOT
1. Introduction
Heterogeneous graphs prevalent in numerous real-world contexts, including academic networks (Wang et al. 2019), biomedical networks (Bai et al. 2021), social networks (Cao et al. 2021) and so on. These graphs serve to represent diverse relationships among various types of nodes. In re- cent years, a range of heterogeneous graph neural networks (HGNNs) has been developed effectively capture the intri- cate structures and abundant semantic information inherent in these graphs, demonstrating commendable performance in the domain of heterogeneous graph representation learn- ing (Wang et al. 2020, Dong et al. 2020). To date, HGNNs have found extensive applications across a variety of fields (Tian al. 2022a, Bansal et al. 2019, Wang al. 2021a).
The majority of current HGNNs rely on supervised or semi- supervised learning methods (Sun et al. 2025, Ji et al. 2024), necessitating the guidance of labeled data throughout the model's learning process. However, acquiring node labels from heterogeneous graph data poses significant challenges and incurs high costs in many applications (Hu et al. 2020), Consequently, in the absence of labeled data, self-supervised learning (SSL. emerges viable alternative, it seeks derive valuable supervisory signals directly from the input data.
In the realm of self-supervised learning applied to heteroge- neous graphs, contrastive learning has emerged as a promi- nent technique. This approach involves carefully designed graph augmentation strategies and the selection of positive and negative samples. The objective is enhance the simi- larity between positive samples while concurrently reduc- ing the similarity between negative samples. Some works (Zhao et al. 2024, Zhu et al. 2022) on heterogeneous graph contrastive learning propose different graph augmentation strategies to better capture the underlying high-order se- mantic structure information. In addition, many methods
'chool of Life Sciences, Tiangong University, Tianjin, China "School of Electrical and Infomation Engineering, Tianjin Uni- versity, Tianjin, China "School of Software, Beihang Univer- sity, Beijing, China. Correspondence to: Zhitao Xiao <xiaozhi- t tiangong. edu>, Xiao Wang < xiao_wang buaa.edu >.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
(Wang et al, 2021a, Zhao et al. 2024) retain the traditional contrastive learning and design the method of selecting the required positive and negative sample pairs from differ- ent views or different structures, and can more effectively obtain high-level factors. Although the contrastive self- supervised learning strategy enhances the quality of node representations in HGNNs and facilitates their application in real-world scenarios devoid of label supervision (Xie et al. 2024, Liu et al. 2024, Chen et al. 2024), two key chal- lenges persist in the contrastive self-supervised learning for heterogeneous graphs:
(branch views) to obtain an aggregated view (central view) which can integrate the semantic information from different meta-paths. We then introduce optimal transport plan to identify the transport relationship between the semantics contained in each branch view and the central view. To accurately capture the matching relationships in the original graph space, the encoders are forced to learn representations that exhibit consistency with the optimal transport plans between the corresponding meta-patt peta-ppah views and aggregated view. In summary, the main contributions of this work are as follows:
C1: More effective self-supervised learning strategy for heterogeneous graphs. On the one hand, existing contrastive self-supervised methods on heterogeneous graph require graph augmentations. The heterogeneous graph data structure is discrete, so even if the graph is slightly perturbed, the properties of the graph may change, which leads to inconsistent labels between the original and augmented graphs. Therefore, it is chal- lenging to design reasonable augmentations that guar- antee the label consistency for heterogeneous graphs. On the other hand, the strategy of contrastive learning is to maximize the similarity for positive pairs of the same anchor and minimize the similarity for those of negative pairs. However, the concept of "maximum or minimum similarity" is difficult to measure since it is vague and lacks a clear indication of how much similarity should be maximized (or minimized) for a given pair of positive (or negative) views.
To our best knowledge, this is the first attempt to apply optimal transport theory to heterogeneous graph. By utilizing the optimal transport to calculate the trans- port plan between the meta-path view and aggregated view, is no longer necessary to perform data augmen- tation and provide positive and negative samples for heterogeneous graph self-supervised learning.
We propose a novel self-supervised Heterogencous Graph neural network with Optimal Transport (HGOT) method, which utilize the optimal transport plan be- tween the local semantics (branch view) and the global semantics (central view) of heterogeneous graphs to align the matching relationship between the graph space and the representation space, thereby obtaining higher-quality node representations.
We conduct extensive experiments on four public benchmarks, demonstrating that the proposed HGOT outperforms the state-of-the-art methods. In partieu- lar, in the node classification task, HGOT achieves an average of more than 6% improvement in accuracy compared with state-of-the-art methods.
C2: More comprehensive heterogencous semantic information aggregation. Heterogeneous graphs typi- cally exhibit complex structural variations, e.g., objects of different types, diverse relationships, and varying degrees of connectivity. Integrating information across these structures while maintaining the integrity of the graph's overall topology (i.e., the interconnections be- tween objects) presents a further difficulty. A sim- plistic aggregation approach could distort the graph's is structural properties, leading to loss of context and nuance.
2. Related Work
Heterogeneous graph self-supervised learning. Heteroge- neous graph self-supervised learning has been a promising paradigm, and it uses the information of the graph itself to learn node representations. For example, HeMuc (Zhang et al. 2023) proposes a contrastive learning method by de- signing a new sampling strategy that combines structure and characteristic information. HeCo (Wang et al. 2021a) ap- plies two network structures to align both local schema and global metapath information. HGCML (Wang et al. 2023) constructs contrastive learning method under multiple views by utilizing meta-paths and proposes new positive sample sampling strategy to reduce sampling bias. In ad- dition to contrastive learning, generative learning is also a good superviee learning method (Kipf and Welling 2016, Park et al. 2019). HGMAE (Tian et al. 2023) applies
To address these challenges, in this paper, we propose HGOT, a novel self-supervised learning strategy for het- erogeneous graphs that does not require graph augmenta- tions and positive or negative samples. We exploits the key concept in optimal transport (OT) theory, which aims to identify the matching relationship between graph and representation. In order to eliminate the diversity of node types, we design the node feature transformation to project all nodes into a unified dimensional feature space. To fur- ther capture more comprehensive semantics, we employ the attention mechanism to aggregate different meta-paths
2

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Self-supervieed Heterogencous Graph Neural Network with Optimal Transport
Meta-path View
91
Encoder
21
mmat θ (T
#2
P6
Hob
P4
Tom
Lily
θ
Encoder
Heterogeneous Graph
Z
Aggregsted View
MR-WX -b
Lmat O (#g.
Encoder
Feature Transformation
6.
Zx
Meta-path View
Figure 1. The overall framework of HGOT: We first project all nodes' features of the original heterogeneous graph into the dimensional space. We then design the heterogeneous semantic information aggregation to obtain the meta-path view and aggregated view. Finally, introduce the optimal transport theory discover matching relationships between each meta-path view and the aggregated view and align them to ohtain higher quality node representations.
masked autoencoders to heterogeneous graphs and obtains better representations through the mask mechanism.
denote sets of object and link types, and (Ty [Tgl > 2. It is homogeneous graph with the same node and edge types when [Tvl = and [Tgl = 1. For example, in the academic network ACM, three node types of Paper, Author, and Subject and two types of link relationships constitute heterogeneous graph.
Optimal Transport. Optimal transport (OT) (Cédric Vil- lani et al. 2009) is a mathematical tool widely used in computer vision (Nicolas Bonneel et al. 201 1), generative adversarial network (Cao al. 2019) and domain adap- tation (Gu et al. 2022) to calculate the distance between two probability distributions or to align two distributions in different spaces. OT aims to derive a transport plan between a source and a target distribution, such that the transport cost minimized. (Facundo Mémoli et al. 2011) proposes the Gromov- Wasssestenn (GW) distance between metrics defined within each space which has been used distance between graphs in several applications (Vayer Titouan et al. 2020, Cédric Vincent-Cuaz et al. 2022, Xu et al. 2019). In this work, we combine the optimal transport theory with heterogeneous graph neural networks to form a new self- supervised learning paradigm.
Definition 3.2. composite relation of several edge types, represented
Meta-paths. A meta-path defines
P Tvi Tej Tv, Te. Tes Tv (abbreviated as Tvi Tv, V,.,. T T which describes composite relation Ts Te, oT o...e, between node types Tv, and T V where o denotes the composition operator on relations. Be- cause meta-path the combination multiple relations, contains high-order structures.
Problem 3.1. Plan and Optimal Transport. The optimal transport problem was first proposed by French mathemati- cian Monge to seek the most cost-effective transportation solution for transporting the shape of one pile of sand into the shape of another (Wang et al. 2024). In particular, given two sets of features X₁ = {Xi > and X₂ = y where n and m are the number of features. μ E R and V Rm are the probability distributions of the entities in the two sets, respectively. It studies how transform distribution μ into distribution y with the minimum total transportation cost (i.e., the optimal transportation distance). This transport plan is called the optimal transport plan n, where the element of π describes the probability of moving mass from one position to another. summary, the goal of optimal transport is find transport plan a that transforms
3. Preliminaries
In this section, we define two concepts related to hetero- geneous graphs and two problems of optimal transport graphs as follows.
Definition 3.1. Heterogeneous Graph. A heterogeneous graph is defined as G (V,E, Tv, Te, sp; v), which con- sists of node set V. a link E, and it is associated with node type mapping function 8 V Ty and a edge type mapping function vb
Tg. where Ty and Te
3

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
the distribution of node set 1₁ into the distribution of node set X2 with the minimum transport distance. In this work, we focus on the case of discrete distributions graphs.
