CaliGCL: Calibrated Graph Contrastive Learning via Partitioned Similarity and Consistency Discrimination
Authors: Yuena Lin
Hao Wei
Haichun Cai
Bohang Sun
Tao Yang
Zhen Yang
Gengyu Lyu
Venue: NeurIPS 2025 poster ()
Keywords: Graph Contrastive Learning
Exponential Partitioned Similarity
Semantics-Consistency Discrimination
Primary area: general_machine_learning

Abstract: Graph contrastive learning (GCL) aims to learn self-supervised representations by distinguishing positive and negative sample pairs generated from multiple augmented graph views. Despite showing promising performance, GCL still suffers from two critical biases: (1) ***Similarity estimation bias*** arises when feature elements that support positive pair alignment are suppressed by conflicting components within the representation, causing truly positive pairs to appear less similar. (2) ***Semantic shift bias*** occurs when random augmentations alter the underlying semantics of samples, leading to incorrect positive or negative assignments and injecting noise into training. To address these issues, we propose CaliGCL, a GCL model for calibrating the biases by integrating an exponential partitioned similarity measure and a semantics-consistency discriminator. The exponential partitioned similarity computes the similarities among fine-grained partitions obtained through splitting representation vectors and uses exponential scaling to emphasize aligned (positive) partitions while reducing the influence of misaligned (negative) ones. The discriminator dynamically identifies whether augmented sample pairs maintain semantic consistency, enabling correction of misleading contrastive supervision signals. These components jointly reduce biases in similarity estimation and sample pairing, guiding the encoder to learn more robust and semantically meaningful representations. Extensive experiments on multiple benchmarks show that CaliGCL effectively mitigates both types of biases and achieves state-of-the-art performance.

CaliGCL: Calibrated Graph Contrastive Learning via Partitioned Similarity and Consistency Discrimination
Yuena Lin lif Hao Wei l.f, Haichun Cai 2 Bohang Sun Tao Yang , Zhen Yang ' & Gengyu Lyu College of Computer Science, Beijing University of Technology, Beijing
College of Computer and Data Science, Fuzhou University, Fuzhou Idealism Beijing Technology Co., Ltd., Beijing yuenalin@126. com, haowei @emails bjut. edu. en, fjsmchc@163. com sunbohang@bjut. edu. cm, yangtaolilxzy en, yangzhen@bjut. edu cn Ayugengyudgmail.com
Abstract
Graph contrastive learning (GCL) aims to learn self-supervised representations by distinguishing positive and negative sample pairs generated from multiple augmented graph views. Despite showing promising performance, GCL still suf- fers from two critical biases: (1) Similarity estimation bias arises when feature elements that support positive pair alignment suppressed by conflicting compo- nents within the representation, causing truly positive pairs to appear less similar. (2) Semantic shift bias occurs when random augmentations alter the underlying semantics of samples, leading to incorrect positive or negative assignments and injecting noise into training. To address these issues, we propose CaliGCL, a GCL model for calibrating the biases by integrating an exponential partitioned similarity measure and a samantics-consistengg discriminator. The exponential partitioned similarity computes the similarities among fine-graines partitions obtained through splitting representation vectors and uses exponential scaling to emphasize aligned (positive) partitions while reducing the influence of misaligned (negative) ones. The discriminator dynamically identifies whether augmented sample pairs maintain semantic consistency, enabling correction of misleading contrastive supervision signals. These components jointly reduce biases in similarity estimation and sample pairing, guiding the encoder to learn more robust and semantically meaningful rep- resentations. Extensive experiments on multiple benchmarks show that CaliGCL effectively mitigates both types of biases and achieves state-of-the-art performance.
Introduction
General graph contrastive learning (GCL) follows a pipeline to realize self-supervised representation learning [1] 2 S which includes: implement augmentations raw data generate multiple views 4. encode the augmented data to produce representations (5), and finally adopt common contrastive supervision that corresponding samples across different augmented views are positives while others are negatives for contrastive learning [6, (7) Even though this GCL framework is well recognized, still incurs similarity estimation bias and semantic shift bias, which may degrade the graph encoder to generate inferior representations.
The similarity estimation bias arises from the fact that representations produced by the encoder feature elements, where the
typically composed of conflicting positively- and
"Corresponding author, f equal contributions.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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positively-synerginn feature elements are those for supporting the alignment of latent positive pairs within representation vectors, while the preaatvely-synerriisse feature elements are those that work against the positively-synerrisiie ones. When computing the similarity (commonly via inner product) between two representations, these conflicting feature elements would cancel each other out even within positive pairs. This undermines the model ability to identify the latent positive pairs, leading to suboptimal representation learning.
