Graph-Theoretic Insights into Bayesian Personalized Ranking for Recommendation
Authors: Kai Zheng
Jianxin Wang
Jinhui Xu
Venue: NeurIPS 2025 poster ()
Keywords: Graph self-supervised learning
Bayesian Personalized Ranking
latent hyperbolic geometry
loss function
Network geometry
Primary area: deep_learning
TL;DR: This paper introduces BPR+, a novel loss function for graph self-supervised learning that extends Bayesian Personalized Ranking (BPR) by incorporating even-hop paths to better capture global connectivity and topological structure.

Abstract: Graph self-supervised learning (GSL) is essential for processing graph-structured data, reducing the need for manual labeling. Traditionally, this paradigm has extensively utilized Bayesian Personalized Ranking (BPR) as its primary loss function. Despite its widespread application, the theoretical analysis of its node relations evaluation have remained largely unexplored. This paper employs recent advancements in latent hyperbolic geometry to deepen our understanding of node relationships from a graph-theoretical perspective. We analyze BPR’s limitations, particularly its reliance on local connectivity through 2-hop paths, which overlooks global connectivity and the broader topological structure. To address these shortcomings, we purpose a novel loss function, BPR+, designed to encompass even-hop paths and better capture global connectivity and topological nuances. This approach facilitates a more detailed measurement of user-item relationships and improves the granularity of relationship assessments. We validate BPR+ through extensive empirical testing across five real-world datasets and demonstrate its efficacy in refining graph self-supervised learning frameworks. Additionally, we explore the application of BPR+ in drug repositioning, highlighting its potential to support pharmaceutical research and development. Our findings not only illuminate the success factors of previous methodologies but also offer new theoretical insights into this learning paradigm.

Graph-Theoreti Insights into Bayesian Personalized Ranking for Recommendation
Kai Zheng Jianxin Wang'2; Xu45,-
School of Computer Science and Engineering, Central South University
Changsha 410083, China
The Hunan Provincial Key Lab of Bioinformatics, Central South University
Changsha 410083, China
The College of Computer Science and Technology, University of Petroleum Qingdao 266580, China
School of Information Science and Technology. University of Science and Technology of China HeFei, 230026, China
Institute of Artificial Intelligence, Hefei Comprehensive National Science Center
HeFei, 230026, China
jxwang@mail csu. edu. en, jhxu@ustc. eet.easttteeeec..o.mo.
Abstract
Graph self-supervised learning (GSL) is essential for processing graph-structured data, reducing the need for manual labeling. Traditionally, this paradigm has exten- sively utilized Bayesian Personalized Ranking (BPR) as its primary loss function. Despite its widespread application, the theoretical analysis of its node relations eval- uation have remained largely unexplored. This paper employs recent advancements latent hyperbolic geometry to deepen our understanding of node relationships from a graph-theoretical perspective. We analyze BPR's limitations, particularly its reliance on local connectivity through 2-hop paths, which overlooks global connectivity and the broader topological structure. To address these shortcomings, we purpose novel loss function, BPR+, designed to encompass even-hop paths and better capture global connectivity and topological nuances. This approach facilitates a more detailed measurement of user-item relationships and improves the granularity of relationship assessments. We validate BPR+ through extensive empirical testing across five real-world datasets and demonstrate efficacy in refining graph self-supervised learning frameworks. Additionally, we explore the application of BPR+ in drug repositioning, highlighting its potential to support pharmaceutical research and development. Our findings not only illuminate the success factors of previous methodologies but also offer new theoretical insights into this learning paradigm.
Introduction
Self-supervised learning (SSL) is revolutionizing deep learning by significantly reducing the reliance on manually annotated labels (I] This approach has gained considerable traction, especially in the application to graph-structured data, referred to as graph self-supervised learning (GSL) [2]. GSL exploits the unique, complex topological structures of nodes (non-Euclidean data space) generate embeddings (2 This contrasts sharply with the more straightforward Euclidean spaces typical of image and language data, presenting unique challenges and opportunities for innovation [[]
"Corresponding authors
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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In GSL, designing an effective loss function capable of navigating the irregularities of graph data is crucial. Bayesian Personalized Ranking (BPR) has been widely adopted in GSL methods to address this need [3]. Originally formulated to solve the personalized ranking problem in recommendation systems, BPR assesses item rankings for a user based on their historical interactions [ The BPR loss function measures the relationships between users and items by calculating the dot product of their embeddings, differentiating connected pairs (positive samples) from unconnected pairs (negative samples) in the graph [5].
Despite BPR loss widespread use, the theoretical analysis of its relations evaluation within GSL remains a relatively unexplored area. In this paper, we analyze the physical underpinnings behind BPR loss from a graph theoretical perspective. Notably, based on the formula for BPR loss, the score yus can be interpreted the computation of elements in the matrix EE" specifically Vuui (EE' )us- If we consider E as an embedding network comprising users, items, and abstract nodes, then Yui can be understood as a statistic of the 2-hop paths from user u to item within this network. However, these 2-hop paths only capture local connectivity information. This coarse- caasse-rriied assessment approach brings about two issues. Firstly, it lacks precision in measuring topological similarity, resulting in single score corresponding to wide range of topological similarities (e.g., from 20% to 50%). Secondly, these scores are influenced by the norm of embedding vectors, higher norm correspond to higher scores with other embeddings.
To address these limitations, it seems intuitive to extend our analysis include longer n-hop paths. However, the selection of these paths and the quantification of their informational content pose formidable challenges. Previous study has shown that when the graph is sufficiently large, 2-hop paths approximate the concept of energy distance defined in latent hyperbolic geometry [6], derived using the principle of maximum entropy 77. This measures the complement of the ratio of the latent distance between nodes in latent hyperbolic space to their chemical potential (a function of expected degree). Previous work indicates that as a measure of energy distance, 2-hop paths are not sufficiently precise. Consequently, we introduce the TopoLa distance to measure energy distance between nodes and, based on this, propose the BPR+ loss. We demonstrate that BPR+ loss more accurately measures the topological structure similarity and global connectivity information between users and items in the embedding network.
In addition to improve BPR loss, the introduction of network geometry (latent hyperbolic geometry) allows us to analyze common graph convolution operations from a new perspective and explain their effectiveness [6]. This not only clarifies the reasons behind the success of previous graph self-supervised learning but also provides a fresh perspective for the theoretical analysis of this learning paradigm. Extensive empirical tests of five existing methods on five real datasets confirm our theory and demonstrated the effectiveness of our new approach. Additionally, we apply BPR+ to the field of pharmaceutical research and development, developing drug repositioning framework, TopoDR, and selecting ten potential therapeutic candidates for four prevalent cancers: colorectal, breast, stomach cancer, and leukemia.
Contributions and paper structure. We analyze Bayesian Personalized Ranking (BPR) loss from a graph-the -hepeetceeeticca perspective, with a particular focus on the physical significance and its impact on embedding representations. In this process, we demonstrate that the measurement of relationships between users and items can be transformed into path problem in graph theory, allowing to use network geometry to aid the analysis. The main contributions are as follows:
Demonstrate the limitations of BPR loss through latent hyperbolic geometry and provide relevant proof's (Section 3).
Explore the effectiveness of common graph convolution operations in existing graph neural networks from the perspective of ropology-encodes latent hyperbolic geometry topologically (Section 4.
Propose novel loss function, BPR+, as an alternative (Section (5)
Develop an approximation technique based on numerical analysis to optimize the computation of loss. (Section 6)
Apply BPR+ to the field of pharmaceutical research and development, introducing a novel computa-
tional framework for drug repositioning (Appendix F).
2

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Preliminaries
Bayesian Personalized Ranking. BPR is pairwise loss designed for one-class collaborative filtering 3. The objective function of BPR encourages higher predictions for observed entries relative to their unobserved counterparts, formulated follows:
M Σ In (Uui - Ouj) + |
LBPR- = Σ ΣΣ> M=LiENu J&N
where Ny. represents the set of items associated with user u. The parameter is utilized to regulate the intensity of L2 regularization. M denotes the number of users. The variable Jai is obtained through the product of the embeddings of user u and item that is Dui e.e,
3 The limitation of BPR loss
In the Bayesian Personalized Ranking (BPR) loss, the score Dus between the user u and the item i is derived from the dot product of the embeddings of user and item i. We find that computing Iis is equivalent to calculating the Gram matrix EET of the embedding matrix E. In graph theory, the Gram matrix EE represents the number of 2-hop paths between user and item During Dos computation, we model matrix E graph comprising users, items, and N. abstract nodes (Figure 1).
original network
embedding network
Figure The illustration embedding network.
Thus, graph self-supervised learning can be understood as the process of constructing embed- ding network with a specified number of abstract nodes along with users and items to preserve the information of the original network. The weights of the edges in this network represent the relationships between the abstract nodes and other elements in the network. In BPR loss, Dus evaluates the interaction between user and item by assessing 2-hop paths via abstract nodes (Figure 2). Krioukov has demonstrated through the principle of maximum entropy that the counting 2-hop paths positively correlates with the complement of the energy distance between nodes in the latent hyperbolic [7]. However, Previous research indicates that relying solely on this local connectivity information is insufficient for accurately measuring the energy distance between nodes in latent hyperbolic space, as stated in Theorem (Appendix A). According to Theorem 1, we demonstrate that there exists more precise method for measuring energy distance, which surpasses the method of counting 2-hop paths.
E
ET
EET
Yui = (EET) Jut
=eeee
#2
represents the statistics of 2-hop paths from node u node
e;
(EET) Ju
embedding network
Figure The physical significance of the Bayesian Personalized Ranking (BPR) loss.

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Theorem 1. Given a network X E RRi+tt)XN,, let (t(x)) denote the expected number of triangles, and (+'()) represent the expected number of weighted triangles. Consequently, a' a, where a' is the logarithm of the thermodynamic activity corresponding to (('( and a is associated with (x))).
Analyzing common graph convolution operations using network geometry
As shown in the previous section, the embedding matrix can be regarded an embedding network. Consequently, we can utilize network geometry to analyze common graph convolution operations and explain their effectiveness. We use lightGCN as an example, a widely utilized foundational module.
Light Graph Convolution (LGC), Within the lightGCN, LGC is designed to focus on the funda- mental elements of graph convolution, specifically information propagation and aggregation, while forgoing superftuous feature transformations and nonlinear activation functions. The operation streamlines the graph convolution process, enhancing the efficiency. The formulation of LGC is defined as follows:
(k+1)) E e
(R)
(k+1)
Σ WEN, VINTIVING
e
(K)
(2)
REN VINilVIW
where the symmetric normalization factor
v Vial in line with the conventional Graph Convolu-
tional Network (GCN) architecture, is employed to curb the increase in embedding scale ensuing from successive graph convolution operations. The embedding of the k-th layer can be calculated as follows:
( 0:k AD- AD E
where 0g 2 0 denotes the importance of the embeddings from the k-th layer in constituting the final embedding. D E 2( represents the degree matrix, where D indicates the degree of the i-th node in the adjacency matrix A. From graph theory perspective, the k-th layer of graph convolution symbolizes the single-step diffusion of the ID embedding ( which is influenced by k-hop paths connecting user and item within A. The symmetric normalization factor weights the statistics of k-hop paths to address their exponential increase.
Layer Combination. In the lightGCN architecture, the sole trainable elements are the ID embeddings: (0) Ca and cQ") Post-initialization, embeddings obtained at k-th layer caleulated via LGC, defined in Equation (3). Subsequently, the final network embedding are constructed through a weighted summation of embeddings from each layer:
K
Cu =
a
(N)
(4)
km
A=
where K is the number of layers. The matrix form of layer combination can be defined as:
E = 000 EgEE (0)
01 (1) + 02 E(2) +...
E(() E(K)
(5)
E (0) +a + AE
+ A - ( + +.... K E
where D is the symmetrically normalized matrix. From graph theory perspective, there are three reasons why layer combination is effective: (1) With an increase in hops, the variance in the number of paths between nodes diminishes, leading to reduction in the disparity between layer embeddings and explaining the tendency for over-smoothing at higher layers. (2) As demonstrated in
previous work, path counts encapsulate information about node degrees and topological similarities between nodes. However, the extent to which these factors are reflected varies with the hop count, and fusing embeddings from multiple layers effectively captures this nuanced information. (3) This