4.2. Aggregated View Generation
Since heterogeneous graphs contain rich semantic informa- tion which are reflected by various meta-paths, we design the aggregated view generation mechanism to aggregate different meta-path views. In particular, we decompose the original heterogeneous graph into multiple homogeneous meta-speeific graphs according to meta meta-paths. Then we con- sider aggregating these homogeneous meta-specific graphs from the perspective of nodes and edges.
Problem 3.2. Optimal Transport for Graphs. On different graphs, we can view node attribute information as different distributions. If the nodes in the graph have no attributes or label information, we can also use the edge structure of the graph calculate the optimal transportation plan between two graphs. Given two graphs 9 (A1, 14) and G2 = (A2, where A and 22 are the adjacency matrix of two graphs. Similarly, we can compute the optimal transport plan т between the edge structures (i.e., adjacency matrix) of the graph. The specific relevant formula for optimal transmission on the graph will be given in detail in the next
For nodes, following most heterogeneous GNNs' ap- proaches, we first generate multiple views and each view corresponding to one meta-path that encodes one aspect of information. Then we leverage an attentive network to compute specific meta-path embedding hi for node v; under the p-th meta-path as
section.
4. Method
h CONCAT y
o W"hh IG (.NN)
Σ
(2)
In this section, we formally propose HGOT, a novel hetero- geneous graph neural network with optimal transport. In particular, HGOT is constructed by three major components: node feature transformation, aggregated view generation, and mueti-view optimal transport alignment. Figure illus- trates the framework of HGOT.
where CONCAT concatenates K standalone node represen- tations in each attention head, .Vp (u;) defines the neighbor- hood of i2 that is connected by meta-path, WP E RRdx is a linear transformation matrix for p-th meta-path, and V (.) is the activation function, such as ReLU (.) = max (0, -). The attention coefficient at can be computed by a softmax function
4.1. Node Feature Transformation
In heterogeneous graph, different node types have features of different dimensions. Even if their feature vectors have the same dimensions, they may not be in the same feature space. Feature vectors of different dimensions are trouble- some when we process them in a unified framework. In addition, considering that nodes are required to have the same dimensions when using the optimal transport calcula- tion later, we project different types of node features into the same latent vector space.
exp
a
h hh a
a
(3)
En (N)))))) Σ
exp
T
where ap E R22d11 a trainable linear weight vector for the p-th meta-path.
Finally, fuse the node representations in all views into aggregated representation. We employ another attentive network to obtain the aggregated representation h that combines information from every semantic space by
Therefore, we apply a type specific linear transformation for each type of nodes by projecting feature vectors into the same latent factor space. Specifically, for the i-th node with type sp. we design a type-specific mapping matrix W Wp: to transform its features into a common latent space as follows:
h
ARR
3" h
(4)
p=1
h;
W o; X; byp
The coefficients are given by
where x; is the original feature vector, h; E Rdx1 is the projected feature of i-th node, and b,, denotes as vector bias.
exp (wP)
(5)
En Epecr exp (wp')'
The node feature transformation addresses the heterogeneity of a graph that originates from node features that are not in the same feature space. All nodes' projected features share the same dimension and are in the same feature space, which facilitates the calculation process of the part of the optimal transport module.
w
q T tanh (Mii, +b).
(6)
2EEV
where q E Rm denotes type-level attention vector. g can be interpreted as the contribution of the p-th meta-path. dm E is a hyperparameter, M E Rdm xd b E T are

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
the weight matrix and the bias vector, respectively. Finally, concatenate and arrange the aggregated representations of all nodes and use a matrix Zagx to represent the represen- tations of all nodes in the aggregated view.
cost of moving H, H we can choose the cosine dis- tance between them calculate. The 1 denotes vector that elements are all 1, In]] = {1, , and (.-) is the inner product operator. The æ E R=*m is called as transport plan. This distance Dn (-5-) is also known as the Wasserstein distance, used to calculate the transport distance between node distribution.
For edges, we create the meta-path-based adjacency matrix Ap for each meta-path E P via meta-path sampling. In order to directly obtain the edge structure information in the fused view, and do not want to introduce repeated edges, we perform a logical OR operation on all adjacency matrices follows:
For graph, difficult to measure the cost between two nodes different graphs without node label (attribute). Even if there is some way to get the cost between nodes, there is no edge information in the above Wasserstein dis- tance formula. Therefore, we need to take the edge informa- tion into account as follows:
Aags- Ai V V ... V A (P],
where Aagg denotes the adjacency matrix of aggregated view. Note that the operation rule here to perform an OR operation on each element in the adjacency matrix. This obviously means that if two nodes are connected in any meta-path view, then there should be an edge between them in the aggregated view.
Dr Ang:) = min Σ
Σ
TETI(U)) iJ[[n]] [.
(11)
Ca (A AA Tak
agg
In this step, we obtain the node representation information Zaga and the edge structure information Aogg of the aggre- gated view Gugg.
abbreviated as
De (Ap. =
min *Tllkr))
(E (Ap: 20 vT, x)),, (12)
4.3. Multi-view Optimal Transport Alignment
In order to achieve information matching from subgraphs induced by meta-paths to aggregated graph, we use the theory of optimal transport to calculate the optimal transport plan between each meta-path view and the aggregated view. Then we align the optimal transport plan between graph and representation to achieve information matching.
where E AaR:))) syRt = CA (A A and it is 4-
D tensor. |na] [m]] x [n]] Since the cost function here involves the adjacency matrix, we directly subtract it and then take the absolute value to calculate, which is A
Ca (A p A
A A This distance D, (:))
is also known as the Gromov- Wasserstein distance, used to calculate the transport distance between edge distribu- tion. Then we consider the complete graph including node features and edge structure Gp Ap. u)) and Gags agg: Augg, v). The fused distance between two graphs can be defined
Our goal is to find the optimal transport plan for the two views, minimizing the transport cost (i.e., optimal transport distance). Therefore, we choose start with the transport distance to calculate the optimal transport plan. Specifically, we obtain the node feature matrix Hp and Hagg in the meta- path view and the aggregated view after projection in the "node feature transformation" step. The formulation of the OT distance is
Dg (gp. Gugs) min T TElI(u.))
Eex
(H
H
Tif
(13)
Dn (H.p H Hagg) min
ΣΣ Cx H
H
mgg))
Tejs
+1- o) Σ
CA
(A
A
ARR)
2
xell(p.))
VEIn] [m[]
kjki
(8)
which is equivalent to
abbreviated
min foF (H,, Hagg) + (1 - 0) (A,, Aass) 2 ng. mg), mell(a.))
Da (Hp. Hags) = min (F (Hp, H xx)
(9)
(14)
xell(p,))
where E [0; 1] represents the trade-off parameter for ad- justing nodes and edges. The formula is a fusion of Equa- tions (8) and (11).
where
II (H, ( E Raxm 1m 14, In v}
(10)
Optimal Transport Plan. For graphs and representation vectors, they are in different spaces and are essentially two different objects. Although it is challenging to compare two different objects in different spaces, the optimal transport
denotes all the joint distributions π with the marginal dis- tribution 1s and v. F (Hp, H mgg))s Cx (H. HJ is the
agg)
5

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
plan provides a solution. In this work, we integrate the node features and topological structure (i.e., edges) the graph and design new objective function to align the optimal transport plan between the graph space and the representa- tion space. Specifically, the fused optimal transport plan "o E R=xm between the two graphs Op can be defined as
4.4. Comparison with Graph Contrastive Learning
A common way of traditional contrastive objective is con- trasting views at the node level. For the representations (positive views) h and h of same node i in two augmented graphs 91 and G2, the pairwise contrastive loss can be de- fined as
and Gagy above
T arg min (II (oF (Hp. H Hapx) +
(15)
Leom = >> log
(1 - o) E AagsA @ ng. Tq):
e 7()
+ Enes 7 (b/ b2)/T
with the fused Gromov-Wass Wasserstein distance. By tuning the parameter o, we can control the deviation of the learned optimal transport plan between nodes and edges.
(20)
where I is a discriminator that maps two views to an agree- ment (similarity) score, such as the inner product. 7 denotes the temperature.
Then we use the backbone model (e.g., HGNNs) to obtain the node representation Zp under each meta-path view. The node representation here contains the semantic information of the p-th meta-path, and the node representation Zagg in the aggregated view calculated in Section 4.2. We can use Equation (8) directly to calculate the optimal plan Tiz E Raxm as
Although our objective function deviates from the con- trastive loss, as outlined in Equation (20), we identify a striking resemblance in the foundational algorithmic prin- ciples governing both methodologies. In this examination, we concentrate the node level. When the attributes and contextual information of a given node Ui demonstrate sim- ilarity across diverse semantic perspectives, the associated cost for these views diminishes, nearly reaching zero. This reduction in cost leads to an optimal transport strategy that favors a high probability of aligning the representations of these nodes. Leveraging Equations (17) and (18), the HGOT model adeptly fine-tunes both the matching probabilities and costs linked to the corresponding node representations within the output domain. Consequently, this ensures that the representations of two analogous nodes are sufficiently congruent. In contrast, for two distinct nodes U; and Uk. the costs linked to their respective node views across varying semantic viewpoints are relatively higher. This discrep- ancy necessitates an adjustment in the opposite direction, ensuring that the representations of two dissimilar nodes are sufficiently differentiated.