To illustrate how negatively-synergistic features suppress similarity, we conduct a statistical anal- ysis on the ratio between vositively-alignes and negatively-all -allgged partitions under different par- tition settings in Figure LI where a partition is the corresponding feature el-
Positively- Aliigee Partition (w/o aug)
84
Negatively-Aligen Partition (w/o aug)
Aligmment Distribution (%)
SaativellyAligned Partition (w/ aug)
positively-aligee
82
Nogatively-Aligna Partition (w/ aug)
ements have the positive inner product, otherwise regatively-aligned Each partition refers to a sub- set of feature elements within representation. In the experiment, we follow the GCL framework to train an encoder on Citeseer dataset and di- vide the generated representations into multiple partitions. With label guidance, calculate the ratios of positively- and negaaively-aalined parti- tions within positive sample pairs. As the num- ber of partitions increases, the ratio of negatively- aligned partitions grows, revealing the prevalence of conflicting feature elements in the learned repre- sentations. This trend suggests that when comput- ing overall similarity via inner product, multiple negatively-alinner partitions collectively suppress posirively -y-laa-lled ones, ultimately leading to the similarity estimation bias.
80
78
76-
2
3
5
Number Partitions
Figure 1: Ratio (alignment distribution) of positively-aligned to negatively-aligned similar- ity partitions under different partition settings. Shaded histogram bars represent results obtained with graph augmentations, while unshaded ones denote results without augmentations.
Apart from the similarity estimation bias, random graph augmentations in GCL easily flip the relationship between sample pairs, which would result in semantic shift bias. Specifically, graph
augmentations as common tools in GCL @ [0] may alter essential node features or graph topology 12] which potentially flips the relationship between sample pairs: semantically consistent sample pairs may be assigned as negatives, while semantically divergent ones may be incorrectly treated as positives 3 This introduces noise into contrastive supervision and misguides the training process Figure [ illluttraess this problem under the condition where the number of partitions is only 1, and the inner product of the partition is equivalent to the inner product of two representation vectors. It could be found that the ratios change when comparing the alignment distributions with and without augmentations, which suggests that augmentations possibly flip the nature of sample pairs (positive ++ negative). Importantly, when these flipped pairs are treated as fixed positives or negatives under conventional contrastive supervision, the error will propagate through the training process, causing the encoder to become biased due to semantic shift.
14 15
To calibrate the biases in GCL, we propose a novel graph contrastive model CaliGCL, which integrates an exponential partitioned similarity measure and a semantics-consistencyy discriminator. The exponential partitioned similarity measure first splits each representation vector into fine-grained partitions and computes the inner product similarity among each partition, then applies an exponential partition similarities while amplifying the positively- feature elements feature elements to alleviate the similarity
function to shrink the negatively- partition similarities, thereby improving the influence of resitieey-ssynrriise and reducing the suppression of negatively-synerriss estimation bias. The semantics- semantics-consistemn consistenccnnnnnncceooo discriminator [16] 17 takes the sample pair information as input and dynamically predicts relationship consistency scores after augmentation, which are used for calibrating the contrastive supervision to enhance the resilience of GCL and combat semantic shift bias. The two components cooperate to realize debiased GCL and guide the encoder to learn more robust and semantically meaningful representations. Our contributions are summarized as follows:
We identify two critical biases in the standard GCL framework, namely similarity estimation bias and semantic shift bias, and propose novel calibrated graph contrastive learning model, CaliGCL, to systematically address them.

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We propose an exponential partitioned similarity measure to restrict the negatively- synergistic features and amplify the estimation bias. Besides, a semantics-consistnccy discriminator is proposed to correct the semantic relationships of augmented sample pairs, effectively debiasing semantic shift.
features to alleviate the similarity
Extensive experiments across multiple datasets show that the proposed model consistently surpasses advanced self-supervised graph models, which demonstrate that our model effec- tively calibrates the biases in GCL and enhances the representation quality.
Related works
= (V,E), where V } is the node set with N nodes and & C V x V is the edge set. The node attribute matrix and adjacency matrix denoted X E RN and A {0, 1] Nx N respecctively. We aim train an L-layer graph encoder f RN x {0, 1] NxN node representations via raw attribute and structure contents
Problem definition and notations. In this paper, a graph is denoted
{VI, U2, ... y
R Nxd to produce low-dimensional
18 where d is the dimension of the
L-th layer, and F is the dimension of the raw attribute.