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operation has the same effect self-connectin in graph convolution. Essentially, self-comnection adds a path connecting each node itself.
5 The Improved Bayesian Personalized Ranking (BPR+) loss.
In the graph self-supervised learning (GSL) framework, abstract nodes are introduced to reconstruct the original network, preserving existing node relationships. The Bayesian Personalized Ranking (BPR) loss evaluates user-item relationships in this restructured network based on 2-hop path statistic. Previous work has demonstrated that 2-hop paths serve as approximate representations of energy distance in the latent hyperbolic space 8 Our research, based on the maximum entropy principle, demonstrates that weighting 2-hop paths aligns more closely with the actual energy distance, revealing BPR loss's s limitations. This led to the introduction of TopoLa distance, defined as follows:
dropo (u, i) = 12 -
X hhol + A 6 - hop
(6)
where À is the scale factor. The BPR+ loss (Figure can be represented
chepo (u, - chopo (u.j) (u.
LBPR+ - Σ E
Σ In
||
NEN JEN
This incorporation of all even-hop paths offers two advantages: (1) global information is introduced to amalgamate relational information of all users and items in the network; (2) The TopoLa distance captures the topological structure information between and items (Theorem and Theorem 3).
See Appendix and C for details.
Theorem 2. Given a user embedding E RN,, the embedding matrix E E x No, and a parameter X, the following optimal solution to the problem is denoted by et in vector form:
min 1 fly cEI? + ||c1
(8)
We have
| -o--cllle
- -
(9)
1
Theorem Given nodes and j. the topological similarity between them is directly proportional to the dopo value.
Clape (1t, i) 7 hopl
1
hopl + + 16- hopl
(1i Bayp (2, t E (enteemeeet 2
LapRe ΣΣN
embedding
n-hop path
IEN.
Figure 3: The illustration of BPR+ loss.
Optimizing loss computation
Direct computation of the loss using the TopoLa distance from the previous section, which involves multiple matrix multiplications, is computationally inefficient. address this, we have optimized the computational approach to enhance efficiency. The matrix power form of the TopoLa distance can be defined as:
1 +
Diopo
2
H
H
(10)
X2
13
A
5

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where H = EE' is the 2-hop path statistic matrix of E. duoo( 1) is an element of D Diopo- denoted as (Diopo .us = dopoo u, t). Due to the necessity of multiple matrix multiplications, the time complexity for calculating matrix Dupo is O (NVz) .N is the batch size. To enhance computational efficiency, we employ matrix decomposition techniques. Firstly, the embedding matrix E undergoes singular value decomposition (SVD), represented E UEV Consequently, the formula for Dupo can be expressed as:
Popo = U -U
-
1 A
Σ
U..
(11)
X
As result, the time complexity is reduced to O(N,N2 + NĀN, + N2). No is the embedding size. Given that GSL models, No significantly exceeds Ne, the proposed calculation form offers a lower time complexity.
Experiments
To confirm the superiority and efficacy of the proposed BPR+ loss, we carried out extensive experi- ments to explore the following research questions:
RQ1: How much improvement GSL models?
proposed loss bring when applied various state-of-the-art
RQQ2: How do the physical properties of our proposed loss differ from those of Bayesian Personalized Ranking (BPR) loss?
RQQ: How can numerical analysis enhance the efficiency of loss computation?
.RQ4: How does proposed technique perform in practical applications?
7.1 Experimental Settings
Evaluation Datasets. In this section, we evaluate our model and several baselines five real-world datasets: Amazon, Gowalla, Yelp. LastFM, and Beer. The Amazon dataset includes implicit feedback from users on books from the Amazon platform. The Gowalla dataset, derived from the Gowalla platform, records user check-ins at different locations, with data provided for January to June 2010. ratings and interaction data for various businesses. The LastFM
The Yelp dataset contains dataset comprises social network interactions, tags, and music artist listening information from the Last.fm online music system. The Beer dataset, sourced from the BeerAdvocate platform, includes beer reviews and is filtered to only retain users and items with at least ten interactions. Details and statisties are provided in Table 1.
Table Statistics of the experimental datasets.
Dataset
User #
Item #
Interaction #
Density
Amazon
76,469
83,761
966,680
1.5 x 10
Gowalla
25,557
19,747
294,983
5.8 x 10
Yelp
42,712
26,822
182,357
1.6 x 10
LastIM
1,892
17,632
92,834
2.8 10 3
Beer
10,456
13,845
1,381,094
9.5 x 10
Evaluation Protocols. We downloaded the datasets from prior work, including its predefined training, validation, and test sets. The evaluation metries used include Recall @ @N and Normalized Discounted Cumulative Gain (NDCG)@N where N = (10, 20), both of which are better when higher.
Baseline Methods. We evaluated the general applicability of the proposed technique by applying BPR+ loss to various baseline models. The details of these baselines are provided below.
It simplifies the traditional Graph Convolution Network (GCN) for collaborative filtering by focusing solely on neighborhood aggregation, eschewing feature transformations and nonlinear activations to enhance training efficiency and improve recommendation performance.
LightGCN 9

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SGL [I0]: It improves LightGCN by using contrastive learning with data augmentation techniques like random walks and node/edge dropout, enhancing accuracy and robustness.
NCL [ It is a neighborhood-eerrched contrastive learning approach that boosts the effectiveness of graph collaborative filtering by incorporating structural and semantic neighbors into contrastive pairs, enhancing the model's ability to handle data sparsity and improve recommendation accuracy.
LightGCL [2]: It is a graph contrastive learning approach that enhances graph-based recommender systems by using singular value decomposition for data augmentation, effectively preserving the intrinsic semantic structures and offering robustness against noise perturbation and data sparsity.
It is a graph contrastive learning framework that improves collaborative filtering through two trainable view generators: a graph generative model and a graph denoising model, which create adaptive contrastive views to effectively address data sparsity and noise issues.
AdaGCL 5]:
7.2 Overall Performance Validation (RQ1)
The effectiveness of the proposed BPR+ loss is validated by applying it to various baseline models and conducting comprehensive performance evaluation across five datasets. We present the experimental result in Table 2. The findings indicate a consistent pattern: algorithms incorporating BPR+ loss exhibit improved performance across all datasets and metries. For instance, in the Amazon dataset, compared to adaGCI, adaGCL achieves an 11.9% improvement in the Recall @ 10 metric; for the NDCG@ 10 metric, the increase reaches 13.4%. The consistent improvement across almost all datasets and algorithms highlights the versatility and robustness of the BPR+ loss. This enhancement not only leads to higher recall rates, meaning more relevant items are recommended, but also improves the NDCG scores, indicating a better ranking quality in the recommendations provided. See Appendix for details. In addition, see Appendix for hyperparameter analysis.
Table 2: Performance comparison on Amazon, Gowalla, Yelp, LastFM, Beer datasets in terms of Recall and NDCG.
Dataset
Metric
adaGCI.
adaGCL.
lightCCL.
LightCL++
NCL.
NCL+
SGL
SGL+
lighGGNN
lighGCNN
eecall
0.0612
0.0685
0.0576
0.0736
0.07883
0,0432
0.0454
0,0535
NDCG0810
0,0307
0.0575
0.476
0.06088
0,049
0.1607
Amazon
Recall
0.0946
0.1032
0.0892
0.0942
0.1125
0.1166
0.0678
0.0690
0.0838
0.1080
NDCG@
0.0620
0.0691
0
0.0621
0.0729
0.0784
0.0343
0.0351
0.0541
0.0727
11
0.1523
0.1536
0.1690
0.1693
0.1755
0.1768
0.1641
0.1650
0.1544
0.1564
Gowalla
NDCG@10
0.1206
0.1208
0.1359
0.1365
0.1409
0.1420
0.1321
0.1323
0.1248
0.1264
0,2253
0.2264
0.2444
0,2451
0.2557
0.2571
0.2369
0.2369
0.2297
0.2317
NDCG@100
0.1427
0.1432
0.1585
0.1592
0.1649
0.1663
0.1540
0.1541
0.1474
0.1489
10
0.0517
0.0529
0.0456
0.0503
0,0510
0.0505
0.0525
NDCG0010
0.0320
0.0334
0.0844
0.0318
0,0554
0,0742
0,0314
0.0325
Yelp
6:
0.0805
0.0837
0,0716
0.0830
0.0794
0.0790
0.0978
0.1055
0.0798
0.0796
2DCG@20
0.0106
0.0425
0.0362
0.0421
0.0398
0.0400
0.0662
0.0863
0.0400
0.0405
Recalle11
0.1688
0.1760
0.1401
0.1412
0.1772
0.1775
0.1655
0.1734
0.1511
0.1715
LasFFM
NDCG@11
0,1521
0,1553
0,1243
0.1239
0.1562
0.1566
0.1470
0.1526
0.1343
0,1529
Recall 20
0,2165
0,2499
0,1919
0.1964
0.2525
0.2538
0.2433
0.2486
0.2206
0.2480
NDCG@10 20
0.1843
0.1860
0.1475
0.1472
0.1877
0.1883
0.1790
0.1836
0.1634
0.1847
0.0448
0.0457
0.0200
0.0729
0.0730
0.0670
0.0757
NDCG@10
0,0584
0.0608
0.0533
0.0338
0.0922
0.0923
0.0922
0.0922
0.0857
0.0937
Recall 20
0.0731
0.0757
0.0489
0.0491
0.1171
0.1175
0.1165
0.1167
0.1084
0.1227
2D
0.06-17
0.0675
0.0542
0.0547
0.1013
0.1014
0.1009
0.1009
0.0938
0.1039
7.3 Physical Property Analysis (RQ2)
In this section, we explore the physical properties of BPR loss and BPR+ loss. To simulate ID embeddings, 1000x64 matrix is constructed with its clements set to 0 and 1. Our first evaluation focused on whether these loss functions could accurately reflect the topological similarity between users and items, as calculated using the Jaccard index. Observations from Figures 4A and 4B demonstrate that the relationships derived by both loss functions proportionally align with topological similarities. However, BPR exhibits coarse granularity, where different topological similarities may correspond to identical scores.
Further analysis involved examining the relationship between the number of neighbors in a user-item pair and the relationship scores. The number of neighbors indicates the norm of the user/item embeddings. The union of neighbors represents the number of non-zero elements in the sum of two
7