Z = arg min T2 (T(( (R (Z,, Zugz) nT),
(16)
where R 7 1j = Cx (Zp, Ragz) Zp and Zuga de- note the node representations corresponding to Gp and Gogg. respectively.
Optimal Transport Alignment. Obviously, if the node embedding learned by the encoder can capture the matching information with other graphs while preserving its own infor- mation, it may obtain better node representations. Therefore, we force the encoder to preserve the matching relationship in graph space by aligning the optimal transfer plan between two graphs with the optimal transport plan between their corresponding node embeddings. We minimize the differ- ence between the optimal transport plans in the two spaces as the alignment loss as follows
Cmat θ (T):T2).
(17)
From the perspective of contrastive learning, the process de- scribed above is in harmony with its core objectives. Never- theless, there is a notable disparity between graph transport alignment and graph contrastive learning. HGOT capitalizes calibration signals sourced directly from the graph space. This unique calibration signal empowers us to circumvent the need to distinguish between positive and negative sam- ples.
where θ (s-)) denotes the discrepancy function, it can be any commonly used metric. For example, we can choose the Frobenius-none normt as (-.-) = -1F
In addition, to guide the encoder to retain the structural infor- mation of the original image while learning node embedding, we correct the cost matrix R (Zp, Zagg) between represen- tations, which implies the implicit structure relationships between nodes, in the representation space as follow
5. Experiments
Let = (oF (Hp: H mgg))
(1 .) E (Ap: Angs) 3
(18)
In this section, we conduct extensive experiments to verify the effectiveness of the proposed HGOT. Specifically, we first demonstrate the advantages of our proposed model over other methods in different tasks, including node classifica- tion, node clustering, and visualization of node represen- tations. Then, we conduct other experiments to verify the effectiveness of the modules in our method, including the
z&. R (Zp, Zagg)).
To this end, we define the overall heterogeneous graph trans- port alignment loss as
Emat + pLotr:
(19)
where p is the trade-off parameter.
6

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
Table I. Experimental results on node classification.
DBLP
ACM
IMDB
Yelp
Metrie
Micro-F1
Macro-F1
Micro-F1
Macro-F1
Micro-F1
Macro-F1
Miero-FI
Macro-F1
GCN
91.71
0.6
90.84
0.7
88.39 i 0.6
88.21
54.64 0.7
51.81 : 0.7
72.16 0.6
52.55 0.4
GAT
91.93 0.7
91.16
0.7
87.
t 0.3
87.37 0.3
56.97 + 0.4
$2.93 + 0.5
72.74 + 0.6
53.02 0.7
HAN
92.02 # 0.8
91.71
+ 0.8
90.55 + 1.1
90.62
1.0
58.55 + 1.4
54.37 + 1.1
72.30 + 0.9
53,56
0.8
GraphSAGE
74.05 + 8.6
73.86
t 8.4
60.72 t 3.5
56.59 199960 6.8
41.33 + 7.1
40.71 1 6.9
63.19
51.97 $..7712 3.2
GAE
90.95 # 0.3
90.08
# 0.3
65.71 + 2.2
61.67 = 2.9
47.62 # 3.1
46.91 = 2.9
71.33 = 1.6
52.34 = 1.7
DGI
90.35 # 0.8
89.19 0
80 15 + 3.2
80.03 3.3
50.56 = 4.0
49.77 = 3.8
72.04 2.2
2.2
DeepWalk
86.23 0.4
84.85 + 0.5
73.87 + 0.6
74.13 0.6
48.35 + 0.8
45.39 = 0.7
69.44 + 1.0
49.81 0.8
Metapath2voc
90.61 0.1
90.81 + 0.1
73.61 + 0.3
73.80 0.4
48.82
1.2
45.18 = 0.8
71.88 = 0.2
50.89 U.6
HERee
91.17 0.3
90.74 + 0.3
73.85 + 0.9
73.92 0.8
50.91
# 0.8
47.62 = 0.8
72.03 = 0.5
$1.35 = 0.3
FeaCon-V
91.04
91.50 1 0.4
87.76 0.3
88.14
53.86 0.8
50.65 4 1.0
72.27 0.7
53.72 0.7
FeaCon-D
91.13
88.82 0.3
57.77 2 0.8
87.27
53.37 0.6
42.32 + 0.8
72.74
52.90
HDGI
91.24
044
50.12 0.8
72.18
$2.79 0.3
HEAD
91.30 0.7
91.08 0.8
87.05
87.28
$7.39
49.01 1.2
74.61 2.0
55.97 2.0
HeCo
92.03 0.2
91.37 0.3
88.79
89.04
57.35 0.8
52.51 1.0
72.48 0.5
53.18 0.5
HGBER-V
92 77 0.
91.80
0.4
92 34 0.7
91 91.55 0
59.97 + 0.6
56.16 1.2
73
0.6
53.48 + 06
HGBER-D
92.93 + 0.2
91.38 + 0.3
88.79 1 0.5
89.03 0.5
$7.33
+ 0.8
52.56 + 1.1
72.41
63.16
HGCML
91.11 0.5
90.47
1 0.3
87.90 1 0.3
88.46 0.4
$7.19
2 0.7
51.03 + 0.4
73.35 0.8
52.90 0.6
HGMAE
91.89 0.3
91.60 + 0.5
89 15 + 0.4
89.29 0.4
60.13
+ 0.4
60.09 + 0.6
72.57 0.6
56.33 # 0.3
HGOT
95.66 0.4
95.14
94,49 1.0
9.860
62.71 1.3
62.34
77.58 0.8
59.12 0.4
ablation study and parameter sensitivity analysis.
5.2. Node Classification
We evaluate different models for the node classification task and report their performances in Table 1. We use Micro-F1 and Macro-FI as evaluation metrics. Results demonstrate the effectiveness of HGOT as achieves the best perfor- mance over these baselines. Compared with the homo- geneous graph model, HGOT can capture the high-order semantic relationship between nodes obtain better node representation. Therefore, our method has significant improvement in the accuracy of node classification tasks compared to homogeneous graph methods, and even im- proves by about 34% compared to GraphSAGE on the ACM dataset. Although methods such as HAN and Metapath2vec have utilized meta-paths, and HeCo has attempted con- trastive learning scheme between meta-paths and network modes, these methods do not study the matching informa- tion between meta-path view and aggregated view. Because our model HGOT attempts to learn the matching informa- tion from the graph space and representation space, the quality of node representation is improved. Thus our HGOT achieves the best classification performance.
5.1. Experimental Setup
Datasets. Four public datasets (DBLP, ACM, IMDB and Yelp) are used to demonstrate the effectiveness of the pro- posed method. The detailed information on datasets is pro- vided in attached page.
Baselines. We compare the proposed HGOT with three categories of baselines: supervised methods: GCN (T. N. Kipf et al. 2016), GAT (P. Veli ckovi al. 2017), a semi- supervised heterogeneous method: HAN (Wang et al. 2019), and unsupervised methods: GraphSAGE (William L. et al. 2017), GAE (T. N. Kipf et al. 2016), DGI (Petar Velickovic et al. 2019), DeepWalk (B. Perozzi et al. 2014), Metapath2vec (Dong et al. 2017), HERec (Shi et al. 2019), FeaCon-V, V, FeaCon-D, HDGI (Ren al. 2019), HEAD (Wang al. 2021), HeCo (Wang al, 2021a), HGBER (Liu et al. 2023), HGCML (Wang et al. 2023) and HGMAE (Tian et al. 2023).
Implementation Details. For baselines, we adhere to the settings described in their original papers and follow the setup in HeCo. For the proposed HGOT, we use HAN as the backbone model obtain the node representation Z,p under each meta-path view. We search the learning rate from le-4 to 5e-3, tune the patience for early stopping from 5 to 20. The Adam optimizer (Diederik P. Kingma et al. 2015) is adopted for gradient descent. For all methods, we set the embedding dimension as 64 and report the mean and standard deviation of 10 runs with different random seeds. Among all the methods, the HGBER model has two variants (HGBER-V V and HGBER-D) and we compare them both.
5.3. Node Clustering
In this task, we utilize hicrarchical clustering algorithm to the learned embeddings of all nodes and the results of the node classification are shown in Table 2. Hierarchical clustering can generate hierarchical clustering structures. Compared with the K-means clustering algorithm, it does not require the number of clusters to be specified in ad- vance. The appropriate number of clusters can be selected as needed, and it can process various types data. Indica- tors such as clustering accuracy (ACC), normalized mutual information (NMI, and adjusted rand index (ARI) (Xia al. 2014) used as metrics to assess the quality of the
7

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Self-supervied Heterogencous Graph Neural Network with Optimal Transport
Table 2. Experimental results on node clustering.
Dataset
DBLP
ACM
IMDB
Yelp
Metric
ACC
NMI
ARI
ACC
NMI
ART
ACC
NMI
ARI
ACC
NMI
ART
GraphSAGE
69.59
51.50
36.40
56
29.20
27.72
29. 10
0.81
0.53
58 04
29. 83
31.36
GAB
83.09
72. 52
77.31
58.93
27.42
24.49
36.64
0.97
0.62
60.90
32.07
36.41
DeepWalk
82.33
70.97
75.61
58. 88.82
29.50
28 86
38.29
0.93
0.66
61.85 83
35.88
38.67
Motapath2vee
75.62
71.33
78.58
58 91
31.17
31.08
38. 74
1.27
1.78
62 96
35.53
40.11
HERco
76.54
71.39
79.47
63 33
42.18
36.70
38.