Graph neural networks. Graph neural networks (GNNs) have been widely adopted for learning graph data by recursively aggregating and combining information from the local neighborhoods to generate expressive node- or graph-level representations [19] [1]. Each layer in a GNN follows the cressage-passing mechanism, always including AGGREGATE and COMBINE two stages, Given graph G, the output h of 1-th GNN layer is defined as:
m (1) = ACGREGATE
(h :u =EN()))) h = COMBINE (1-11) .m
where ho is the representation of v at [-th layer, and (vv) denotes its neighbors. The AGGREGATE stage gathers neighbor information and outputs aggregated information m!!, while the COMBINE stage integrates it with the self-information h (1111) A graph-level representation is derived with READOUT function 21
hg READOUT (h.) u EE
Graph contrastive learning. A typical GCL framework begins by implementing two augmentation functions To and Ty the input graph g (X, A) obtain two augmented views U To (9) and V = Tv( These augmented graphs are processed by GNN encoder f(.) to generate representations Hy = f(U) and Hy f(V).
To train the model, a contrastive objective follows the contrastive supervision that the corresponding samples in the two views are positive, while other samples negative. For a sample pair (u, v,), the contrastive loss based on the InfoNCE principle [22] is defined as:
exp exp(S(nu; hw,)/T) Re ho.) = log Σ, exp (S(hu, hn.)/T) + En exp (S(hug, (u,,,, 3/T)
(3)
where -(.,) denotes a similarity function, and a temperature coefficient. The overall objective sums the contrastive losses over all nodes:
Les
Σ 2N
1
(Ceelhun + (ee hu.))
Though this simple-yat-effectivee idea that maximizes the mutual information 23
followed by con-
siderable works 24 251, recent studies have raised concerns regarding the rationality of contrastive
supervision, particularly in how positive and negative samples are selected. NCLA (66 notices the
fact that neighboring nodes are viewed as negatives is contradictory with the homophily principle, and introduces neighbor contrastive loss to include additional positives. HomoGCL (27 finds that part of connected nodes belonging to different classes would degrade the model performance when treating them as positives, then leverages Gaussian mixture model to calculate the sample similarities for assigning weights potential positives. While these methods have made important
3

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Encoder Training
Discriminator Training
Decompisse
Decompon
Zu
Gg
Top-k Posibives
Z
G
Expomentiall Patitioned
Calibrated Contrastive
CHOEEE
Predicted
Consistency Labels
Simillrity parrt
Loss figt part
Negabives
Indices of Positives
Indicas of Negabives
Semandios contstenny
Disciininator
Connected Positives
Pair and Fuse
Negatives
Hy
Fusion Repredandation
Fusion Representation Pairs
Balanced Softmax
Decomposi
Loss Lis
Zy
Relatlonshijp
Cv
Consistancy
Labels
Mask Attributes
Concatenation
Indersection
Positives
Negatives
!I Anchor Node
Figure Training process for ClaiGCL. During encoder training, input graph G is transformed as views U and V with augmentations To and Tv to abstract feature information Hy and Hv, and structure information Cy and Cv for constructing fusion representations Zy and Zv. The fusion representations are constructed as inter-view and intra-view pairs that are then fused and input into semantics-consistengg discriminator to predict the relationship labels calibrating the contrastive loss. During discriminator training, is used to abstract feature information Hg and structure information Cg for constructing the fusion representation Zg. where Hg is also employed to select positives and negatives pre-defined relationship labels with the k-NN algorithm and the homophily principle (i.e., connected positives). The positives and negatives are selected from Zg with corresponding indices, which input into the discriminator for obtaining the consistency scores for calculating balanced softmax loss along with the relationship labels.
strides refining contrastive supervision, most of them primarily focus on local neighborhood relationships. In contrast, our proposed CaliGCL expects to calibrate the wrong relationships from global view, which offers a principled solution to reduce similarity estimation bias and semantic shift bias beyond local structural corrections.
3 CaliGCL
The training process of CaliGCL includes encoder training and discriminator training. When training the encoder, CaliGCL implements the exponential partitioned similarity measure into the contrastive loss to relieve the conflicts between positively- and greatively-synerginne introduces semantics-consisters discriminator calibrate the semantic shift bias caused by the fixed contrastive supervision in GCL. Figure shows the overview of both training processes.
features. Besides, it
3.1 Exponential partitioned similarity measure
When gauging the similarity of a positive pair, the positively-synerrisiie feature elements dominate to capture semantic consistency. However, the representation vectors are always composed both positively- and negatively-synergisti feature elements, which easily cause conflicts in similarity calculation and obscure the semantic relationships between sample pairs.