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(4)
(b)
(d) b.30
Sior 5%
Sim-10% 10%
0.3
Sim-11%
20
0.20
DPPM
0.15
10
0.10
0.1
0.05
0.2
01.4
0L6
0..8
0.2
014
0.6
0.5
20
25
30
35
30
35
40
45
Topological simallarit
Topological similariit
union
The union neighborr
Figure 4: Comparison of the physical properties of BPR loss and BPR+ loss.
embedding vectors. A user connected numerous abstract nodes will have larger norm, leading to higher dot products with all item embeddings. Our experimental validation supports this analysis: The BPR loss scores relationships, increasing with the expanding union of neighbors. Additionally, when classifying these scores by varying degrees of topological similarity (within a range of +2%), observe that BPR delineates less distinet boundaries between different similarity levels with narrower range of scores. In contrast, BPR+ offers a more nuanced evaluation, assigning distinct scores different degrees of similarity, thus enhancing the fidelity of relationship assessments.
7.4 Efficiency Study (RQ3)
Due to the inefficiency of calculating loss directly via matrix exponentiation, we incorporate matrix factorization techniques to accelerate the loss computation process. To analyze computational efficiency in practical scenarios, we select the NCL as an example and compared the computation times of three loss computation methods: BPR, BPR+ (MM), and BPR+ (MF), across five datasets, as illustrated in Figure 5A. We record the time required to complete one epoch for each loss function. BPR+ (MM) denotes BPR+ calculated using matrix multiplication, whereas BPR+ (MF) refers to BPR+ calculated using matrix factorization, both considering 40-hop paths. The results indicate that using the optimized loss computation methods significantly reduces computation time, comparable to that of BPR. For instance, on the Amazon dataset, the computation time for BPR is 30.5s, for (MR+(MF) is 36.5s, and for BPR+(MM), significantly increases to 227.7s. Moreover, the increase in data scale further enhances the improvement in computational efficiency. Similarly, the results for LightGCL on the same datasets are consistent (Figure 5B).
250
300
200
RPR-
250
HPR-(MF) (PR-
200
150
1.50
160
Gowalla
Gowalln
Lasti'M
(4) NCL
(b) lightGCL.
Figure Comparison of computational efficiency between BPR, BPR+(MM), and (PR+(MF).
7.5 Case Study (RQ4)
In this section, we apply our proposed technique specifically to the domain of drug repositioning to validate its performance in practical scenarios. We develope a new framework to aid pharma- ceutical research and development (TopoDR), integrating multimodal drug and disease information encompassing chemical structures, side effects, drug-drug interactions, drug target profiles, disease phenotypes, and disease ontologies.
To verify the validity of the framework, we compare TopoDR with four leading drug repositioning methods (deepDR [13), DRHGCN [4], DDAGDL (15), and AdaDR (16)) using their default settings
8

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from public codes, validated on gold standard datasets (Fdataset and Cdataset), Performance was assessed via 10-fold validdani,,,,,,, with results detailed in Figure We use six evaluation metries to compare the performance of TopoDR with other methods: the area under the receiver operating characteristic (ROC) curve (AUC), the area under the precision-reeall curve (AUPR), accuracy (Acc.), precision (Pre.), recall (Rec.), and Matthew's correlation coefficient (MCC). In these comparisons, particularly on the Fdataset, TopoDR consistently outperformed the other models in metrics assessing overall model performance. Specifically, in the Fdataset, TopoDR's s AUC, AUPR, and MCC are 0.9594, 0,9630, and 0.7440, respectively, surpassing AdaDR, the next best method, by 2.89%, 1886% and 7.34%. Similarly, on the Cdataset, TopoDR exceeds AdaDR by 1.82%, 1.18%, and 2.64%. These results confirm TopoDR's 's superior ability to accurately predict potential indications for drugs. Furthermore, select ten potential therapeutic candidates for four prevalent cancers (colorectal, breast, stomach cancer, and leukemia). Table 6 displays the top ten candidate drugs for these cancers, with confirmed therapeutic drugs from the Comparative Toxicogenomies Database (CTD) emphasized in bold. See Appendix F for details.
(b):
Compariiee
(c)
80-
Troe Fustiv Ten
dopiR (AUI (20611
-eqURR (AEPR -L877))
8-7
DRHIGN (AUC
CALIPE
BOAGOL (.960))
Adadst (ABI- E.530
COLOECAPIE PR (.354))
TyySDDEBPRS 11
(xpaDRISFFE CAUPH (.0.00
ACC
Prodidan
Rocall
Fabe Pasliie
(d)
BOC Cuena Compar hore
(e)
AUPR Cuen Comperbon
0.0
Tre Protioo
8.0
Tresidan
1.0
DEERGON (ABIC - LAB1D
DROGGN
65024)
DBAGOL (ALISR
6.2
MdBBR (AUI (ATIC E5450
EapatliBP10 (ALIC- (3900)
TXpaDR/EPR1 (ALIFR -23640
Tipalttc
9.6
412 0.4
0.6
0.5
8,5
ADC
Truride
Reall
Pabee Fartine Hat:
Bocall
Figure 6: The performance of drug repositioning methods on Fdataset and Cdataset. Fdataset: (A) The ROC curve. (B) The precision-recall curve. (C) The accuracy, precision, recall, and Matthew's 's correlation coefficient of all methods. Cdataset: (D) The ROC curve. (E) The precision-recal curve. (F) The accuracy, precision, recall, and Matthew's 's correlation coefficient of all methods
8 Conclusion
Although Bayesian Personalized Ranking (BPR) loss has been widely applied in graph superviee learning with considerable success, it currently lacks robust theoretical support and analysis for node relations evaluation. this paper, we explain from a graph-theoretical perspective that BPR essentially counts the 2-hop paths from users to items within the embedding network. Analysis based on the principle of maximum entropy reveals that while BPR loss can represent the energy distance between users and items, it is not sufficiently precise. Based on these limitations, loss function, BPR+, as an alternative. Our work not only elucidates the success of previous graph self-supervised learning efforts but also provides fresh perspective for theoretical analysis in this field.
propose new
This work holds broad prospects. Foremost and most challenging is the inspiration derived from our theoretical analysis to design superior graph neural network modules. Additionally, our work represents a step towards deeper understanding of the relationships between nodes, which is central to the utility of graph neural networks. However, computational time remains an unresolved issue. Therefore, accelerating computation will be crucial direction for our future work.

---

Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (62350004, 62332020, U22A2041, 62502540, 6250072467), Shandong Provincial Natural Science Foundation (2R20250C1546, ZR20250C1540), Fundamental Research Funds for the Central Universities (25CX06033A), the Project of Xiangjiang Laboratory (No. 23XJ01011) 1), the Science and Technology Major Project of Changsha (No. (No.kh2402004), Hunan Provincial Postgraduate Scientific Research Innovation Project (CX20230234), the start-up funds from USTC and IAI.
References
[1] Kaiming Hc, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 9729-9738, 2020.
[2] Yixin Liu, Ming Jin, Shirui Pan, Chuan Zhou, Yu Zheng, Feng Xia, and S Yu Philip. Graph self-supervised learning: A survey. IEEE transactions on knowledge and data engineering, 35(6):5879 5900, 2022.
[3] Xiangnan He, Zhankui He, Xiaoyu Du, and Tat-Seng Chua. Adversarial personalized ranking for recommendation. In The 41st International ACM SIGIR conference on research development in information retrieval, pages 355-364, 2018.
[4] Steffen Rendle, Christoph Freudenthaler, Zeno Gantner, and Lars Schmidt-Thieme. Bpr: Bayesian personalized ranking from implicit feedback. arXiv preprint 1rX5v:1205.2618 2012.
[5] Yili Wang, Kaixiong Zhou, Rui Miao, Ninghao Liu, and Xin Wang. Adagel: Adaptive subgraph contrastive learning to generalize large-scale graph training. In Proceedings of the 31st ACM International Conference Information & Knowledge Management, pages 2046-2055, 2022.
[6] Marian Boguna, Ivan Bonamassa, Manlio De Domenico, Shlomo Havlin, Dmitri Krioukov, and
M Ángeles Serrano. Network geometry. Nature Reviews Physics, 14-3:11 2021.
[7] Dmitri Krioukov. Clustering implies geometry in networks. Physical review letters,
I 116(20):20830, 2016.
[8] Kai Zheng, Qilong Feng. Yaohang Li, Qichang Zhao, Jinhui Xu, and Jianxin Wang. Topola: novel embedding framework for understanding complex networks. arXiv preprint arXiv:2405. 16928, 2024.
[9] Xiangnan He, Kuan Deng, Xiang Wang, Yan Li, Yongdong Zhang, and Meng Wang. Lightgen: Simplifying and powering graph convolution network for recommendation. In Proceedings of the 43rd International ACM SIGIR conference on research and development Information Retrieval, pages 639-648, 2020.
[10] Jiancan WU, Xiang Wang. Fuli Feng, Xiangnan He, Liang Chen, Jianxun Lian, and Xing Xie. Self-supervised graph learning for recommendation. In Proceedings of the 44th international ACM SIGIR conference on research and development in information retrieval, pages 726-735,
2021.
[11] Zihan Lin, Changxin Tian, Yupeng Hou, and Wayne Xin Zhao. Improving graph collaborative filtering with neighborhood-enriched contrastive learning. In Proceedings of the ACM web conference 2022, pages 2320-2329, 2022.
[12] Xuheng Cai, Chao Huang. Lianghao Xia, and Xubin Ren. Lightgel: Simple yet effective graph contrastive learning for recommendation. arXiv preprint arXiv:2302.08119 2023.
[13] Xiangxiang Zeng, Siyi Zhu, Xiangrong Liu, Yadi Zhou, Ruth Nussinov, and Feixiong Cheng- deepdr: network-based deep learning approach to silico drug repositioning. Bioinformatics, 35(24):5191-5198 2019.
10