1.29
1.58
63.84
35.09
40.18
FeaCon-V
68.21
45.26
40.49
68.22
45.18
40.44
42.88
4.47
3.92
63.29
36.61
40.27
FcaCon-D
80.47
56.89
36.50
78.62
53.66
$2.09
43.67
3.55
3.90
63.37
37.02
39.94
HDGI
88.12
71.58
76.64
71.80
54.43
49.61
44.34
3.72
4.19
63.89
37,52
40.16
HEAD
38.93
71.60
73
80. 92
56.71
43.55
44.19
3.44
4.20
63.89
38. 04
42.25
HeCo
90.33
74.51
80.17
88.72
56.87
56.94
44.73
4.58
4.36
65.60
37.61
42.37
HGBER-V
89.97
72
79.60
10
65.88
70
44.77
4.72
4.63
65.37
38. 75
42.56
HGBER-D
88.45
71.82
78.59
89. 89.30
65,99
70.71
45.12
4.88
4.69
65.82
37.50
42.19
HGGMML
90.36
73.28
79.61
89.21
65.13
71.05
50.66
9.34
9.15
65.72
37.71
42.49
HGMAE
91.47
76.92
82.34
89.83
66.68
71.51
$3.90
11.36
12.03
62.10
39.04
42.88
HGOT
93.41
78.05
84.00
89 77
66. 12
71.94
55.80
14,28
13.04
66.23
39.15
43.07
HGd0--ice1
HGG Teebos
HOOT wisd?
HCOT w/uk
50
HG00
Nome-3 --000
DGE
60
(5)
Figure 3. Ablation experimental results on four datasets.
LNGOY
Figure 2. Visualization results of different methods ACM.
5.5. Ablation Study
In this section, we compare HGOT with its variants on four datasets to verify the effectiveness of three compo- nents. HGOT-w/o#1 does not have "aggregated graph generation" module. We directly match the two different meta-path views without calculating any information the aggregated view. HGOT-w/o#2 represents HGOT without implicit structure loss L HGOT-w/ofe drops the align- ment of the optimal transport plan, but instead optimize the transport distance directly. In order to make the optimal transport distance between the graph space and the represen- tation space consistent, we replace the matching loss with the following:
clustering results. It can be seen that HGOT achieves the best performances on most datasets in node clustering task. Notably, HGOT demonstrates an average performance im- provement of approximately 2.5% on the DBLP, IMDB, and Yelp datasets.
5.4. Visualization
In order obtain more intuitive evaluation results, we con- duct embedding visualization ACM dataset. We use the I-SNE (Van Der Maaten L. 2014) dimensionality reduction method to experiment on different models and the results are shown in Figure 2, in which different colors mean dif- ferent labels. As we can see, the Metapath2vec and DGI method does not classify nodes well, the visualization re- sults show that they cannot make the boundaries between different types of nodes clearer because they cannot fuse all kinds of semantics. Although DMGI and HeCo have achieved the ability to clearly distinguish between various types of nodes, they are still mixed to some degree. The proposed HGOT correctly separates different nodes with relatively clear boundaries, demonstrating the effectiveness of HGOT.
L'mat = IDy (gp, Gugs) DD, (Zp, Zoge)
(21)
where / denotes absolute value. The results are shown in Figure 3.
From Figure 3, conclusions can be drawn follows: (1) The results of HGOT are consistently better than three variants, indicating the effectiveness of the heterogeneous semantic information aggregation, the implicit structure loss and the alignment of the optimal transport plan strategy. (2) HGOT w/o#33 the worst one, which makes us realize the necessity
8

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
96
Wamo-1 Rccro12
Vatrcc 60
Micro-F1
94
52
93
HUaeLV
0.0 0.2 0.2 0.3
0.6
0.7 08
02 10
50
0.0
01
0.2
04
0.5 0/ 0.7
10
(8) DBLP
IMDB
92
B
HIGHF
Muro+1 Ml
Mupe1 F1
HEAD
91
0.2
Training time (per epoch) /
04
12
Figure 5. Micro-FI scores and time consumption on DBLP dataset.
0.0 0.1
0.2
0.3
07
08 09
0.0
0.4 055 DE
time complexity of the model.
(6) DBLP
(b) IMDH
Figure 4. Analysis of hyperparametes p and a on DBLP and IMDB datasets.
5.7. Complexity Analysis
We select seven heterogeneous graph self-supervised learn- ing methods and evaluate running time per training epoch on the DBLP dataset. According to the experimental result in Figure 5, we can see that HGOT has the best Micro-F1 score and requires the least training time, because it abandons the complex graph enhancement methods and the positive and negative sample selection process. The time complexity of the HGOT model mainly hinges on the optimization meth- ods used for Equations (15) and (16). For Equation (15), which includes the fused Waanoo-Wasssstteinn term, we use a conditional gradient (CG) solver for optimization. At each step of this solver, we need to calculate a gradient, and the time takes for this calculation grows roughly in cubic way the number of nodes in the graph increases. On the other hand, Equation (16), which contains the Wasserstein term, can be optimized using the Sinkhorn- Sinkhorn-Knopp algorithm. This algorithm is quite efficient because its time complexity grows almost quadratically with the graph size.
of the optimal transport plan alignment strategy. (3) The per- formance of HGOT-w/o#t is also very competitive, which demonstrates that the strategy of aggregating all semantic information makes relatively little contribution to the overall model.
5.6. Parameter Sensitivity
In this section, we investigate the sensitivity of hyperpa- rameters p and a on two datasets. The experimental results are shown in Figure 4. We firstly fix the value of param- eter a 0.5 and analyze the effect of parameter p on the prediction of performance. We can see that as the param- eter P increases, HGOT shows an overall growth trend on both datasets. HGOT achieves optimal performance when p == 1. However, without the implicit structure loss, HGOT performs not well, which is also shown in the ablation study section. In addition, we fix the value of parameter o to and analyze the effect of parameter a on the prediction of performance. When 0, HGOT only considers the matching information between edge structures, while when = 1, it only considers the matching information between nodes. Figure 4 reports the surprising outcome: With only node attributes for the calculation of the plan, the model achieves outstanding performance on both datasets. How- ever, with both nodes and edges, the result is not optimal. At the same time, our model HGOT performs not well when only edges are considered. These experimental results give us an inspiration: In heterogeneous information networks, the number of edges may far exceed the number of nodes, so we can appropriately abandon some edge information. In this way, performance can be optimized while reducing the
6. Conclusion
In this paper, propose a novel self-supervised heteroge- neous graph neural networks with optimal transport, named HGOT. The attention mechanism is employed to obtain an aggregated view which can integrate the semantic informa- tion from different meta-paths. Then, HGOT exploits the optimal transport theory to discover the optimal transport plan between the meta-path view and the aggregated view. By aligning the transport plans between graph space and representation sapce, HGOT enforces the backbone model to learn node representations that precisely preserve the matching relationships. Extensive experiments conducted on multiple datasets demonstrate the ofate-of-the-ar art perfor- mance of the proposed HGOT.

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
Acknowledgements
115.
This work supported in part by National Natural Sci- ence Foundation of China (No. 62371340, 62172052, 62322203), Tianjin Natural Science Foundation Project (No. 24IC2DIC00800, 23JCYBJC00520) Tianjin Measurement Science and Technology Project (NO.2024TJMTT061,
Ji Z, Kong D, Yang Y, et al. ASSL-HGAT: Active semi- supervised learning empowered heterogeneous graph at- tention network. Knowledge-Bad Based Systems, 2024, 290: 156-167.
Weihua Hu, Bowen Liu, Joseph Gomes, Marinka Zitnik, Percy Liang, Vijay S. Pande, and Jure Leskovec. 2020. Strategies for Pre-training Graph Neural Networks. In ICLR.
Impact Statement
This work provides a novel self-supervised learning ap- proach for heterogeneous graphs real-world scenario, and it aims to promote the development of heterogeneous graph neural networks in machine learning. There are many po- tential societal consequences of our work, none which we feel must be specifically highlighted here.
Wang, X.; Liu, N.; Han, H.; and Shi, C. 2021a. Self- supervised heterogeneous graph neural network with co- contrastive learning. In KDD.
Zhao Z, Zhu Z. Liu Y, al. Heterogeneous Graph Con- trastive Learning with Augmentation Graph. IEEE Trans- actions Artificial Intelligence, 2024.
References
Zhao M, Jia A L. Dahgn: Degree-aware heterogeneous graph neural network. Knowledge-Based Systems, 2024, 285: 135-145. 145.
Wang, X.; Ji, H.; Shi, C.; Wang, B.; Ye, Y.; Cui, P:; and Yu, P. 2019. Heterogeneous Graph Attention Network. In Www
Xie Y. Yu C, Jin X, et al. Heterogeneous graph contrastive learning for cold start cross-domain recommendation. Knowledge-Basse Based Systems, 2024: 2054-2065.
Bai, Y:; Ying, Ren, H.; and Leskovec, J. 2021. Modeling heterogeneous hierarchies with relation-specife hyper- bolic cones. In NeurIPS.
Liu Y, Xia L, Huang C. 2024. Selfgnn: Self Self-supervised graph neural networks for sequential recommendation. In SIGIR.
Cao, Y.; Peng, H.; Wu, J.: Dou, Y.: Li, J.; and Yu, S. 2021. wwwwledge-preserviing incremental social event detection via heterogeneous gnns. In www.