Theorem Given a positive representation pair (he: h2) with d feature elements, where each element h., and h. is independent and identically distributed (i.i.d)) following normal distribution N Nu, o²). Suppose that the expectation of the inner product of h and h is a positive value M. then for partition consisting of m (m d) features, the probability that the inner product over

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this partition a negative value is 4 function of the standard normal distribution.
-mp²
where ((-) is the cumulative distribution
Vmlu22 20253
This theorem provides the possibility that negatively-synergiin features exist in a positive pair. To address this, we introduce an exponential partitioned similarity measure, specifically designed to reduce the impairments of regativell-syyergiise
feature elements while enhancing the influence of positively-synergiss ones, thereby mitigating the conflicts between the two types of features and strengthening semantic alignment in representation learning.
In our design, a representation is split into fine- -eeee-rriiddd partitions, with each containing a subset of the feature elements. Thus, for the representation h., f(;) E Re of node vi can be represented as h., = [ho. || he. || ... || hek) where || is the concatenation operator, and (8)) K (ho.. ]==1 represents the different partitions of hu..- Among these partitions, each of the first (K - 1) partitions contains m feature elements, and the last partition has d - (K - 1) Im feature elements, where is hyperparameter to control the partition size.
We use Spart (fe, he,)) to represent the exponential partitioned similarity between a sample pair (us, 0y), which is mathematically defined as the sum of inner product similarity over K partitions:
1
K
K heg
(k)T
(k) Reng
Sport (he Ru,)
exp
(5)
K
T
where T is the temperature coefficient. Unlike the commonly-used exponential inner product in contrastive learning, i.c., exp(ha h.., hT, our method computes the inner product for each partition independently before applying the exponential sum on the results. This design naturally suppresses the impact of negatively-alignn ones. For instance, given a positive pair (v;, ej) having positive inner product divided into two partitions where the first is negatively-aligner and the second pnsitively-aligned it is easy to find the following conclusion:
partitions by inducing their exponentials to approach zero while
amplifying positively-aligned
2 2-
(1)T (1)
hus
2 h
(2)T
h
(2)
h levs
exp
O, and exp
2 exp
(6)
T
T
T
This holds since h,,, ho, partitioned similarity is more influenced by positively-syneriisee components. Furthermore, we provide the following theorem to explain that the proposed exponential partitioned similarity is larger than the standard exponential inner product across scenarios.
(L) he: h
(1) + h. he and h h::
(2)
h. he, Thus, the overall
Theorem 2
The exponential partitioned similarity provides an upper bound on the standard expo-
nential inner product similarity between two representations.
This similarity measure will be used in the contrastive loss for debiasing the similarity conflicts.
3.2 Semsnties-consisseney discriminator
During the contrastive training, random augmentations may flip the semantic consistency between sample pairs, which implies that positive and negative sample pairs convert each other potentially. This is difficult to perceive with the fixed contrastive supervision that the corresponding samples across different augmented views are positives, while others are negatives. To alleviate the harm from the strong assumption, we propose a semantics-consistenn semantic relationships among sample pairs during the training process. The discrimination process includes two parts: (1) feature and structure encoding, and (2) semantics-consistency discrimination.
discriminator to dynamically evaluate the
Feature and structure encoding. To provide sufficient pair information to support semantics- consistency judgements, we incorporate both feature and structural information as the input for the discriminator. For sample pair (v;, u2),, we employ GNN encoder generate the corresponding representation pair (h,,, h,,)) as the feature information. Meanwhile, we decompose the graph Laplacian matrix obtain the eigenvectors as node structural information (28), which is defined as:
À = UAUT.
5

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A is the normalized adjacency matrix as [29]. and it is defined as A = D- AD 3, where A = A IN. D is a diagonal matrix, and its diagonal element is D;: Σ, orthogonal matrix constructed with eigenvectors, and A is diagonal matrix filled with eigenvalues. We select t eigenvectors corresponding top-t greatest eigenvalues as the final node structure embeddings, which is denoted as C' E Ik: and denote the structure embeddings for and Vj as Cu, and Coys respectively.
A.j. U is an
Semantics-consistencyy discrimination. In this process, the feature encoding (h he,)) and structural encoding (Cu,, Cuz) are concatenated to form new fusion representation pair as (ze, hu, || Cv,, Re; Re, Cv,) Then the representation pair (v,) containing both the feature and structural information is used for evaluating the semantic consistency. Specifically, the representation pair (Ze:: 2vy) is first fused and then input into the camantics-consiseens discriminator D(-) that is defined as multi-layer perceptron (MLP) with non-linear actívation function. The process is defined as:
Po,,,, Dzz,, ① Zuz).