---

[14] Lijun Cai, Changcheng Lu, Junlin Xu, Yajie Meng, Peng Wang, Xiangzheng Fu, Xiangxiang Zeng, and Yansen Su. Drug repositioning based on the heterogeneous information fusion graph convolutional network. Briefings in bioinformatics, 22(6):bbab319, 2021.
[15] Bo-Wei Zhao, Xiao-Rui Su, Peng-Wei Hu, Yu-Peng Ma, Xi Zhou, and Lun Hu. A geometric deep learning framework for drug repositioning over heterogeneous information networks. Briefings in Bioinformatics, 2bb6):bba3384, 2022.
[16] Xinliang Sun, Xiao Jia, Zhangli Lu, Jing Tang, and Min Li. Drug repositioning with adaptive graph convolutional networks. Bioinformatics, 40(1):btad7433 2024.
[17] Can-Yr Yi Lu, Hai Min, Zhong- Qiu Zhao, Lin Zhu, De-Shuang Huang, and Shuicheng Yan. Robust and efficient subspace segmentation via least squares regression. In Computer Vision-ECCV 2012: 12th European Conference on Computer Vision, Florence, Italy, October 7-13, 2012, Proceedings, Part VII 12, pages 347-360. Springer, 2012.
[18] Xubin Ren, Lianghao Xia. Yuhao Yang. Wei Wei, Tianle Wang, Xuheng Cai, and Chao Huang. Ssirec: A self-supervised learning framework for recommendation. In Proceedings of the 17th ACM International Conference on Web Search and Data Mining, pages 567-575, 2024.
[19] Ted T Ashburn and Karl B Thor. Drug repositioning: identifying and developing new uses for existing drugs. Nature reviews Drug discovery, 3(8):673-68) 2004.
[20] Natalia Novac. Challenges and opportunities of drug repositioning. Trends in pharmacological sciences, 3465)267-272, 2013.
[21] Vijay P Kale, Hasan Habib, Robert Chitren, Milan Patel, Kartick Pramanik, Subash C Jonnalagadda, Kishore Challagundla, and Manoj K Pandey. Old drugs, new Drug repurposing in hematological malignancies. In Seminars in cancer biology, volume 68, pages Elsevier, 2021.
[22] Vijay Pralhad Kale, Shantu G Amin, and Manoj K Pandey. Targeting ion channels for cancer ther- apy by repurposing the approved drugs. Biochimica et Biophysica Acta (BBA)--iomembranee 1848(10):244727555 2015.
[23] Joel Dudley, Tarangini Deshpande, and Atul J Butte. Exploiting drug relationships for computational drug repositioning. Briefings in bioinformatics, 12(4)3303-311, 2011.
[24] Hanqing Xue, Jie Li, Haozhe Xie, and Yadong Wang. Review of drug repositioning approaches and resources. International journal of biological sciences, 14(10):123) 2018.
[25] Nansu Zong, Andrew Wen, Sungrim Moon, Sunyang Fu, Liwei Wang, Yiqing Zhao, Yue Yu, Ming Huang, Yanshan Wang. Gang Zheng, et al. Computational drug repurposing based on electronic health records: a scoping review. NPJ digital medicine, 5(1):77, 2022.
[26] Xiao Liu, Fanjin Zhang, Zhenyu Hou, Li Mian, Zhaoyu Wang, Jing Zhang, and Jie Tang. Self-supervised learning: Generative contrastive. IEEE transactions knowledge and data engineering, 35(1):857-876, 2021.
[27] Ming Gao, Leihui Chen, Xiangnan He, and Aoying Zhou. Bine: Bipartite network embedding. In The st international ACM SIGIR conference research & development in information retrieval, pages 715-724, 2018
[28] Assaf Gottlieb, Gideon Y Stein, Eytan Ruppin, and Roded Sharan. Predict: a method for inferring novel drug indications with application personalized medicine. Molecular systems biology, 7(1):496, 2011.
[29] David S Wishart, Yannick D Feunang, An C Guo, Elvis J Lo, Ana Marcu, Jason R Grant, Tanvir Sajed, Daniel Johnson, Carin Li, Zinat Sayeeda, et al. Drugbank 5.0: a major update to the drugbank database for 2018. Nucleic acids research, 46(D1))D174--11022 2018.
[30] Ada Hamosh, Alan F Scott, Joanna S Amberger, Carol A Bocchini, and Victor A McKusick. Online mendelian inheritance man (omim), a knowledgebase of human genes and genetic disorders. Nucleic acids research, 33(suppl_
(ithppp__1:D51111 1):D5144DD D517, 2005.

---

[31] Cheng Yan and Guihua Duan. Pmdags: predicting ear diffusion convolution network and similarities. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2024.
associations with graph nonlin-
[32] Huimin Luo, Jianxin Wang, Min Li, Junwei Luo, Xiaoqing Peng, Fang-Xiang Wu, and Yi Pan. Drug repositioning based on comprehensive similarity measures and bi-random walk algorithm. Bioinformatics, (2(17)2664426777 2016.
[33] Mengyun Yang. Lan Huang, Yumpei Xu, Chengqian Lu, and Jianxin Wang. Heterogeneous graph inference with matrix completion for computational drug repositioning. Bioinformatics, 36(22-23):5456-54455 2020.
[34] David Weininger. Smiles, chemical language and information system. introduction to methodology and encoding rules. Journal of chemical information and computer sciences, 28(1):31-36, 1988.
[35] Christoph Steinbeck, Yongquan Han, Stefan Kuhn, Oliver Horlacher, Edgar Luttmann, and Egon Willighagen. The chemistry development kit (cdk): An open-source java library for chemo-and bioinformatics. Journal of chemical information and computer sciences, 43(2):493-500, 2003.
[36] Philip Resnik. Using information content to evaluate semantic similarity in a taxonomy. arXiv preprint emp-Ig/95 1007, 1995.
[37] Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pages 855-864, 2016.
[38] Jian Tang. Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large- scale information network embedding. In Proceedings of the 24th international conference world wide web, pages 1067-1077, 2015.
[39] Daixin Wang, Peng Cui, and Wenwu Zhu. Structural deep network embedding. Proceedings of the 22nd ACM SIGKDD international conference Knowledge discovery and data mining, pages 1225 1234, 2016.
[40] Chunyu Zhang, Chao Xu, Xueyun Gao, and Qingqiang Yao. Platinum-based drugs for cancer therapy and anti-tumor strategies. Theranostics, 12(5):2115, 2022.
[41] Allan Peter Davis, Cynthia J Grondin, Robin Johnson, Daniela Sciaky, Jolene Wiegers, Thomas C Wiegers, and Carolyn J Mattingly. Comparative toxicogenomics database (ctd); update 2021. Nucleic acids research, 49(D1):D1138 -D1443, 2021.
[42] M Riis. Modern surgical treatment of breast cancer. Annals of medicine and surgery, 56:95-107, 2020.
[43] Frederic Buxant, Nadège Kindt, Guy Laurent, Jean -hrisChrrttophe Noël, and Sven Saussez. An- tiproliferative effect of dexamethasone in the mcf-7 breast cancer cell line. Molecular Medicine Reports, 12(3):405144054 2015.
[44] Jian Zhang. Zhonghua Wang, Xichun Hu, Biyun Wang, Leiping Wang. Wentao Yang, Yang Liu, Guangyu Liu, Genhong Di, Zhen Hu, et al. Cisplatin and gemcitabine as the first line therapy in metastatic triple negative breast cancer. International journal cancer, 136(1):204-211, 2015.
12

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A Proof of Theorem 1
We restate Theorem 1: Given a network X E (R(++))NN,, let (t(z)) denote the expected number of triangles, and (+)) represent the expected number of weighted triangles. Consequently, a' where of the logarithm of the thermodynamic activity corresponding to {t' (z)),, and is associated with (t(x))
Proof. network geometry, the expected numbers of triangles can be obtained through the integration of the graphon 7
(I(x)) //2 //g p(a, s)p(v, =))(2,
(22
Under the assumption of redundancy the current statistics, the non-redundant expectation (z)), is anticipated to be less than its redundant counterpart, (t(z)). In this context, t(x) denotes the quantity of triangles associated with node r. Utilizing the maximum-entropy principle (21), we derive the following formula:
(f())) = / ///e (l (v. (s, = <t.
(13)
Therefore,
1 pl (z, w)p' (y, =)p (z, xddydz //e p(r, u)p(y, 2)p(z, x) d)dddz,
(14)
When the network size is sufficiently large, the approximate solution for the graphon that maximizes entropy is the Fermi-Dirac graphon:
i < r S 1,
p" (z (z. y)
Ite ()-3)
(15)
if > 1,
1tca
where a and are the rescaled inverse temperature and energy distance, respectively, Inserting the terms from Formula [17] into Formula [16], we obtain:
/1 1 11+e20 (c {)
| dydz < // 11 11+ 22(
3
1 [
3
dyd2.
(16)
Consequently, deduce that a' > a. Based on prior research, the common neighbor integral corresponding to a' provides a more accurate representation of the latent space distance than that associated with 77.. From this, we demonstrate that the current triangle statisties exhibit redun-
dancy. Given that the common-neighhor integral is defined as Ja
dz, the 2-hop paths
(-
(common neighbors) are also found be imprecise when representing distances within the latent space and Theorem 1 holds.
B Proof of Theorem 2
We restate Theorem 2: Given user embedding R.N, the embedding matrix E B(m+a)xNe and a parameter X, the following optimal solution to the problem is denoted by c* in vector form:
1 1 CEI + ||c|
min x
(17)
We have
|| - 1f
x le_ e,
(18)
1
13

---

Proof. Inspired by prior work, we conducted the following inference [17]. Let L(c) = b/1 cEIP 2 + ||c|| Since et is the optimal solution to Equation (17), it satisfies
@T(c)
(19)
8ck
c==
Thus, we have
2
(y - -c²E) + 2c; = 0,
(20)
A
2
x o (y - c² E) + 2c =0
(21)
Equation (20) and (21) give us
c - = x
e - -ee) (y - c² E)
(22)
Since c* is optimal to equation (17), we get
] yl c² El
=== < L(0) = 1/y1?
(23)
Thus, we have 1 11 Eli < 11 Then equation (23) implies
| -sil |
$ lle S
1
e,
(24)
The optimal solution C, of optimization problem (17) corresponds to the i-th row of matrix Dropo From the above equation, we discern that the difference in distances between two nodes in the latent space relative to other nodes correlates with the divergence in their topological structures. That is, nodes with highly similar topological structures occupy proximate positions within the latent space and Theorem 2 holds.
C Proof of Theorem 3
We restate Theorem 3: Given nodes i and j. the topological similarity between them is directly proportional to the deopo value.
Proof. Our way to exploring the relationship between Topola distance and the topological similarity is to differentiate various types of paths connecting nodes and j. By counting the number of each type of paths, it allows us to take into consideration not only local and global connectivity, but also node degrees in an implicit manner. Particularly, we focus the classification of n-hop paths connecting nodes é and j. Due to the possible existence of loop or loops on these paths, we establish classification system for all such n-hopp paths, based the length of path actually traversed (i.e., the number of hops after removing all the loops on the path). Clearly, there are a total of (n - 1) types. To facilitate a clear differentiation, we represent these path types with polygons. For example, path that actually traverses 2-hop is defined as a P2 path, which can be viewed as a triangle adding direct edge back from node node i. Similarly, a path that actually traverses 3 hops can be defined as Pa (represented as a quadrangle), and a path that actually traverses 4 hops can be defined as P4 (represented as a pentagon). The quantity of these types can be expressed as PP loop (at). where is the set of loop-free I-hop paths connecting nodes and j and loop (-) represents the number of different ways of adding loops to paths in a form n-hop paths between nodes and j. While there is correlation between |P/| and lay], accurately measuring this relationship remains challenging. Nevertheless, our observations suggest that a proportional relationship exists between the quantity of Pt. characterized by solely loops between node and its neighbors, or between node j and its neighbors, and |az). Paths exhibiting such characteristics are defined as b,, (I). The interplay between Jasl and [b, (1) can be calculated as follows:
14