Chen L, Cao J, Liang W, et al. Geography-awaz Heteroge- neous Graph Contrastive Learning for Travel Recommen- dation. ACM Transactions on Spatial Algorithms and Systems, 2024.
Wang, H.; and Leskovec. J. 2020. Unifying graph con- volu tional neural networks and label propagation. arXiv preprint arXiv:2002.06755
Dong, Y. Hu, Z.; Wang, K.; Sun, Y:; and Tang. J. 2020. Heterogeneous Network Representation Learning. In IJCAL.
Yang. C.; Xiao, Y.; Zhang, Y.; Sun, Y.; and Han, J. Het- erogeneous Network Representation Learning: A Unifed Framework with Survey and Benchmark. IEEE Transac- tions on Knowledge and Data Engineering, 2020, 34(10): 4854-4873.
Tian, Y.; Zhang, C.; Guo, Z.; Huang, C.; Metoyer, R.;
and Chawla, N. V. 2022a. RecipeRec: A Heterogeneous Graph Learning Model for Recipe Recommendation, In IJCAI.
Wang, X.; Bo. D.; Shi, C.; Fan, S.; Ye, Y.; and Philip. S. Y. A survey on heterogeneous graph embedding: methods, techniques, applications and sources. IEEE Transactions on Big Data, 2022, 9(2): 415-436.
Bansal, T.; Juan, D.-C.; Ravi, S.; and McCallum, A. 2019. A2N: Attending neighbors for knowledge graph infer- ence. In Proceedings of the 57th annual meeting of the association for computational linguistics, 4387-4392.
Fan, Y.; Ju, M.; Zhang, C.; and Ye, Y. 2022. Heterogeneous temporal graph neural network. In SDM.
Wang, S.; Wei, X.; Nogueira dos Santos, C. N.; Wang, Z.;
Fu, X.; Zhang, J.; Meng, Z.: and King, I. 2020. Magnn: Metapath aggregated graph neural network for heteroge- neous graph embedding. In wwW.
Nallapati, R.; Arnold, A.; Xiang, B.; Yu, P. S.; and Cruz, 2021a. Mixed-curvature multi-relational graph neu- ral network for knowledge graph completion. In WWW, 1761-1771.
Zhang Q. Zhao Z, Zhou H, et al. Self-supervised contrastive learning on heterogeneous graphs with mutual constraints of structure and feature. Information Sciences, 2023, 640: 119026.
Sun Y, Zhu D, Wang Y, et al. GTC: -ransaaasfrrrer co- contrastive learning for self-supervised heterogeneous graph representation. Neural Networks, 2025, 181: 106-
Kipf, T. N.; and Welling, M. 2016. Variational graph auto-
10

---

Self-supervieed Heterogencous Graph Neural Network with Optimal Transport
encoders. arXiv preprini 1rXiv
auto-encoders. arXiv preprini arXiv:161 (2016).
Park. J.; Lee, M.; Chang, H. J.; Lee, K.; and Choi, J. Y. 2019. Symmetric graph convolutional autoencoder for unsupervised graph representation learning, In ICCV.
Petar Velickovic, William Fedus, William L. Hamilton, Pietro Liò, Yoshua Bengio, and R. Devon Hjelm. 2019.
Deep Graph Infomax. In ICLR.
Wang Z, Li Q. Yu D, et al. 2023. Heterogeneous graph contrastive multi-view learning. In SDM.
Perozzi, R. Al-Rfou, and S. Skiena, 2014. DeepWalk: Online learning social representations. KDD.
Tian Y, Dong Zhang C, et 2023. Heterogeneous graph
Y. Dong, N. V. Chawla, and Swami, 2017. Metapath2vee: Scalable representation learning for heterogeneous net- works. In KDD.
masked autoencoders. In AAAI.
Cédric Villani et al. Optimal transport: old and new, volume 338. Springer, 2009.
C. Shi, B. Hu, W. X. Zhao, and P. S. Yu. Heterogeneous information network embedding for recommendation. IEEE Trans. Knowl. Data Eng., vol. 31, no. 2, pp. 357-370, 2019.
Nicolas Bonneel, Michiel Van De Panne, Sylvain Paris, and Wolfgang Heidrich. Displacement interpolation using lagrangian mass transport. In Proceedings of the 2011 SIGGRAPH Asia Conference, pages 1-12, 2011.
Y. Ren, B. Liu, C. Huang, P. Dai, L. Bo, and J. Zhang, Heterogeneous deep graph infomax. 2019, arXiv: 1.Xi:/991.00836
Jiezhang Cao, Langyuan Mo, Yifan Zhang, Kui Jia, Chun- hua 20e9.2019. and Mingkui Tan. Multi-marginal wasser- stein gan. In NeurIPS.
R. Wang. C. Shi, T. Zhao, X. Wang, and Y. F. Ye, Heteroge- neous information network embedding with adversarial disentangler. IEEE Trans. Knowl. Data Eng., Jul. 13, 2021.
Xiang Gu, Yucheng Yang, Wei Zeng, Jian Sun, and Zongben Xu. 2022. Keypoint-guided optimal transport with appli- cations in heterogeneous domain adaptation. In NeurIPS.
Liu Y, Fan L, Wang X, al. HGBER: Heterogeneous graph neural network with bidirectional encoding repre- sentation. IEEE Transactions on Neural Networks and Learning Systems, 2023.
Facundo Mémoli. Cromov-wasserrterr distances and the metric approach to object matching. Foundations of Com- putational Mathematics, 2011, 11:417-487.
Vayer Titouan, Nicolas Courty, Romain Tavenard, and Rémi. 2019. Flamary. Optimal transport for structured data with application graphs. In ICMI..
Diederik Kingma and Jimmy Ba. 2015. Adam: A Method for Stochastic Optimization. In ICLR.
R. Xia, Y. Pan, L. Du, and J. Yin. 2014. Robust multi-view spectral clustering via low-rank and sparse decomposi- tion, In AAAL
Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan. 2022. Vayer, and Nicolas Courty. Semi- relaxed gromov-wasserstein divergence and applications on graphs. In ICLR.
Van Der Maaten L. Accelerating t-SNE using tree-based algorithms. The Journal Machine Learning Research, 2014, 15(1): 3221-3245.
Hongteng Xu, Dixin Luo, and Lawrence Carin. 2019. Scal- able grosov-wasserstiin learning for graph partitioning and matching. In NeurIPS.
Wang Y, Zhao Y, Wang D 7, et al. 2024. GALOPA: graph transport learning with optimal plan alignment. In NewrlPS.
N. Kipf and M. Welling. Semi-supervised clas- sification with graph convolutional networks, 2016, 1rXiv:1609.02900
P. Veli ckovi's c, G. Cucurull, A. Casanova, A. Romero, P. Lid, and Y. Bengio. Graph attention networks, 2017, arXiv: 1710110000.0003
William Hamilton, Zhitao Ying, and Jure Leskovec. 2017. Inductive Representation Learning on Large Graphs. In NeurIPS.
Thomas N. Kipf and Max Welling, 2016. Variational graph

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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
A. Datasets Details
The following four public datasets are used to verify the effectiveness of the proposed method.
DBLP: DBLP is an integrated database of English-language literature in the field of computer with the results of research as the author as the core. In this article, subset of DBLP which consists of 14 328 papers (P), 4057 authors(A), 20 conferences(C), and 8789 terms(7) is extracted. We divide the authors into four categories according their research areas: database, data mining, machine learning, information retrieval. Depending the conferences linked to the authors, we classify the authors into the appropriate research areas. The attribute information of authors is extracted from keywords. Three meta-paths, APA, APCPA, and APTPA are extracted for use with experiments.
ACM: ACM a paper network contrived from the ACM dataset using papers published in ACM Knowledge Discovery and Data Mining (KDD), SIGMOD, SIGCOMM, Mobi-COMM, and VLDB. ACM dataset consists of 3025 papers (P), 5835 authors (A), and 56 subjects (S). The papers are divided into three categories: database, wireless communication, data mining. Depending the conferences linked to the papers, we classify the papers into the appropriate categories. The attribute information of papers is extracted from keywords. Two meta-paths PAP, PSP are extracted for use with experiments.
IMDB: IMDB is a link dataset built with permission from the Internet Movie Data (IMDB). A subset of IMDB which contains 4780 movies (M), 5841 actors (A), and 2269 directors (D) is extracted. We divide the movies into three categories: Action, Comedy, Drama according to their genre. The attribute information of movies is extracted from plots. Two meta-paths MAM, MDM are extracted for use with experiments.
Yelp: The Yelp dataset contains 2614 businesses (B), 1286 users (U), two reviews (R), two services (S), and nine rating levels (L). The business nodes are labeled by their category. The node features are constructed by the bag-of-wordd words representation of the related keywords. Four meta-paths, BUB, BRB, BSB, and BLB, are extracted for use with experiments.
B. Dimension Analysis
We further perform dimension sensitivity analysis to show the impact of hidden dimensions in Figure 5. In particular, we search the number of hidden dimensions from {64, 128, 256, 512, 1024), shown the inflection point of the discount in the figure. It can be seen from the figure, the optimal hidden dimension can be different across different datasets. As shown in the Figure 6, the best hidden dimension on the DBLP dataset is 256, while on the IMDB is 512. In addition, as the hidden dimension increases, the model effect may be better. For example, when the hidden dimension only 64 dimensions, the model effect is not good, because using a small hidden dimension could prevent the model from fully capturing the knowledge. However, too wide a hidden dimension could decentralize the model's focus on meaningful information. Therefore, the model under the standard latent dimension will capture more complex and useful information, and the effect can be optimal.