(8)
where the Hadamard product @ is used to fuse Zv, and žuy- Po,,,, the predicted consistency score of pair (Z Zn;) by the discriminator. Finally, Puam; is used to judge the semantic relationship between and vy according to the following rule:
ten., = =(0, f1, if Poris 2
(
if Per.es
where v) is the threshold. In,,o, defined as the predicted consistency label, where Teyoy = means the pair (e;, Uz) is positive, and Te,.r, 0 denotes the pair is negative. This label is critical for recognizing the varying relationships of sample pairs.
3.3 Training strategy for CaliGCL
Training overview. The training strategy for CaliGCL is composed of a pre-training process and fine-tuning process. The pre-training process trains the GNN encoder to produce similarity-debiase representations by integrating the exponential partitioned similarity into the contrastive learning and develops the ability the semantics of sample pairs. The fine-tuning process adopts an alternating algorithm: the discriminator employed to calibrate the contrastive loss by correcting false positive or negative sample pair assignments caused by ietpeenttatonnnnnueee semantic flips. Then the calibrated encoder generates more reliable representations to further enhance the discriminator in distinguishing the semantics- consistency relationships. The training strategy for improving both representation quality and semantic discrimination is detailed in the following.
discriminator for capturing the changing
3.3.1 Pre-training process
Pre-training for similarity-debisse GNN. Following the standard GCL paradigm, we implement the augmentations on the raw graph to generate two augmented graphs and V, where each node u, U and its counterpart U V form positive pair, while others negatives. The key difference in our framework lies replacing the exponential inner product similarity function with the proposed exponential partitioned similarity in the contrastive loss, which is called the partitioned contrastive loss here. And the partitioned contrastive loss for u; is defined as
Spart (Bug: hw:)) lat la--part (u:) log E, S Spart (hr hv,)) +Efes Spart hug
(10)
To ensure sufficient interactions among feature dimensions across partitions, shuffle the feature columns representations before loss computation. And the overall loss is given by:
1
Lel-part
Σ (et-par((u))) + let-pare(U;)].
(11)
2N
Theorem Assume the similarity of each partition for positive pairs Stare obeys N (pi, of), and the similarity of each partition for negative pairs S Sport obeys N (H2, 02). If o > of and the partition

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similarities are i...d. then the partitioned contrastive objective serves as tighter lower bound on mutual information, and yields an upper-bounded estimate compared with the contrastive objective.
The assumption that of o2 is reasonable based on the fact that the positive samples similar because of the existence of positively-synergistic feature elements, while other negatively-syneriiste feature elements would cause great divergence. In contrast, negative pairs tend to be uniformly dissimilar, leading to lower overall variance in their feature alignment. This variance divergence across partitions results in sharper separation under exponential partitioned similarity, thus achieving a more ideal contrastive objective.
Pre-training for semantics- conssteeeyy discriminator. The key to training the discriminator depends on adopting reliable positive and negative pairs with pre-defined relationship labels. Specifi- cally, a sample pair (i,, U2) is assigned positive relationship label lo.es, = 1 only if the samples are both directly connected (topology proximity) and k-NN neighbors (feature closeness). Otherwise, we assign fe,,,, and treat the pairs as negative. Since the pre-trained GNN encoder still suffers from the semantic shift bias, the k-NN algorithm is applied in the original feature space but not the representation space.
Let pos and Nneg denote the total number of pre-defined positive and negative pairs, respectively. Due to Npos < Nneg in general, the imbalance leads the discriminator to overly favor negative predictions. Thus, we employ the Balanced Softmax loss 30 31 to mitigate the imbalance problem:
Mer,o 1) exp(Pr.., + log Npos) + The,,,y = 0) exp(Pe.nn + log Nneg)
Los Σ log i.j
exp(Pe..,, + log Npos) + exp(Pe..., + log Nneg)
(12)
where Po..yy is the predicted consistency score for the pair (u;, 1z) according to Eq. 88 and II()) is the indicator function. This loss effectively prevents the prediction bias and enables the discriminator to more accurately distinguish semantically consistent pairs.