---

n-1
3
bb (1)
K
|ai
(25)
h=0
where (1)] is the quantity of bn (1) in n-hop Pa paths. Ky and Ky represent the degrees of node i and node j. Thus, n-hopp paths between node and node j in the network are represented as:
hop
|an + [bxl + lcal
la,1
+
Σ
(Pi)
+ lcal
Picn hop
(26)
if n is even if odd where Cn denotes the paths distinct from an and by n- hopl refers the counts of n-hop paths. Our analysis for 2-hop, 4-hop, and 6-hop paths reveals that |c, does not exhibit a direct correlation with K2 and Kj. Equation (26) reveals that dn and b, for even-hop paths consist of polygons with an odd number of sides, while for odd-hop paths, they comprise polygons with an even number of sides. Subsequently, we delve further into the physical significance Dupp which can also expressed in the form of set of paths (note that the following formulas reveal the physical meanings of D upo and duopo (i. j), but will not be used for calculation):
aaze
|cal
E Ex
2
K
-
a22+1
|cal
D Dropo AA T
1 AA AATAAA X
1 + 1 AA AA A
(27)
The emphasis on even-hop paths arises from their ability to capture the degree information and the global connectivity information of nodes and j. Furthermore, the TopoLa distance depo is demonstrated to correlate with the topological structure of nodes. Specifically, a can be classified into two types: those overlapping with 2-hop paths (gt) and the remaining ones (sr), with |ay] [9i] + |sr|- The |5y] relates to the topological structure
nodes and j. For instance, identical topological structure between i and j renders |sy] to zero (which indicates that it measures the topological similarity of nodes and j Thus, we can express deopo as:
k -t-
lazel
C(2))
deopo (i,j) = lim
2-000
(->) 3 [++]
X
(=1 h=0
1=1
K// ()))
32
I22d
32c
C(2t)
lim
n
4=0 (--)))+
A)1-1
(-))--1
(28)
lozd where Σ Et (-xjt-1 and
Cr2)) are unaffected by node's topological structure, while
Isgel Ei - acts as a aypology-dependen penalty term. We find that the topological similar- ity between and j is directly proportional to the depo value and Theorem 3 holds.
D More details about overall performance validation
The code for all models originates from SSLRec, with each model's embedding size fixed at 32 and the batch size set to 4096. To ensure a fair comparison, we employed a grid search method to determine the optimal parameter combination for each model, with search range set (le-3, Ic-4, le-5, le-6, le-7).
The Amazon, Gowalla, and Yelp datasets were downloaded from SSLRec [8] and have been pre- divided into training, validation, and test sets The LastEM and Beer datasets were obtained from the corresponding code and data provided by adaGCL (5]. and have also been mre-pariitioeed
Experiments were conducted on a high-performance hardware platform comprising an Intel Xeon Platinum 8352V processor, NVIDIA RTX 4090 with 24 GB of memory, and system running
15

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Ubuntu 20.04. The software environment included PyTorch version 1.11.0, Python 3.8, and CUDA 11.3.
To ensure statistical significance, we trained models using BPR loss and BPR+ loss, calculating the p-values. The results are presented in Table 3.
Table 3: Statistical significance of overall performance comparison.
Dataset
adaGCL
lightGCL
NCL
SGL
lightGCN
Amazon
1.97E-12
1.20E-18
1.34E-08
1.31E-13
7.37E-12
Gowalla
6.13E-11
3.29F-16
1.76F-13
3.72F-19
3.80F-14
Yelp
1.99E-10
5.31E-13
4.90E-11
2.78E-10
8.82E-08
LastFM
4.24E-12
2.38E-30
9.18E-10
1.65-114
1.74E-11
Beer
7.56E-11
3.91E-10
8.44E-14
2.19E-15
3.17E-12
E Hyperparameter analysis
In this section, explore the sensitivity of the adaGCL model the critical hyperparameter Ain the BPR+ loss formulation, which governs the influence of n-hop paths on relationship assessments. To ascertain the optimal X, systematically search across range (1e-3, le-4, 1e-5, 1e-6, le-7), aiming to determine how it impacts adaGCL's performance. As illustrated in Figure 7, the model's performance across five datasets with varying values is analyzed. The data suggests that the optimal performance can occur at any point within the tested À range, highlighting the varying influence of n-hop paths on different datasets. This variability underscores the need for dataset-specific tuning of to achieve the best results in adaGCL.
0.070
Amazon
0.105
Gowalla
0.055
Yelp
0.050
0.065
0.100
0.045
0.060
0.095
0.040
0.055
0.035
0.090
0.030
0.050
0.025
0.085
0.045
0.020
Tc-07
1c-06
1e-05 1c-04 Tc-03
1c-07
Ic-06 Ic-05
1c-04 Ic-03
1c-07
1c-06
1e-05
Tc-04 1c-03
LastFM
0.13
0.07
recallig
0.06
adegie10 10
0.12
0.05
0.11
0.10
0.03
0.02
0.09
0.01
0.08
1c-07
1c-06
Te-05
Te-04
1e-03
Ic-07 lc-06 lc-05 1c-04 Ic-03
Figure 7: The impact of À
F More details about case study
Drug repositioning, encompassing strategies like repositioning and therapeutic switching, is an evolving paradigm in pharmaceutical research, recognized for its time and cost efficiency compared
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to traditional drug development [0]] It leverages existing drugs, bypassing initial development phases, and accelerating market entry. Furthermore, drug repositioning emerges as a pragmatic way for addressing complex diseases like cancer [2] 22 Computational drug repositioning utilizes molecular, clinical, and biophysical data to ascertain the potential of approved drugs for new clinical applications [23]. Such strategy not only expedites the repositioning process but also minimizes associated costs [24]. Researchers have actively applied this strategy in identifying and developing therapeutic agents against COVID-19 (25).
With the advancement of artificial intelligence technology and the accumulation of large-scate scale biomedical data, drug repositioning based on deep learning (DL) has shown significant advantages over traditional computational approaches. For instance, DeepDR employed a collective variational autoencoder to integrate diverse network data, enhancing drug repositioning by effectively capturing non-linear structures and mitigating data sparsity [3]. Another innovative technique, DRHGCN, integrated data from various networks to refine drug repositioning [14] It extracted both intra- domain and inter-domain features, improving drug and disease representations through sophisticated layer attention mechanisms, thereby enhancing repositioning accuracy. Additionally, DDAGDL, a framework that incorporated diverse drug-related networks, employed non-Euclidean domain knowledge and an attention mechanism to strengthen feature representation [15. This framework
further used XGBoost for robust prediction tasks. Lastly. AdaDR employed graph neural networks and attention mechanisms to integrate node features with topological structures [16]. Self-Supervised Learning (SSL) represents a potent paradigm that diminishes the dependency on manually annotated labels, facilitating training on extensive unlabeled datasets via self-supervised tasks [26]. It has achieved notable success across multiple domains, encompassing visual representation learning, pre-training of language models, and beyond 26). Within the sphere of recommender systems, graph self-supervised learning (GSL) has gained prominence (2]. This paradigm has evolved from its initial unsupervised methods to sophisticated models that utilize network embeddings and random-wall proximity for self-supervision (27). LightGCN, recognized for its simplicity and efficacy, has emerged as a favored graph self-supervised learning method, and forms the foundation of numerous graph contrastive learning (GCL) methods [2]. Bayesian Personalized Ranking (BPR) loss, employed in LightGCN, is also prevalent loss function in latest GCL methods 4.. However, within the context of complex networks, BPR loss relies on 2-hop path statistics for assessing sample relationships. Previous research has proved that relying solely on this local connectivity information may not suffice for accurately measuring the energy distance between nodes in the latent hyperbolic space. This limitation might neglect critical, broader network patterns, potentially limiting its effectiveness in biological analysis.
In this section, we develop a graph self-supervised learning framework using Topology-encoded Latent Hyperbolic Geometry for drug repositioning. By assigning ID embeddings to drugs and diseases and refining them across the drug-disease network. we aggregate these embeddings at various propagation layers to form the final network embedding. Multisource information about drugs and discases is then integrated for drug repositioning tasks.
To summarize, this framework makes the following main contributions:
. We build TopoDR, a new computational drug repositioning framework based on lightGCN+, which integrates multimodal information.
Empirical analyses on benchmark datasets reveal that our approach surpasses current drug reposi- tioning algorithms in predictive accuracy.
F1 Materials and methods
In this section, we first delineate the benchmark datasets for drug repositioning, along with the multimodal representations related to drugs and diseases. Subsequently, we present the framework of the drug repositioning model, TopoDR, which primarily comprises two modules: the network embedding module and the prediction module. The primary contribution of this work lies in the net- work embedding module, developed through the graph self-supervised learning method, lightGCN+ This method is grounded in Topology-encoded Latent Hyperbolic Geometry. The prediction module integrates multimodal information to forecast alternative indications of drugs.
17

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F111 Materials
The Fdataset and Cdataset are benchmark datasets in drug repositioning research. The Fdataset, recognized as the gold standard, comprises 1933 verified drug-disease associations, integrating 593 drugs from DrugBank and 313 diseases from the OMIM database 128 29 30 [1]. The Cdataset, turn, widens the data collection spectrum, encompassing 663 drugs, 409 diseases, and 2352 drug-disease associations (32] Together, these datasets offer a rich and diverse foundation of data, indispensable for the development and stringent validation of drug repositioning models.
Furthermore, we computed multimodal representations for drugs and diseases, informed by prior research [33]. For drugs, five types of similarities were calculated: chemical structure similarity Res using SMILES and Tanimoto score [4]; ATC code similarity Rate based on therapeutic effects and chemical characteristics [35; side effect similarity Rge employing data from the SIDER database [3)] drug- drug interaction similarity Rad: derived from pharmacokineties [29]; and target profile similarity Rit based on known drug targets (2). For diseases, phenotype similarity Cn was assessed using phenotypic scores from the MimMiner database and dísease ontology (DO) similarity Ceo based on gene ontology algorithms. We integrated these multimodal information sources by computing a weighted average of these similarity measures, described as follow:
R Res + Ratc + Ros + Rads + Ratt
(29)
C Oph ph + Coo
(30)
F.1.2 TopoDR for drug repositioning
TopoDR comprises three main components: a multimodal data integration component, a network embedding component, and a prediction component (Figure 8). We use ligthGCN+ as the network embedding component of the To framework. The prediction module utilizes both network embedding and multimodal data to depict drug-disease pairs, which be defined as follows:
Descriptor (r, d) = E,, Ed, R, Cal
(31)
We identify known associations as positive samples and correspondingly select an equal number of unacknowledged associations serve as negative samples, thereby forming training set. To predicting potential drug-disease associations, we employed a random forest algorithm as the classifier in this study.
F.2 Experiments
F.2.1
Performance evaluation
The 10-fold cross-validatte was conducted to assess the performance of the drug repositioning model. In this process, all known drug-disease pairs were randomly divided into ten subsets, with each subset sequentially used as the test sample, while the remaining associations served as training samples. This cross-valin validation process was repeated ten times, and the averaged results were obtained. Additionally, six evaluation metrics were selected for assessing predictive performance: the area under the receiver operating characteristic curve (AUC), the area under the rreiision-recall curve (AUPR), accuracy (Acc.), precision (Pre.), recall (Rec.), and Matthew's 's correlation coefficient (MCC).
F.2.2 Baseline model
To evaluate the performance of our proposed framework, we compared TopoDR with four existing drug repositioning methods listed below.
deepDDR [3] is a siwwork-basedd deep learning model, integrating heterogeneous networks through
mudti-modal deep autoencoder. It uses variational autoencoder encode and decode low- dimensional representations of drugs, identifying new uses for approved drugs.
DRHGCN 14 is approach using graph convolutional network to integrate drug-drug, disease- disease similarities, and drug-disease associations. It combines intra-domain and inter-domain feature extraction with layer attention mechanism for enhanced drug and discase embeddings.
18