100
100
std
90
Micro-Fl
80
Micro-F1
80
70
70
60
DBLP
std
50
50
200
400
600
800
1000
200
400
600
800
1000
Dimension
Dimension
(a)
Figure 6. Performance HGOT under different hidden dimensions.
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Self-supervised Heterogencous Graph Neural Network with Optimal Transport
C. General Settings
Environment. The environment in which we run experiments is:
Operating system: Microsoft Windows 11 64-bit
CPU information: 12th Gen Intel(R) Cor)(TM) 19-12900H (20 CPUs), 2.5GHz
GPU information: NVIDIA GeForce RTX 3070 Ti Laptop GPU
Resources. The address and licenses of all datasets are as follows:
DBLP: https://dbippuui-rrired.d
ACM: http://dl.acm.orgg
IMDB: http:///oomarixoor/aad////
Yelp: http://wwwwyell..om//attsst
13


Reviews:
[{"review_id": "hIOXwJS3GM", "reviewer": "Reviewer_xm87", "summary": "This paper proposes a novel self-supervised learning framework for heterogeneous graphs that leverages optimal transport theory to align meta-path views with an aggregated central view, eliminating the need for graph augmentation or explicit positive/negative sampling. The method achieves state-of-the-art performance on node classification, clustering, and visualization tasks across four real-world datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Theoretical Claims:\n\nQ: A concern is why the fused Gromov-Wasserstein distance is the optimal choice for heterogeneous graphs? A theoretical or empirical comparison with alternative OT metrics (e.g., Wasserstein barycenters) is missing.\nR: Thank for your comments. The fused Gromov-Wasserstein distance considers both nodes and edges, the information in the heterogeneous graph is more comprehensive. Therefore, compared with other OT distances, the fused Gromov-Wasserstein distance is the best choice for optimal transmission on heterogeneous graphs. We will present experimental analysis with other OT distance metrics in the final version.\n\nExperimental Designs Or Analyses:\n\nQ: My concern about this part is that the parameter sensitivity analysis (Section 5.6) lacks depth. For example, the claim that \"abandoning edge information improves performance” (p. 24) conflicts with the fundamental role of edges in graph learning. This requires further exploration (e.g., edge sparsity analysis).\nR: Thank you. The results of this experiment can be inferred that: Although the edge information on the graph is important, in some cases (redundant and complex edges), we may abandon the connection relationship of some edges and only consider the node feature information, which may lead to better results. We will discuss this issue more in the final version.\n\nOther Strengths And Weaknesses:\n\nQ: The method section (Section 4) lacks intuitive explanations. Visualizations of the OT alignment process would aid understanding.\nR: Many thanks. We will try to provide the visualization of the optimal transport to help understand the transporting and matching process in the final version.", "claims_and_evidence": "The claims made in the submission are well-supported by clear and convincing evidence.  The authors propose a novel self-supervised heterogeneous graph neural network with optimal transport (HGOT) and provide detailed algorithm design.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are appropriate and well-justified for the problem at hand.", "theoretical_claims": "The paper primarily focuses on methodological innovation and empirical validation rather than formal theoretical proofs. The authors provide detailed formulations of the optimal transport-based alignment mechanism and its integration into heterogeneous graph learning.  \n\nA concern is why the fused Gromov-Wasserstein distance is the optimal choice for heterogeneous graphs? A theoretical or empirical comparison with alternative OT metrics (e.g., Wasserstein barycenters) is missing.", "experimental_designs_or_analyses": "I have checked the soundness of the experimental designs and analyses, and they appear appropriate and valid. The authors evaluate HGOT on four widely used heterogeneous graph datasets (DBLP, ACM, IMDB, Yelp), covering diverse application domains. They compare against a comprehensive set of baselines, including both supervised and self-supervised methods, which ensures fair and relevant comparisons. \n\nMy concern about this part is that the parameter sensitivity analysis (Section 5.6) lacks depth. For example, the claim that \"abandoning edge information improves performance” (p. 24) conflicts with the fundamental role of edges in graph learning. This requires further exploration (e.g., edge sparsity analysis).", "relation_to_broader_scientific_literature": "The paper builds on and extends prior work in self-supervised heterogeneous graph learning and optimal transport. It addresses key limitations of existing contrastive self-supervised methods (e.g., HeCo, HDGI) that rely on graph augmentations and positive/negative sample selection, by introducing an optimal transport-based alignment mechanism.", "essential_references_not_discussed": "Most of key references are cited.", "other_strengths_and_weaknesses": "The method section (Section 4) lacks intuitive explanations. Visualizations of the OT alignment process would aid understanding.", "other_comments_or_suggestions": "No", "questions_for_authors": "Please refer to above sections.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "O4x8of4j8R", "reviewer": "Reviewer_LkeL", "summary": "This paper proposes a novel self-supervised heterogeneous graph neural network (HGOT), which aims at addressing the limitations of existing contrastive learning methods on heterogeneous graphs by combining Optimal Transport method. HGOT avoids data augmentation and the construction of positive and negative sample pairs, and proposes a new matching mechanism that performs optimal transmission matching between local (branch view) and global (center view) semantics to improve the quality of node representation. Extensive experiments on four public datasets have validated the effectiveness of HGOT.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Experimental Designs Or Analyses:\n\nQ1: The paper does not provide sufficient explanation or analysis for the selection of certain parameters (such as the σ and ρ parameters in optimal transport), particularly lacking a comprehensive discussion on the impact of different parameter values on model performance. As is well known, the training cost of deep learning is very high, and it is not feasible to manually adjust these hyperparameters through repeated experimentation. The paper needs to explain the specific values of these parameters (such as σ and ρ) and provide guidance for their reasonable selection in practical applications.\nR1：Thank you. In the parameter experiment in Section 5.6, we can see that the model performs better when the parameterρis larger, indicating the importance of implicit structural loss. Similarly, when the parameterσis larger, the model performs better, indicating that the transport information between nodes is more important in the fused Gromov-Wasserstein distance formula. We will explain the selection of these hyperparameters and discuss the impact of different parameter values on model performance in the final version.\n\nQ2: Due to the significant improvement in some metrics in this paper, it is recommended to provide significant analysis results such as t-test in Table 3.\nR2: Thank for your comments. We will add variance analysis to the clustering results experiment in Table 3 to demonstrate the stability of our method in the clustering experiment. \n\nQ3: Computational complexity: Solving fused Gromov-Wasserstein distances involves cubic time complexity in node count, limiting scalability to large graphs.\nR3: Thanks. Although the fused Gromov-Wasserstein distance requires more calculations, it does not take up too much memory or too much running time. In Appendix D, we give the complexity analysis. We employ the OT method instead of traditional contrastive learning to significantly reduce complexity. Consequently, compared to other self-supervised learning methods, our approach maintains a slight edge in terms of complexity.\n\nQ4: Meta-Path Dependency: Performance may hinge on pre-defined meta-paths, yet the paper does not explore automated meta-path selection or robustness to suboptimal choices.\nR4: Thank you. Inspired by your suggestion, we will discuss how to automatically select meta-paths or directly decompose the heterogeneous graph into several subgraphs to drop the dependence on meta-paths.\n\nOther Comments Or Suggestions:\n\nQ1: The abstract is overly verbose. For example, the implementation details of the method can be omitted. The abstract should clearly and concisely convey what the paper has done and what problem it addresses, rather than focusing too much on the specific details of the method. The author spends almost half of the abstract discussing the method's implementation, which makes the abstract appear redundant and detracts from its clarity and focus.\nR1: Many thanks. We will revise the abstract based on your valuable suggestions, and delete unnecessary parts and made the abstract more concise.\n\nQ2: The symbol definitions in the formulas are quite complex. The authors can add specific definitions for each symbol in the appendix and check if there are any inconsistent definitions (such as σ in Eq.13 and Eq.2).\nR2: Many thanks. According to theσsymbol in formula 2 and 13 you mentioned,σin formula 2 represents the activation function, andσin formula 13 is an adjustable hyperparameter. To avoid confusion, we will replace the activation function's σ with a capital Greek letter. In addition, we will give the detailed meaning of each symbol in the appendix in the final version.\n\nQ3: About parameter sensitivity: the experimental results shows that the parameter σ (node and edge weights) has a significant impact on the results, and the best effect is achieved when it only relies on node attributes, which may mean that it is lacking in processing edge information.\nR3: Thank you. Our experiments show that in some cases (edge information is redundant and complicated), we can appropriately abandon the connection of edges and only consider node features, which can reduce the complexity of the model. It will be clarified in the final version.\n\nQ4: Experiment incomplete: The proposed method is an algorithm for heterogeneous graphs, but only experiments on point classification and point clustering are conducted, without explanation for other graph problems such as link prediction.\nR4: Thanks. Inspired by your advice, we will study deeper into this issue and discuss in the final version.", "claims_and_evidence": "Yes.\n\nS1: Applying optimal transport to self-supervised learning on heterogeneous graphs is a unique contribution. This method helps bypass the challenges associated with graph augmentation and the selection of positive and negative samples in traditional contrastive learning.\n\nS2: HGOT optimizes the self-supervised learning strategy of heterogeneous graph neural networks from the perspective of information matching, avoiding the dependence of contrastive learning on data augmentation and positive negative sample pairs.", "methods_and_evaluation_criteria": "Yes.\n\nS3: The paper has complete experiments and visually demonstrates the effectiveness of HGOT.\n\nS4: This work integrates OT with heterogeneous graph learning, offering a principled alternative to contrastive SSL;", "theoretical_claims": "yes.", "experimental_designs_or_analyses": "Yes.