3.3.2 Fine-tuning process
After the pre-training process, the initially calibrated graph encoder could produce relatively high- quality representations, while the gllaionship-consisterr detect semantic changes among sample pairs. It is proper to further enhance both components and drive them towards mutual reinforcement. Considering this, we adopt an alternating training strategy between the GNN encoder and the discriminator.
discriminator possesses a certain ability to
Encoder fine-tuning with calibrated supervision. In each iteration, we first obtain two augmented graph views U and V of the input graph with graph augmentations, then follow the procedure in Section 3.2 to yield the fusion representation matrices Zy and Zv for the two augmented views. These matrices are used for constructing the intra-view pairs (Zu; Zu), (Zv, Zv) and inter-view pair (Zy, Zv) as the input of the discriminator. The discriminator then estimates the consistency score of each sample pair among these fusion representation pairs to further calibrate the encoder through the calibrated contrastive loss:
(cali(u;)
Exturoy Spart (eur: hwy) + Ejjts Jurny Spare Shurs Kuy)) - -log Eglus,o Spart (hrug h.,)) + Eg Mer.os Spart(hu, huy) + Ejzstie 11 Merin Spart (hrus hug)
(13)
where lur.o, I(Par,e 2 1) represents the consistency label between u, and vy by inputting the corresponding fusion representation into the discriminator, and Pu = 1 1- Iur.yy. Additionally, we only add the positive sample pairs with the highest confidence in the denominator to reduce computation cost, where an adopted sample pair is judged as positive by the discriminator and includes corresponding samples between the augmentation views. This calibrated contrastive loss promotes alignment between consistent pairs while suppressing misleading or semantically flipped pairs that arise from graph augmentations. Similarly, we have the overall loss for fine-tuning GNN as
N
Leati 2N
(us) + Eoats (v:))
(14)
7

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Table 1: Empirical results of self self-supervised representation learning in node classification. The best results are highlighted in bold, and the second-best results are highlighted with underline.
Dataset
Cora
Citeseer
Pubmed
DBLP
Photo
Computers
AFGRL
83,34 1 0.87
71.49 + 0.78
85.21 = 0.22
83.08 = 0.14
93.22 : 0.28
89.88 : 0.33
SUGRL
85.37 + 0.55
73.49 + 0.66
86,68 = 0.21
83,03 # 0.23
93.20 2 0,40
88,90 0.20
NCLA
7
85.47 # 0,40
83.97 0.17
93.48
89.14
GREET
0.46
86.95 0.62
8313
9.31
87.94 0.35
HomoGCL
0.44
72.34
86.31 0.18
0.13
03.25
90.09 = 0.32
S2GAE
0.28
0.477
86,46 0.09
84.08
93.5
89.99 = 0.14
iGCL
0.443
72.85 0.85
86.17
83.74
93.10
90.06 = 0.41
PIGCL
73,51 0.78
8.23
84.30
93.14
89.25 = 0.27
Bandana
73.04 0.84
86.35
8.02
89.62 = 0.09 93.44
SGRL
8.23
73.20 + 0.22
86.17 + 0.01
8.01
9.08
90.02 = 0.02
CaliGCL
85.87 # 0.62
74.13 + 0.54
87.16
85.32 =5.32 0.13
93.72
90.79 = 0.28
Discriminator enhancement with updated representations. Discriminator fine-tuning is similar to the pre-traininn pre-training. but differs in the sample selection strategy. Since the semantic shift bias of the encoder is alleviated by the discriminator, the produced representations of the encoder are more reliable. Considering this, we apply the k-NN algorithm in the representation space to pick the positives and negatives rather than the original feature space. These pairs are used to update the discriminator with Eq. (12), allowing it to accurately model semantic consistency based progressively refined representations.
4 Experiments
In this section, we evaluate our proposed model across multiple widely used graph datasets, including citation networks (Cora, Citeseer, Pubmed, and DBLP), co-purchase networks (Amazon Photo, and Amazon Computers), social networks (COLLAB. REDDIT-BINARY. REDDIT-MULTI-SK IMDB-BINARY) and biochemical networks (NCI1 PROTEINS, DD, and MUTAG), The detailed dataset information and hyperparameter settings for these datasets are provided in the Appendix B. In the comparative experiments, the node classification and graph classification are employed to evaluate the model expressiveness. For fair comparison, we compare our model with recent state-of -frt self-supervised graph models for each task. All the experiments are implemented in Pytorch and conducted on server with RTX 3090 (24 GB).
4.1 Node classification
In node classification, we compare our model against both contrastive and generative graph rep- resentation learning methods, including AFGRL (II), SUGRL (32) NCLA (26), GREET [33). HomoGCL (27). S2GAE [3], iGCL (35) PiGCL [36), Bandana (37. and SGRL 38] In our
evaluation pipeline, the raw graph first encoded to produce node representations, which are then employed for training linear classifier. Due to the weakness of the linear classifier, the performance mainly depends on the representation quality. In our experiment setting, we split 10% representations for training the classifier, 10% for validation, and the remainder for testing. We report the mean FI with standard deviation to evaluate the performance, and repeat experiments 10 times to provide reliable outcomes in Table II
The comparison results reveal that our model competitive even surpasses other baselines, which validates the model effectiivnesss. Additionally, we have the following observations:
(1) GCL models that solely rely on cross-view positives, such as GCA and PiGCL, show inferior performance to other models incorporating the extra neighbor positives, e.g., NCLA, GREET, and HomoGCL. suggests the significance of calibrating the contrastive supervision.