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embedding network
daaso Cr. d,)
Improved Bayesian Personalized
Ranking (BPR+)
Layer Combination (weighted sum)
4
Layer 3
Layer 3
Layer
2
Layer 2
Layer
Layer
Normalized Sum
Normalized Sum
Cr
1-1)
9-1)
U-1)
d
0-1
IT-1)
neighbors of d,
neighbors of
Light Gruph Convolution (LGC)
Figure 8: The flowehart of lightGCN+ model.
DDAGGL is novel framework that integrates attention-basee Graph Deep Learning with heterogeneous biomedical networks, enriched with biological knowledge. Utilizing non-Euclidean geometrie priors and an attention mechanism, DDAGDL efficiently learns refined feature representa- tions of drugs and diseases.
AdaDR [16] is adaptive GCN framework, leveraging attention mechanism fuse drug and disease features from topological and feature spaces for enhanced embeddings.
F.2.3 The sensitivity analysis of parameters
In this section, we mainly examine the sensitivity analysis of parameters À and μ within -fold cross-validation. TopoDR achieves its optimal performance when x=le-5 and μ = le-7. By altering one parameter while maintaining the other constant, we explore how such adjustments benefit the AUC values. It is noteworthy that we employed identical parameters for both the Edataset and Cdataset.
Figure 9A depicts the performance trends of TopoDR with varying À values. As İ varies, the AUC demonstrates a pattern of initial increase followed by a decrease. downtrend in AUC is observed when À increases from le-5 to 1e-7, indicating the necessity of assigning higher weight to global topological structure similarity information. Similarly, Figure 9B presents the performance trends of TopoDR across different 7 values, with the optimal AUC occurring at r=1e-7. A continuous decrease in AUC values is observed as is reduced from 1e-7 1e-9, suggesting that the regularization terms
in Equation (7), specifically
E should not be excessively reduced.
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A
B
0.9586
0.95995
0.9585
ALC
0.958
0.9575
0.9578
165
1c6
Figure 9: Effect of different and on TopoDR performance. (A) Variation of the AUC values with the different settings of X. (B) Variation of the AUC values with the different settings of T.
F.2.4 Ablation study
In this section, the efficacy of TopoDR is evaluated through comparative training across datasets with diverse feature types. Two scenarios are analyzed, each predicated the utilization of distinct modalities within the proposed framework:
Wo-N: AdaDR without network information.
TopoDDR-wo-AA AdaDR without attribute information.
Table 4 reports the performance of each module in TopoDR. It clearly demonstrates that both modali- ties of TR enhance model performance, with the network information providing considerable predictive accuracy. TopoDR-w/o-A shows notable precision, achieving 0.9677 in the Fdataset and 0.9804 in the Cdataset, playing a crucial role minimizing false positives. TopoDR- wwooNN exhibits balanced performance across various metrics. In the Fdataset, it achieves an AUC of 0.9397, accuracy of 0.8471, and recall of 0.7454, whereas the Cdataset, scores an AUC of 0.9444, accuracy of 0.8544, and recall of 0.7555. These results indicate its strong overall predictive capability, enhancing the model's effectiveness various aspects of prediction, from true positive rate to general accuracy and consistency. The synergistic interplay of these modalities enhances the model's 's robustness and adaptability across various predictive scenarios.
Table 4: Performance of models associated with different modality across all datasets.
Dataset
Method
AUC
AUPR
Acc.
Pre.
Rec.
MCC
TopoDR-w/o-A
0.9312
0.9453
0.7998
0.9677
0.6239
0.6430
Edataset
TopN
0.9397
0.9455
0.8471
0.9412
0.7454
0.7120
TopoDR
0,9597
0.9581
0.8618
0.9683
0.7511
0.7440
TapaDR-wwo-A
0.9377
0.9529
0.8323
0.9804
0.6793
0.6989
Cdataset
TopoDR-w/o-N
0.9444
0.9511
0.8544
0.9431
0.7555
0.7240
TopoDR
0,9629
0.9686
0.8679
0.9753
0.7555
0.7556
F.2.5
Comparison with graph self-supervised learning methods
To assess lightGCN+'s performance, we conducted comparisons with existing graph self-supervised learning methods (Node2vec (37), LINE (38], SDNE [39], and lightGCN (2) As shown in Table lightGCN+ consistently outperforms the other methods in almost all metrics, particularly in AUC and AUPR. These metrics are crucial as they represent the model's ability to distinguish between classes effectively. In the case of the Fdataset, lightGCN+ shows superior performance, leading in AUC and AUPR, and displaying high accuracy and MCC. These results indicate not only effectiveness in classification tasks but also its reliability in handling imbalanced datasets, as suggested by the high MCC values. the Cdataset, lightGCN+ again stands out, particularly in AUC, AUPR, and Precision, suggesting its strong ability to predict positive classes accurately while reducing false
20

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positives. The slight trade-off between Recall and Precision in favor of the latter is noticeable for LightGCN+, suggesting a tendency towards conservative classification strategies that prioritize precision. Comparatively, other methods like Node2vec, LINE, SDNE,, and lightGCN show varying degrees of effectiveness. Node2vec and lightGGN, for instance, exhibit balanced performance across most metrics, suggesting their versatility. However, SDNE tends to lag slightly, particularly in the Fdataset, indicating possible limitations in capturing complex graph structures or generalizing across different datasets.
In summary, strengths of lightGCN+ in handling graph-structuee structured data with strong emphasis on precision and overall classification effectiveness. Its consistent performance across different datasets underscores its robustness and potential for broader applicability in graph-related machine learning tasks.
Table 5: The performance of existing graph self-supervised learning methods on Fdataset and Cdataset.
Dataset
Method
AUC
AUPR
Acc.
Pre.
Rec.
MCC
Node2vee
0.9483
0.9472
0.8570
0.9698
0.7382
0.7360
LINE
0.9448
0.9399
0.8469
0.9514
0.7362
0.7144
Fdataset
SDNE
0.9256
0.9254
0.8244
0.9508
0.6875
0.6764
lighIGCN
0.9540
0.9523
0,8495
0.9583
0.7336
0.7203
lightGCN+
0.9597
0.9581
0.8618
0.9683
0.7511
0.7440
Node2vec
0.9537
0.9539
0.8685
0.9668
0.7650
0.7546
LINE 0.9516
0.9588
0.8687
0.9601
0.7713
0.7529
Cdataset
SDNE
0.9509
0.9542
0.8588
0.9711
0.7413
0.7395
lightGCN
0.9602
0.9592
0.8640
0.9655
0.7584
0.7469
lightGCN+
0.9629
0.9686
0.8679
0.9753
0.7555
0.7556
F.2.6 Case studies
To verify the practical applicability of TopoDR in identifying potential therapeutic drugs for complex diseases, we conducted case studies. Using TopoDR on Fdataset, we predicted unknown drug- disease associations. For each disease, candidate drugs are ranked in descending order based on their predictive scores. In recent years, the development of anti-tumor drug has received increasing attention [0). In this context, we selected four common cancers (colorectal cancer, breast cancer, gastric cancer, and leukemia) for case studies and retrieved evidence for candidate drugs from the CTD database [41] Table 6 lists the top 10 candidate drugs for these cancers ranked by the TopoDR, with the confirmed therapeutic drugs from the CTD database highlighted bold. In the United States, approximately 12% of women are estimated to develop breast cancer over their lifetimes, statistie underscored by the recording of over 250,000 new cases in 2017 alone [42] Our drug candidates have undergone preliminary biological validation for breast cancer treatment. For instance, the study by Buxant et al. investigates the impact of dexamethasone (Dex) on MCF-7 breast cancer cells. Their findings indicate that Dex inhibit cell proliferation, potentially via a pro-apoptotic mechanism, suggesting its therapeutic potential 43). Similarly, the emergence of cisplatin as treatment for metastatic triple-negative breast cancer (mTNBC) is attributable to its role as a DNA-damaging agent [44]. The rationale for using cisplatin in m TNBC derived from the subtype's 's unique molecular characteristies, notably its aberrant DNA repair mechanisms and extensive genomic instability. These features make mTNBC cells especially susceptible to DNA damage, thereby explaining the efficacy of cisplatin in this context. In summary, the outcomes of case studies indicate the practical significance of To in identifying potential therapeutic drugs for complex diseases.
21

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Table 6: Top 10 candidate drugs for colorectal cancer, breast cancer, gastric cancer, and leukemia.
Diseases (OMIM IDs)
Top 10 candidate drugs (DrugBank IDs)
Doxorubicin (DB00997); Teniposide (DB00444); Prednisone (DB00635); Cisplatin
Colorectal Cancer (DI 14500)
(DB00515), Bleomycin (DB00290)) Busulfan (DB01008); Imatinib (DB00619):
Conjugated Estrogens (DB00286): Vincristine (DB00541): Zoledronic Acid (DB00399).
Bleomycin (DB00290); Cisplatin (DB00515); Dacarbazine (DB008S1); Salmon
Calcitonin (DB00017); Mechylpreenisoooone (DB00959); Teniposide (DB00444);
Cancer (D114480)
Vincristine (DB00541); Alendronic Acid (DB00630): Risedronic Acid (DB00884):
Fludarabine (DB011033).
Prednisone (DB00635); Vincristine (DH00541); Bleomycin (DBB02290); Teniposide
Gastric Cancer, Hereditary Diffuse (D137215)
(DB00444); Cisplatin (DB00515); Imatinib (DB00619); Dacarbazine (DB00851);
Zoledronic Acid (DB00399); Methyyprecuisooooe (DB00959); Azathioprine (DB00993).
Dexamethasone (DB01234). Preduisone Cisplatin (DBMI555)), Doxorubicin
(DB0W997; Triamcinolone (DB00620): Daunorubicin (DB0664)); Zoledronic Acid
Breast Cancer (D109543)
(DB00399); Vinblustine (DB00570); Pamidronic Acid (DB00282); Methylprednisolone (DB00959).
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29