\n\nS5: Achieves SOTA results on node classification (6%+ accuracy improvement) and clustering across four datasets, demonstrating robustness.\n\nW2: The paper does not provide sufficient explanation or analysis for the selection of certain parameters (such as the σ and ρ parameters in optimal transport), particularly lacking a comprehensive discussion on the impact of different parameter values on model performance. As is well known, the training cost of deep learning is very high, and it is not feasible to manually adjust these hyperparameters through repeated experimentation. The paper needs to explain the specific values of these parameters (such as σ and ρ) and provide guidance for their reasonable selection in practical applications.\n\nW3: Due to the significant improvement in some metrics in this paper, it is recommended to provide significant analysis results such as t-test in Table 3.\n\nW5: Computational complexity: Solving fused Gromov-Wasserstein distances involves cubic time complexity in node count, limiting scalability to large graphs.\n\nW6: Meta-Path Dependency: Performance may hinge on pre-defined meta-paths, yet the paper does not explore automated meta-path selection or robustness to suboptimal choices.", "relation_to_broader_scientific_literature": "S6: Eliminates cumbersome graph augmentation and sample selection, simplifying self-supervise training pipelines.", "essential_references_not_discussed": "N.A.", "other_strengths_and_weaknesses": "Please refer to the above comments.", "other_comments_or_suggestions": "W1: The abstract is overly verbose. For example, the implementation details of the method can be omitted. The abstract should clearly and concisely convey what the paper has done and what problem it addresses, rather than focusing too much on the specific details of the method. The author spends almost half of the abstract discussing the method's implementation, which makes the abstract appear redundant and detracts from its clarity and focus.\n\nW4: The symbol definitions in the formulas are quite complex. The authors can add specific definitions for each symbol in the appendix and check if there are any inconsistent definitions (such as σ in Eq.13 and Eq.2)\n\nW7: About parameter sensitivity: the experimental results shows that the parameter σ (node and edge weights) has a significant impact on the results, and the best effect is achieved when it only relies on node attributes, which may mean that it is lacking in processing edge information\n\nW8: Experiment incomplete: The proposed method is an algorithm for heterogeneous graphs, but only experiments on point classification and point clustering are conducted, without explanation for other graph problems such as link prediction.", "questions_for_authors": "Please refer to the above comments.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4U0x5LmiG6", "reviewer": "Reviewer_nStG", "summary": "This paper presents HGOT, a self-supervised heterogeneous graph neural network that harnesses optimal transport theory to establish an optimal transport plan between the meta-path and aggregated views. By compelling the model to learn node representations that faithfully preserve the intrinsic matching relationships between these two views, HGOT significantly enhances the quality of representation learning. Comprehensive experiments conducted on multiple benchmark datasets validate the superiority of the proposed approach, consistently achieving state-of-the-art performance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Other Strengths And Weaknesses:\n\nQ: However, the complexity of Gromov-Wasserstein optimization raises scalability concerns for large graphs. The paper lacks a discussion on runtime and memory overhead, which are crucial for practical deployment. Furthermore, interpretability could be improved with case studies or attention weight analysis.\nR: Thank for your comments. We discuss the complexity issue in Appendix D, and we will move this section to the main text in the final revision of the paper. On the other hand, we will provide a discussion of running time and memory overhead, and perhaps we will provide the result in the form of charts in the experiments.\n\nOther Comments Or Suggestions:\n\nQ: In the caption of figure 3, explanations of different ablation settings could be provided for better readability.\nR: Many thanks. We will provide the detailed settings of the ablation experiments in the caption of figure 3 in the final version.\n\nQuestions For Authors:\n\nQ1: In the ablation study (Section 5.5), the parameter sensitivity analysis indicates that outperforms the combined node-edge alignment. Does this suggest that incorporating edge information may introduce noise or redundancy in certain scenarios? Furthermore, would this effect vary depending on the dataset characteristics?\nR1: Thank for your comments. It can be inferred from the experimental results: In some cases (edge information is redundant and complicated), we can appropriately abandon the connection of edges and only consider node features. Too much edge information may bring noise and redundancy, the experimental results have verified this to a certain extent.. In addition, we only shows results on two datasets, but the same results are obtained in the other two datasets. It will be clarified in the final version.\n\nQ2: More intuitive explanations are expected to intuitively understand the effectiveness of equations 13-16. Could the authors provide further insights or illustrative examples to clarify the reasoning behind their derivations?\nR2: Thank you. Formula 13 is the fused Gromov-Wasserstein distance (considering both nodes and edges) commonly used for OT. Formula 14 is a shorthand for the right side of the equal sign in Formula 13. Formulas 15 and 16 are used to calculate the optimal transport plan Π from the corresponding OT distances. We will provide a more detailed explanation of the formula in the final version.", "claims_and_evidence": "The paper claims the integration of optimal transport with heterogeneous graph neural networks (HGNNs) is novel. Specifically, the alignment between graph-space and representation-space transport plans provides a fresh perspective for self-supervised learning, circumventing the limitations of traditional contrastive learning. Experimental results validate the effectiveness of the proposed method, showing state-of-the-art performance on benchmark datasets.", "methods_and_evaluation_criteria": "This paper formulates the alignment problem using Gromov-Wasserstein optimal transport and employs a self-supervised learning framework to refine node representations. The proposed method is technically sound.\n\nThe evaluation metrics used to benchmark HGOT’s performance include accuracy and clustering-based evaluation on multiple real-world heterogeneous graph datasets. The evaluation criteria are widely used and appropriate.", "theoretical_claims": "This work provides mathematical definitions of heterogeneous graphs and optimal transport. The proposed method itself has no theoretical claims.", "experimental_designs_or_analyses": "The proposed method is evaluated on four real-world datasets under different tasks, including node classification, node clustering, and visualizations. The experimental designs and performance comparisons are sound.", "relation_to_broader_scientific_literature": "This work may be helpful for researchers interested in molecule structure learning.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "One of the key strengths of HGOT is its novel integration of OT with HGNNs, providing an alternative to contrastive learning. However, the complexity of Gromov-Wasserstein optimization raises scalability concerns for large graphs. The paper lacks a discussion on runtime and memory overhead, which are crucial for practical deployment. Furthermore, interpretability could be improved with case studies or attention weight analysis.", "other_comments_or_suggestions": "In the caption of figure 3, explanations of different ablation settings could be provided for better readability.", "questions_for_authors": "In the ablation study (Section 5.5), the parameter sensitivity analysis indicates that outperforms the combined node-edge alignment. Does this suggest that incorporating edge information may introduce noise or redundancy in certain scenarios? Furthermore, would this effect vary depending on the dataset characteristics?\n\nMore intuitive explanations are expected to intuitively understand the effectiveness of equations 13-16. Could the authors provide further insights or illustrative examples to clarify the reasoning behind their derivations?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YygpYMJo70", "reviewer": "Reviewer_4Sje", "summary": "The paper proposes a self-supervised heterogeneous graph neural network (HGNN) coined ``HGOT''. It first incorporates optimal transport (OT) into heterogeneous graphs to better facilitate the learning of a more semantic accurate similarity measure between graph instances and structure. The method introduces three components: node feature transformation, which projects different types of node features into one latent space; aggregated view generation, which constructs multiple meta-path-based views and an aggregated view on the entire graph; and multi-view OT alignment, which calculates the optimal transport plan and minimizes the difference between views and costs between feature and structural representations.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "R(1): Different from other self-supervised learning methods, optimal transport (OT) can capture the matching information from the original graph space to the representation space, obtaining node representations that exhibit consistency with the optimal transport plans. Second, reconstruction-based methods usually involve mask-reconstruction mechanisms, which require a certain degree of damage to the input information. While our method directly extracts useful supervisory signals for matching information based on the characteristics of the heterogeneous graph itself.\nR(2): The Wasserstein distance is a metric used to quantify the difference between nodes in the graph, measuring the minimal cost required to transform one distribution into another. The Gromov-Wasserstein distance is a metric used to quantify the difference between edges in the graph. The fused Gromov-Wasserstein distance is the fusion of above two distances through a trade-off parameter to achieve optimal transport over the entire graph. The reasons for choosing CX and CA as the cost matrix are as follows: The cosine similarity (CX) is popularly used in most work. The absolute adjacency matrix difference (CA) is convenient to reduce the computational complexity for the calculation of four-dimensional tensors. \nR(3): The aggregated view, as a center view containing all semantic information, enables each branch view to capture the transport relationship between semantic information through optimal transport. By transporting each branch view to the center view, the transport relationship from each semantic to the comprehensive semantic is obtained, which can better promote heterogeneous representation learning. In summary, although the contribution of this part to the overall model is not very large as shown in the ablation study results, the performance of the algorithm is still improved, and the aggregated view plays an important role in the optimal transport process.