(2) Generative models like S2GAE and Bandana, which focus on reconstructing graph topology, achieve performance on par with powerful contrastive methods. This supports the homophily principle and further explains why those GCL models focusing on local consistency perform well.
(3) The proposed CaliGCI. effectively calibrates the similarity estimation bias and semantic shift bias in GCL, showing best performance nearly all the datasets. It signifies that CaliGCL has sufficient
8

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Table 2: Comparisons with different graph models in graph classification. The best results highlighted in bold, and the second-bes results are highlighted with underline.
Biochemical Molecules
Social Networks
Datasets
NC11
DD
MUTAG
COLLAB
RDT-B
RDT-M5K
IMDB-B
InfoGCI.
=
72.9 # 1.8
89.0 1.1
70.7 1.1
82.5 1.4
53.5 1.0
73.0 =
GraphCI.
7.4
7..6
1..
1.2
89.5 = 0.8
56.0
71.1 0.4
JOAO
7.51
0.5
87.4
6.4
55.7 0.6
7.12
JOAOv2
7.5
1.24
8..7
0.3
1.5
5..0
70.8
AD-GCL
6.57
7.55
1.
0.6
8.8
53.0
71.6
SimGRACE
79.1
1.1
89.0
71.7
0.8
8.9
0.3
71.3
SPAN
0.54
0.5
89.1
0.5
8..6
0.5
73.7 0.7
GPA
8.4
79.9
89.7
0.1
8..3
5.2
74.6
DRGCL
7.47
0.74
0.6
0.8
0.3
56.3
7.50
CaliGCL
80.8 2.0
2.5
90.4588
1.7
9.77
56.5
76,9 5.00
competence to distinguish its positives and negatives, benefiting the downstream classification task. (4) While CaliGCL performs well on smaller datasets such as Cora and Citeseer, it exhibits even stronger improvements on larger datasets like DBLP and Computers. This highlights its ability to uncover and utilize latent positives that are often neglected by the conventional contrastive loss.
The superior performance of CaliGCL stems from its ability to calibrate both the similarity estimation bias and semantic shift bias GCL, which allows CaliGCL to identify latent positive pairs and remove spurious negatives, ultimately leading to more discriminative representations.
4.2 Graph classification
In this experiment, we evaluate the performance of CaliGCL on the graph classification task under an unsupervised learning setting, following standard protocols introduced in [39] and (40). Different from the node classification, the positives based on the k-NN algorithm are difficult to obtain since graph datasets may lack feature information. Thus, the discriminator is not pre-tree mre-trained and ignores the structure information.
For the experiment setting, we first train GIN encoder and then train SVM classifier to classify the produced graph representations via 10-fold cross-validation. Our proposed model compared with
recent self-supervised graph models, including InfoGCL [41]. GraphCL [39], JOAO and JOAOv2
[42) AD-GCL [43), SimGRACE [43]. SPAN [44) GPA [40], and DRGCL 45). According to the results in Table we have the following observations:
(1) When compared with the augmentation- model SimGRACE, CaliGCL preserves the augmen- tations to realize sample diversity, which is erucial for effective GCL learning. Moreover, CaliGCL mitigates the adverse effects of the augmentations by calibrating the biases inherent in GCL.
(2) CaliGCL still outperforms models that employ learnable augmentations, such GPA. This superiority is attributed to the competence of debiasing semantic shift, which is also neglected by models that only optimize augmentation strength combination strategies, without considering their potential to distort semantic consistency between sample pairs.
Overall, the graph classification results confirm that CaliGCL could also boost the graph-level task by adapting a contrastive learning framework to work on graph-level representations. This highlights the flexibility and robustness of the proposed framework in self-supervised learning settings.
4.3 Hyperparameter analysis
we conduct sensitivity analysis on the temperature coefficient and the partition size and report the mean F1 as the model performance Figure 33 Figure B]a) and (b) show that the performance changes over different values of r, indicating that the temperature coefficient is critical for distinguishing the positives and negatives. Figure c)) and (d) demonstrate that partitions with an appropriate number of feature elements can improve the model performance. This supports the effectiveness of partitioning the representation vectors to calibrate fine-grained semantic structures. However, more partitions may introduce variance and degrade the performance.