Reviews:
[{"review_id": "sBl61BmgBJ", "reviewer": "Reviewer_fhF3", "summary": "This paper focuses on the field of graph self-supervised learning, analyzing the widely used but rarely examined BPR (Bayesian Personalized Ranking) loss function. It identifies and verifies its weaknesses in capturing global connectivity and topological similarity from multiple perspectives. Building on the perspective of latent hyperbolic geometry, the authors propose a more effective distance metric called TopoLa and its corresponding loss function BPR+. The improvements brought by BPR+ are validated on five public datasets, enhancing the performance of LightGCN and other contrastive learning models. Furthermore, in the field of drug repositioning, the proposed model achieves significant improvements over existing approaches.\nThe main contributions of this paper are as follows:\n1.\tRevealing the limitations of traditional BPR loss.\n2.\tAnalyzing the effectiveness of common graph convolution operations under the latent hyperbolic framework.\n3.\tProposing a novel loss function, BPR+, which enhances model performance.\n4.\tOptimizing the computational complexity of BPR+.\n5.\tConducting extensive experiments to demonstrate the superiority of proposed method, establishing a new framework for drug repositioning.", "questions": "Firstly, the paper acknowledges the over-smoothing phenomenon in higher layers of LightGCN while proposing the new method involving multi-layer aggregation. I would appreciate clarification on whether the TopoLa-based approach is susceptible to over-smoothing, or if specific design optimizations have been implemented to prevent this issue.\nThen, the proposed methodology demonstrates that the addition/removal of a single node can influence representation learning across the entire graph. This characteristic raises concerns about its adaptability to real-world scenarios with dynamic user/item pools. I would like to understand the authors' analysis and potential improvements regarding this aspect.\nThird, beyond time complexity, space complexity represents another critical consideration for graph algorithms, as storing intermediate node states often requires substantial memory resources. Has the new method been evaluated in terms of its space complexity requirements?\nThese three points constitute my primary questions after careful reading. Satisfactory explanations to these concerns could lead to an 'Accept' rating.", "limitations": "Yes, the authors have addressed the limitations in their paper, and I have provided further suggestions in the previous part.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "First of all, regarding the quality of proposed TopoLa technique, its derived methods BPR+ and TopoDR demonstrate statistically significant superior performance over existing models in experiments, confirming the effectiveness of the approach. Furthermore, this work provides comprehensive theoretical foundations for the technique's validity from the perspective of latent hyperbolic geometry, including mathematical proofs and physical property analysis, making it a thorough and well-substantiated research contribution.\nSecondly, regarding the clarity, this paper begins with a systematic analysis of BPR's limitations, employs latent hyperbolic geometry as a robust analytical framework, and subsequently develops novel methods with computational complexity optimizations. The work culminates in comprehensive experimental validation of the proposed approach's effectiveness. The logical progression is complete and well-structured, with particularly clear presentation. Furthermore, the inclusion of source code and datasets in the appendix enhances reproducibility, enabling verification of the reported results.\nThen, regarding the significance of this work, the proposed novel metric represents a breakthrough in traditional loss functions through cross-domain knowledge integration by the research team. Notably, it exhibits considerable model-agnostic characteristics, enabling broad applicability across various graph-structured algorithms. These features suggest substantial potential for future utilization and development in the field, but it still potentially possesses a bit of complexity issue which will be addressed in the Question part.\nFinally, the proposed even-hop paths based method in this paper can be regarded as an extension of traditional BPR loss that incorporates insights from latent hyperbolic geometry. By condensing multi-level topological structure information into multi-hop path representations, this approach achieves more accurate capture of topological similarity and global connectivity. Thus, the underlying originality and reasoning is well-justified.", "was_revised": true, "final_justification": "I confirm that all my questions have now been satisfactorily addressed.", "rebuttal": "We thank reviewer fhF3 for his/her thorough and insightful comments on our method as well as its theoretical analysis. We are highly appreciative of the positive rating. Below are our responses to each raised point.\n\n**Question 1:**  \n> \"Firstly, the paper acknowledges the over-smoothing phenomenon in higher layers of LightGCN while proposing the new method involving multi-layer aggregation. I would appreciate clarification on whether the TopoLa-based approach is susceptible to over-smoothing, or if specific design optimizations have been implemented to prevent this issue.\"\n\n**Response:**  \n  We are grateful to the reviewer for raising this subtle and important issue. Compared to BPR, the introduction of latent hyperbolic geometry in BPR+ alleviates the negative effect of over-smoothing on relationship estimation.\n\n  Over-smoothing refers to the convergence of node features as the network depth increases, leading to indistinguishable embeddings across nodes. In the case of BPR, which relies on inner product to evaluate pairwise relationships, this convergence causes the scores between different sample pairs to become numerically similar. As illustrated in Figure 4a, when dot products are concentrated around a constant value (e.g., 20), the corresponding topological similarities can still vary significantly (e.g., from 20% to 50%). This implies that BPR could yield coarse-grained and uninformative relationship scores under over-smoothed conditions.\n\n  Contrarily, BPR+, by incorporating latent hyperbolic geometry, is more sensitive to subtle structural differences in the graph. Even if node features are similar, the TopoLa distance is still able to reflect the variation in topological similarity between nodes (Figure 4b). As a result, BPR+ provides a more fine-grained and discriminative measurement of node relationships, even in the presence of over-smoothing.\n\n**Question 2:**  \n> \"Then, the proposed methodology demonstrates that the addition/removal of a single node can influence representation learning across the entire graph. This characteristic raises concerns about its adaptability to real-world scenarios with dynamic user/item pools. I would like to understand the authors' analysis and potential improvements regarding this aspect.\"\n\n**Response:**  \n  We sincerely thank the reviewer for this insightful and important question. We acknowledge that the TopoLa distance is conceptually defined over the entire abstract network and thus reflects a form of global structural modeling. However, in practice, our method does not compute TopoLa distance over the full user-item graph at each step. Instead, the BPR+ loss is calculated within each mini-batch, and the corresponding abstract network is constructed only from the users and sampled items in that batch. This means that the actual computations are localized, and the full graph structure is not directly involved in each training step.\n\n  Taking LightGCN as an example, with a batch size of 1,500 and a network containing 40,000 items, introducing or removing a single item changes its sampling probability by approximately $1500/40000 = 3.75\\%$. This change is minor and only affects whether the node is sampled into a future batch. As a result, the composition of nodes in any specific batch remains relatively stable, and the structure of the corresponding abstract network exhibits minimal variation.\n\n  Therefore, while the TopoLa distance can reflect how structural relations change with node addition/removal, such changes only take effect when the node is actually sampled into the batch. Since loss computation is restricted to the batch, the representation learning process remains stable and unaffected by a single node's presence or absence unless it is sampled.\n\n  In summary, the proposed method benefits from structural awareness via TopoLa, but the training is performed on localized, batch-level networks, which makes it naturally robust to small-scale perturbations in the overall user/item pool.\n\n**Question 3:**  \n> \"Third, beyond time complexity, space complexity represents another critical consideration for graph algorithms, as storing intermediate node states often requires substantial memory resources. Has the new method been evaluated in terms of its space complexity requirements? These three points constitute my primary questions after careful reading.\"\n\n**Response:**  \n  We are very thankful for the constructive comments and suggestions. As the main modification of BPR+ lies in replacing the inner product with the TopoLa distance, the space complexity primarily depends on how the relevant matrix representations are stored and computed.\n\n  In BPR, computing the inner product requires storing the embedding matrix $\\boldsymbol{E} \\in \\mathbb{R}^{(m+n) \\times N_e}$, where $m$ and $n$ are the numbers of items and users, respectively, and $N_e$ is the embedding size. This leads to a space complexity of $O((m+n) N_e)$.\n\n  In contrast, BPR+ introduces an additional step of SVD decomposition on $\\boldsymbol{E}$, producing three matrices:   $\\boldsymbol{U} \\in \\mathbb{R}^{(m+n) \\times r}$, $\\boldsymbol{\\Sigma} \\in \\mathbb{R}^{r \\times r}$, and  $\\boldsymbol{V} \\in \\mathbb{R}^{r \\times N_e}$, where $r$ is the rank of $\\boldsymbol{E}$ satisfying $r < \\min(m+n, N_e)$. Therefore, the space complexity for computing the TopoLa distance is $O((m+n)r + r^2)$. Since $r$ is typically smaller than $N_e$, the overall space cost of BPR+ remains manageable and, in some cases, even lower than that of directly storing the full matrix $\\boldsymbol{E}$. \n\nTo provide a concrete example:\n\n  In our experiments with LightGCN, each batch typically contains $m = 1500$ users and $n = 3000$ items, with embedding size $N_e = 64$. The SVD rank $r$ is also set to $64$ for simplicity.\n\n- **The space cost of BPR is**:  \n  $(m+n) \\times N_e = 4500 \\times 64 = 288000$\n\n- **The space cost of BPR+ is**:  \n  $(m+n) \\times r + r^2 = 4500 \\times 64 + 64 \\times 64 = 292096$\n\n\n\nThus, BPR+ introduces only an additional 4,096 units of memory, corresponding to a 1.4% increase in space complexity compared to BPR.\n\n  Considering the performance improvements gained through TopoLa and the small overhead incurred, we believe this trade-off is acceptable and practical."}, {"review_id": "HKpAsDuHPS", "reviewer": "Reviewer_HFzz", "summary": "The paper re-interprets the common BPR loss function in recommendation under the lens of hyperbolic geometry to work on its current limitations and improve them. \n\nConcretely, the authors start by observing that the calculation of the predicted user-item interaction score, namely can be also re-interpreted as the $(u, i)$ entry of the Gram matrix of the node embeddings. This allows to go from the original user-item graph topology to another network involving user and item nodes, along with some dummy nodes representing the embedding latent elements. Thus, calculating $\\hat{y}_{ui}$ eventually implies assessing the number of 2-hops connecting any user and item in the graph through the dummy embedding node. While the number of 2-hops connections is positively correlated to the energy distance between nodes in the hyperbolic space, the authors claim only using 2-hops distances is not sufficient. \n\nThus, they propose an improved version of the BPR loss, dubbed BPR+, where the computation of $\\hat{y}_{ui}$ is replaced with the calculation of the topological distance between the two nodes, computed at even-numbered hops (i.e., 2-4-6-... hops). This improved loss has the advantage of incorporating more global information and capturing the topological structure between users and items in the system. \n\nFinally, as the computation of the topological distances is computationally expensive (because it involves a matrix power repeated multiple times), the authors propose to work on the decomposed representation of node embeddings through SVD. \n\nExtensive experiments on five recommendation datasets, when testing the BPR+ loss on five baselines, demonstrate the improved recommendation performance with respect to the shallow models. Further analyses considering the relation between the topological similarity and the loss, and the union of neighbors and loss, provide another important perspective on the proposed loss. Moreover, an efficiency study shows how the integration of the SVD decomposition can greatly improve the training time of the model. Finally, the authors efficiently apply the same proposed methodology to another similar domain and task, namely, drug repositioning, demonstrating how BPR+ can outperform other baselines on this task.", "questions": "My questions are mainly referred to the last three weaknesses I raised.\n\nQ1) To which power the computations in equations 10-11 are computed?\n\nQ2) Can the proposed approach be potentially applied to monopartite graphs (i.e., graphs with nodes from one single partition, such as citation networks)? And, in case, which would it be the implication of that? Could it be beneficial also to them?\n\nQ3) The literature has shown how computing more than 1 negative for each positive item in the BPR loss can be beneficial to improve the performance [\\*]. How this would impact on your loss function? Would this increase the computational complexity?\n\n**References**\n\n[\\*] Steffen Rendle, Walid Krichene, Li Zhang, and John R. Anderson. 2020. Neural Collaborative Filtering vs. Matrix Factorization Revisited. In RecSys. ACM, 240–248.