\n\n\nR(1): By using node feature transformation, the feature dimensions of all nodes are unified, which facilitates subsequent optimal transport calculations. Different from HeCo and HAN, which only use the attention mechanism to aggregate node-level representations, we aggregate nodes and edges separately to obtain a complete aggregated graph structure. Then, the relationship between the aggregated view and each meta-path view is obtained through optimal transport to promote heterogeneous representation learning.\nR(2): Heterogeneous graph contains different semantic information. Meta-paths are used to capture these semantic information in our work. By using OT, we can capture the optimal transport plan from each meta-path view to the aggregated view, and calibrate the encoder by aligning each plan to obtain a better node representations. Therefore, OT promotes heterogeneous representation learning and captures rich semantic information in heterogeneous graphs. It will be clarified in the final version.\n\nTheoretical Claims: We will discuss the connection between OT and other self-supervised learning objectives such as the InfoMax principle in the final version. \n\n\nExperimental Designs Or Analyses: We will make revisions based on your comments. It is worth mentioning that, different from COLES, HGOT uses OT theory to capture rich information in heterogeneous graph to obtain better node embedding. Our method does not require consideration of complex data enhancement and the selection of positive and negative samples.While COLES is based on the Laplace eigenmap method, it still requires the selection of positive and negative samples. \n\n\nSupplementary Material~Other Comments Or Suggestions: We will make revisions based on your comments.\n\nQuestions For Authors:\nThe responses to the four questions you raised are as follows: (1)We will explain the three OT formulas in more details in the final version and will give the relevant definitions in the introduction, and add several references into this section. Formulas (8)-(13) are formally the original definitions of these distances. We  modified them according to some related works (only some symbolic definitions in the formulas have been modified, so the complete physical meaning remains unchanged). The symbol ⊗ is defined for the Hadamard product (element-wise multiplication of matrices), which will be added in the final version. (2)This is a mistake in our writing. We meant to say \"we first project the node features in the heterogeneous graph into the same feature space\". The node feature X is defined as the original feature of the node in the heterogeneous graph. We will make the corresponding changes in the final version. (3)In our experiment, we divide each dataset into 8 (training set): 1 (validation set): 1 (test set) and report the results. Same as HeCo, we also conducted experiments with other ratios, but they were not presented in the paper. We will add them into the final version.\n(4)We will open source our code in the future.", "claims_and_evidence": "Most parts of the paper are clearly written and easy to read, despite some grammatical errors. The authors have clearly demonstrated the semantic advantage of local-global alignment between the meta-path view and the aggregated view in their proposed method. However, I have some concerns about the clarity and persuasiveness of the methodology section.\n(1) The authors do not explicitly clarify the significance of using OT for graph self-supervised learning. What are the advantages of calculating the distribution distance between views compared to other graph self-supervised learning methods? For instance, why is the OT-based approach superior to reconstruction-based methods (e.g., HGATE [1], HGMAE [2], RMR [3]) when both of them do not require augmentation methods and negative sampling?\n(2) The explanation of the OT distance is unclear. Authors should provide an intuitive explanation for the OT distances designed for feature and edge information, and it would be better with theoretical or empirical comparisons to other metrics. Additionally, a more convincing analysis for choosing cosine similarity and absolute adjacency matrix difference as $\\mathcal{C}_X$ and $\\mathcal{C}_A$ is expected. \n(3) The ablation study results cannot validate the effectiveness of the aggregated view, as it only provides relatively little contribution to the overall model.\n\n[1] Wang, W., Suo, X., Wei, X., Wang, B., Wang, H., Dai, H. N., and Zhang, X. 2021. HGATE: heterogeneous graph attention auto-encoders. TKDE.\n[2] Tian, Y., Dong, K., Zhang, C., Zhang, C., and Chawla, N. V. 2023. Heterogeneous graph masked autoencoders. In AAAI.\n[3] Duan, H., Xie, C., and Li, L. 2024. Reserving-masking-reconstruction model for self-supervised heterogeneous graph representation. In KDD.", "methods_and_evaluation_criteria": "On the methodology side, I believe that the proposed method is an application of existing self-supervised learning techniques to heterogeneous graphs, rather than presenting innovative insights or designs.\n(1) Node feature transformation and attention-based meta-path representation aggregation are very common practices in HGNNs, as seen in HAN [1] and HeCo [2].\n(2) Self-supervised learning using Wasserstein distance across different contrastive views has already been proposed in COLES [3].\nAlthough the authors claim that they are the first to apply OT to heterogeneous graphs, they do not explain or analyze how OT facilitates the learning of heterogeneous representations. Instead, heterogeneous information is only considered in feature transformation and the construction of contrastive views, while the OT-based loss itself seems unrelated to heterogeneous graph learning. For example, OT could have been used to compute the distance between different node/edge type distributions to facilitate the learning of cross-type heterogeneous representations. However, this paper only focuses on learning the distribution distances of node features and adjacency matrices under the homogeneous transformation, which raises doubts about the necessity of the OT strategy for HGNNs. Therefore, I find the innovation in this paper to be limited.\nOn the evaluation side, the authors selected four heterogeneous benchmark datasets and equipped each downstream task with multiple evaluation metrics. The overall evaluation setup is sound to me except for the absence of dataset split information.\n\n[1] Wang, X., Ji, H., Shi, C., Wang, B., Cui, P., Yu, P., and Ye, Y. 2019. Heterogeneous graph attention network. In WWW.\n[2] Wang, X., Liu, N., Han, H., and Shi, C. 2021. Self-supervised heterogeneous graph neural network with co-contrastive learning. In KDD.\n[3] Zhu, H., Sun, K., and Koniusz, P. 2021. Contrastive Laplacian Eigenmaps. In NeurIPS.", "theoretical_claims": "The  theoretical analysis is insufficient. I suggest that the authors include some theoretical analyses of the OT-based learning objective, including but not limited to the connection with other self-supervised objectives such as the InfoMax principle, generalization bounds, and cross-domain transferability. Particularly, contributions of OT in heterogeneous representation learning would be analyzed to support the claims and establish a solid theoretical foundation for the proposed method.", "experimental_designs_or_analyses": "I appreciate that the authors conducted experiments on both node classification and node clustering to demonstrate the generalization performance of HGOT across different scales of graph structure. \nDespite that HGOT achieves significant performance improvements on all datasets and multiple evaluation metrics, I believe that its performance does not reach the state-of-the-art claimed by the authors, as only one baseline method published after 2020 (HGBER) is included. More recent works should be incorporated.\nBesides, the paper fail to provide an in-depth empirical analysis of HGOT's advantages. Some additional experiments need to be included:\n(1) More cutting-edge reconstruction-based and attention-based methods should be added as baselines, such as those mentioned in Claims and Evidence [1], [2], [3]. Considering the similarity between HGOT and COLES [4], the authors should also elaborate on the difference and advantages of the proposed OT loss over COLES by comparative experiments.\n(2) Since OT plays a positive role in domain adaptation, I recommend that the authors design transfer learning-based experiments to evaluate HGOT's transferability across different data domains or to out-of-distribution samples.\n(3) Considering the complexity of calculating distribution distances for large-scale networks, the efficiency and scalability of HGOT are not guaranteed, considering that it utilizes multiple meta-path views during training. Although the authors provide a rough efficiency analysis of HGOT in Appendix D, I suggest they further present a complexity comparison and efficiency evaluation of HGOT against other self-supervised models. This would provide a clearer validation of HGOT's efficiency without augmentation methods and negative samples.\n\n[1] Wang, W., Suo, X., Wei, X., Wang, B., Wang, H., Dai, H. N., and Zhang, X. 2021. HGATE: heterogeneous graph attention auto-encoders. TKDE.\n[2] Tian, Y., Dong, K., Zhang, C., Zhang, C., and Chawla, N. V. 2023. Heterogeneous graph\nmasked autoencoders. In AAAI.\n[3] Duan, H., Xie, C., and Li, L. 2024. Reserving-masking-reconstruction model for self-supervised heterogeneous graph representation. In KDD.\n[4] Zhu, H., Sun, K., and Koniusz, P. 2021. Contrastive Laplacian Eigenmaps. In NeurIPS.", "relation_to_broader_scientific_literature": "Self-supervised learning on heterogeneous graphs is one of the key approaches for exploring heterogeneous data structure. This field can provide meaningful insights for building graph foundation models across different graph types in the future.", "essential_references_not_discussed": "See Claims And Evidence & Experimental Designs Or Analyses.", "other_strengths_and_weaknesses": "See other parts of the review.", "other_comments_or_suggestions": "(1) The references format is inconsistent, making the citations difficult to retrieve.\n(2) The mixed use of certain symbols, such as $\\sigma$ and $\\mathcal{E}$, can lead to confusion.", "questions_for_authors": "(1) The authors should provide clear definitions of Wasserstein distance, Gromov-Wasserstein (GW) distance, and fused GW distance in the preliminary section, along with appropriate citations, to help readers unfamiliar with OT. Are eq(8)--(13) the original definitions of these distances? What does the symbol $\\otimes$ represent?\n(2) Caption of Figure 1 \"We first transform all nodes of the original heterogeneous graph into node features\" is ambiguous. Does HGOT utilize special encoding methods to generate node features $X$?\n(3) HeCo conducted experiments under different dataset splits. What is the dataset split in this work?\n(4) Will the code implementation of HGOT be open-sourced?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes a Heterogeneous graph neural network with Optimal Transport (HGOT), a self-supervised learning model without graph augmentation. It employs optimal transport theory to align meta-path views with an aggregated central view. It avoids the construction of positive and negative sample pairs. Experimental evaluations show its high performance, but can be strengthened.
Decision: Accept (poster)