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n$\\bullet$ The paper has the merit to question the goodness of one of the most established loss methodologies in the literature, namely, BPR, in the realm of graph self-supervised learning. Thus, it can hold great significance in the current literature on the topic. \n\n$\\bullet$ The proposed methodology seems adequately sound, being supported by theoretical proofs for each introduced theorem (as also evidenced in the Supplementary Material part).  \n\n$\\bullet$ It is nice to see how the proposed approach can be applied to any graph-based recommender system without necessarily modifying its structure and methodology underneath.  \n\n$\\bullet$ The experimental section is quite clear and shows the efficacy of the model under different perspectives. \n\n$\\bullet$ Remarkably, the authors apply the proposed approach for another task and domain, namely, drug repositioning, demonstrating how the approach can go further the scenario of personalized item recommendation.\n\n**Weaknesses**\n\n$\\bullet$ To the best of my understanding, the code is not released at review time, thus not ensuring the complete reproducibility of the work. While the authors acknowledge the usage of the framework SSLRec, I think it would have been beneficial for them to provide the code for their implemented work.  \n\n$\\bullet$ To the best of my understanding, it is not clear to which power the matrix computation (equations 10 and 11) is actually performed. The authors should have better clarified this aspect.  \n\n$\\bullet$ I am wondering which would be the implications of applying the proposed strategy to monopartite graphs (instead of bipartite graphs as in the paper's case).  \n\n$\\bullet$ Moreover, I am also wondering how the computational complexity of the model would change if, in the BPR schema, we sample more than 1 negative item per each positive item. The literature has shown how this could potentially improve results, and I think that analyzing this aspect would be beneficial to improve the quality of the work.", "was_revised": true, "final_justification": "After the rebuttal phase, after reading the other reviews, and authors' responses to those, I decide to rise my initial score I gave to the paper to \"Accept\".\n\nSpecifically, all my concerns have been discussed and addressed by the authors.\n\nWhatever the final outcome for this paper, I think it will benefit quite a lot from the fruitful discussion we had along with the authors, the reviewers, and the Area Chair. I believe the paper will improve its quality a lot in any case.", "rebuttal": "We are grateful for the reviewer's thoughtful engagement with our work and their constructive suggestions. Below are our responses to each raised point.\n\n**Question 1:**  \n> \"To which power the computations in equations 10-11 are computed?\"\n\n**Response:**  \nWe thank the reviewer for raising this valuable point regarding the efficiency of computing TopoLa distance. In our formulation, Eq.10 represents a direct matrix-based calculation of even-hop path counts between nodes. It is designed primarily to intuitively illustrate how structural information is aggregated across multiple hops. While it captures all even-length paths, repeated matrix multiplications (e.g., $\\boldsymbol{A}^2$, $\\boldsymbol{A}^4$, ..., $\\boldsymbol{A}^{2k}$) can indeed lead to high computational cost, especially for large $k$. This limitation is also reflected in the runtime analysis shown in Figure 5.\n\nTo address this, we introduce Eq.11, which is derived from Eq.10 via singular value decomposition (SVD). By applying SVD to the embedding matrix, the even-hop contributions are encoded into the diagonal matrix $\\boldsymbol{\\Sigma}$, and since only its diagonal elements are non-zero, the cost of computing high powers of $\\boldsymbol{\\Sigma}$ becomes constant-time, i.e., $O(1)$ for each exponentiation. This dramatically reduces the overall computational burden of accumulating even-hop contributions.\n\nBecause of this optimization, the cost of computing even-hop path contributions becomes independent of the hop count, allowing us to incorporate an arbitrary number of hops without any significant increase in runtime. In our experiments, we observed that contributions from paths beyond 40 hops have a negligible effect on the final TopoLa distance. Therefore, we truncate the expansion at 40 hops to balance accuracy and efficiency.\n\n**Question 2:**  \n> \"Can the proposed approach be potentially applied to monopartite graphs (i.e., graphs with nodes from one single partition, such as citation networks)? And, in case, which would it be the implication of that? Could it be beneficial also to them?\"\n\n**Response:**  \nThanks for raising this subtle question. Our method is inherently compatible with monopartite graphs. As shown in the definition of the embedding matrix $\\boldsymbol{E}$ in the paper, we concatenate user and item embeddings into a single matrix. This formulation treats users and items as nodes of the same type within a unified latent space, which aligns well with the concept of a monopartite graph.\n\nMoreover, the proposed TopoLa distance is defined based on the connectivity patterns (i.e., multi-hop paths) between nodes, regardless of their original semantic roles or partitions. Whether the graph consists of user-item pairs or homogeneous nodes (e.g., in citation networks), our method captures the global structural relationships through even-hop connectivity and thus remains applicable and effective in such scenarios.\n\n**Question 3:**  \n> \"The literature has shown how computing more than 1 negative for each positive item in the BPR loss can be beneficial to improve the performance [*]. How this would impact on your loss function? Would this increase the computational complexity?\"\n\n**Response:**  \nWe thank the reviewer for asking this interesting question. Our current implementation of BPR+ follows the original BPR setting, which samples one negative item per positive instance. However, as the reviewer observed, sampling multiple negatives per positive can often lead to improved model performance.\n\nThe proposed BPR+ framework can naturally accommodate this extension. In fact, the TopoLa-based formulation remains unchanged under multiple negative sampling, as the abstract network is constructed per batch and the TopoLa distance is computed based on all sampled pairs. The only difference lies in the increased number of user-item pairs within each batch.\n\nThis extension would linearly increase the computational cost with respect to the number of negatives per positive, but the additional overhead remains tractable, especially given the fact that our method uses an SVD-based optimization in TopoLa computation. We appreciate the suggestion and will consider including this variant in future work."}, {"review_id": "FaVZJ6rpU6", "reviewer": "Reviewer_uD2G", "summary": "This paper analyzes the limitations of BPR loss in graph self-supervised learning (GSL) from a graph-theoretic perspective. The author argues that BPR relies solely on 2-hop paths for node-relation evaluation, ignoring global topology. They propose BPR+, a novel loss function incorporating even-hop paths via TopoLa distance (derived from latent hyperbolic geometry) to capture global connectivity.", "questions": "See the Weaknesses", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n1. This paper is solid, and most of the theories are accompanied by corresponding theoretical analyses.\n\n2. This paper conduct complete experimental verification to demonstrate the rationality of its analyses and methods, and the proposed BPR+ has shown promising performance improvements.\n\nWeaknesses\n\n1. The inclusion and meaning of the abstract node need to be more clearly presented, since this is the reason why the calculation method of the original BPR loss can be understood as a 2-hop path. This is also one of the core issues addressed in this paper.\n\n2. Although the paper provides reasonable insights, the novelty of the proposed BPR+ has limitations.\n\n3. There is a lack of comparison with related work. More relevant work would help readers understand the background and the contributions of this paper.", "was_revised": true, "final_justification": "After carefully considering the authors’ rebuttal and the other reviewers’ comments, I have decided to keep my original score unchanged. The authors have satisfactorily addressed most of my specific concerns—especially regarding missing baselines and clarifications on experimental settings. However, the overall contribution to the broader field remains incremental; the proposed method offers only modest improvements over existing approaches and does not substantially advance the community’s understanding of the problem. Consequently, I maintain my initial rating.", "rebuttal": "We thank reviewer uD2G for the concise summary of our paper. Below we address his/her concerns.\n\n**Question 1:**  \n> \"The inclusion and meaning of the abstract node need to be more clearly presented, since this is the reason why the calculation method of the original BPR loss can be understood as a 2-hop path. This is also one of the core issues addressed in this paper.\"\n\n**Response:**  \nWe are appreciative to the reviewer for the thoughtful suggestion. The concept of \"abstract node\" is used as a theoretical tool to reinterpret the inner product in BPR from a graph perspective. Specifically, the user embedding $\\mathbf{e}_u$ and item embedding $\\mathbf{e}_v$ are both vectors in the same latent space, and their inner product $\\langle \\mathbf{e}_u, \\mathbf{e}_v \\rangle$ can be viewed as a projection through a shared set of latent dimensions. Each dimension of this space can be interpreted as an abstract node that connects to both users and items.\n\nTo make this more interpretable in topological terms, we consider each latent dimension as an \"abstract node\" that connects to all users and items. Under this view, the inner product corresponds to counting 2-hop connections: $user \\rightarrow abstract node \\rightarrow item$. \n\nThis analogy allows us to model the BPR loss as a function of 2-hop connectivity in a user-item-abstract network. This formulation reveals that BPR only captures coarse structural similarity (i.e., 2-hop paths) and motivates us to extend it to TopoLa distance, which incorporates higher-order even-hop paths to better reflect global graph topology.\n\n**Question 2:**  \n> \"Although the paper provides reasonable insights, the novelty of the proposed BPR+ has limitations.\"\n\n**Response:**  \nWe are grateful to the reviewer for the thoughtful comments on our method. The primary novelty of our work lies in re-examining the widely used BPR loss from a graph-theoretic and network geometry perspective. To the best of our knowledge, this line of thinking has not been explored before. This reinterpretation not only reveals the limitations of the conventional inner product formulation, which effectively corresponds to 2-hop connectivity, but also provides a foundation for integrating latent hyperbolic geometry with graph neural networks.\n\nBased on this theoretical understanding, we introduce the TopoLa distance as a fine-grained topological metric derived from multi-hop path structures, going far beyond BPR's coarse similarity modeling. Building upon this insight, we propose BPR+, which can be viewed as a significantly generalized framework. Its relevance extends beyond BPR, as many other widely-used techniques in graph representation learning, including InfoNCE, self-attention, and contrastive learning, also rely on inner product similarity. Our work highlights the limitations of such similarity measures in theory and provides a new direction for rethinking them from a topological perspective.\n\nWe believe this shift offers valuable insight for the community, and hope that future research will further build upon this line of thought.\n\n**Question 3:**  \n> \"There is a lack of comparison with related work. More relevant work would help readers understand the background and the contributions of this paper.\"\n\n**Response:**  \nWe sincerely thank the reviewer for this helpful suggestion. We review many representative extensions of BPR. Most of them focus on improving input features, loss functions, or specific application scenarios, while keeping the core scoring function, namely the inner product, unchanged. For example, ABPR [1] models feedback uncertainty, VBPR [2] integrates visual features, DPR [3] addresses item recommendation bias, and LkP [4] focuses on ranking diversity. These methods significantly improve BPR from different perspectives, but none of them question or modify the use of the inner product itself.\n\nIn contrast, our work focuses on this fundamental assumption. We revisit BPR from a geometric and graph-theoretic perspective and analyze how the inner product may fail to capture meaningful topological relationships, especially under conditions such as over-smoothing. Based on this observation, we propose using a network geometry-based distance (TopoLa) to replace the inner product as a new way to compute sample similarity.\n\n\n[1] Pan W, Zhong H, Xu C, et al. Adaptive Bayesian personalized ranking for heterogeneous implicit feedbacks[J]. Knowledge-Based Systems, 2015, 73: 173-180.  \n[2] He R, McAuley J. VBPR: visual bayesian personalized ranking from implicit feedback[C]//Proceedings of the AAAI conference on artificial intelligence. 2016, 30(1).  \n[3] Zhu Z, Wang J, Caverlee J. Measuring and mitigating item under-recommendation bias in personalized ranking systems[C]//Proceedings of the 43rd international ACM SIGIR conference on research and development in information retrieval. 2020: 449-458.  \n[4] Liu Y, Walder C, Xie L. Learning k-determinantal point processes for personalized ranking[C]//2024 IEEE 40th International Conference on Data Engineering (ICDE). IEEE, 2024: 1036-1049."}]

Author rebuttal:

Meta-review: This paper furthers the understanding of Bayesian Personalized ranking and develops new algorithms. 
At the heart of the new results is a new measure of Topological distance through the use of hyperbolic geometry. There was overall appreciation of the work. Concerns raised by the reviewers was addressed by the authors during the rebuttal. One of the key concerns was the comparison with other global strategies such as Topola. Authors pointed out that the proposed algorithm was different than Topola. There was consensus that this paper may be considered as accepted. 
However during the discussion phase no reviewers championed this paper. 
Overall assessment is this paper could be of interest to graph neural networks community attending to Neurips community.
Decision: Accept (poster)