---

77.0
91.0
77.0
26.3.
90.5-
Retormmne thd
26.0
13.3
75.5
68.5
75.0
75.0
0.8
89.0
1.2
1-
25.
128
Partition Size
66
89
125
Puttilos
(a) IMDB-B
(b) RDT-B
(c) IMDB-B
(d) RDT-B
Figure 3: Hyperparameter analysis the temperature coefficient and the partition size m.
Conclusion
In this paper, we identify two biases in GCL: similarity estimation bias caused by conflicts between positively- and pegatively-synergiste features, and semantic shift bias introduced by random augmen- tations. To mitigate the issues, we propose CaliGCL to enhance the influence of pesitively-synerge features while suppressing negative ones, and correct semantic inconsistencies in contrastive pairs. However, the exponential partitioned similarity measure is not adaptive, and it could be further enhanced by automatically distinguishing the positively- and negatively-synerge features.
Acknowledgments and Disclosure of Funding
This work was supported by the National Natural Science Foundation of China (No. 62306020), the Young Elite Scientist Sponsorship Program by BAST (No. BYESS2024199), Beijing Natural Science Foundation (No. L244009), the National Key Research and Development Program of China (No. 1023YF31007000 and partially supported by the Central Guidance for Local Scientific and Technological Development Fund (No 204220242)0)24) The first author is funded by the China Scholarship Council (CSC) from the Ministry of Education, China.
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13

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A Theorem proofs
A.1 The proof of Theorem
Proof. Let h and hug be the i-th elements of h2 and hgs respectively. Since h and h are ii.d., then the expectation of ho, is
[he,a = n.². Based on this, the expectation of the inner product over m (m S d) features is
B = h (
(15)
E huull mp².
(16)
For hu,s, N( we have
Elhi,d Var(hu..)) + (EE (un 0² + p²² Similarly, we have E,h,, 0² + n.2. Thus, the expectation of h2 i is
(17)
= (o² + 11²)2-
(18)
Then the variance of hh is calculated Var(hu,,hh..) = B[h hE hv x) = (o² 12/22 1====²= =
(19)
Since f and h are independent, we have
Var
hu Mus mly + 22220))
(20)
Therefore, by the Central Limit Theorem, we have Em Munthr N(mpp, m(n² + 20²)2). Finally, the probability that
Ex ii=l hun < is
mm² mmla² + 20023
P
hustovie 0 x0==
+
(21)
A.2 The proof of Theorem 2
Proof. We consider the temperature coefficient for both the exponential partitioned similarity and the exponential inner product similarity. For pair (hu, hw), we have the standard exponential inner product similarity defined as:
K
S(hus lw.) = exp(h. - exp
Σ
h
(B)T
(k) h
where S(.) denotes the exponential inner product similarity, and h h (b) are the k-th partitions of representations h, and h., respectively. The exponential partitioned similarity measure is defined as
K
Spare(huus
Σ
exp
h
(E)T
h
(k)
(22)
K
By Jensen's 's inequality and the convexity of the exponential function, for all S S,, E R (each representing the inner product of partition), have
exp
Sk
exp(Sk),
(23)
which leads
K
K
exp
Σ
Sτ
<
exp (KSt),
(24)
K
k=
This inequality corresponds to
S(bu, hv) Spare kw).
(25)
14

---

A.3 The proof of Theorem 3
To prove the theorem, we first provide the following Lemma
Lemma 4 Let h and h- be positive sample pairs drawn from joint distribution p(h, h",, and let Spart (a = K Et= exp ( h h((h)) denote the exponential partitioned similarity between two representations. Define the partitioned contrastive loss as:
Sport (te, htt)
Spart -Bpatt
loog
Sparp Chari(,,.. h!)
Then, the the following lower bound on mutual information holds:
T(h; h
2 log N Lepart
where N is the number of samples.
Proof. This result follows from the lower bound of mutual information derived in 22 which shows that any InfoNCE-style loss of the form:
p(h+ [h) LNCE -nn, log En p(h'(h))
(26)
satisfies the lower bound I(h: 2 log N LNCE for any positive scoring function p(h (h) approximating the true density ratio. In our case, we define p(h*1h) x Spart ht), which is strictly positive and normalized over the dataset via softmax. Therefore, the loss Lpart conforms to the InfoNCE formulation and inherits the same mutual information lower bound.
Based Lemma 4, the proof of Theorem 3 is transformed to prove that Part is the lower bound of LNCE- Therefore, we provide the following proof.
For positive pair, its inner product similarity S obeys the following distribution:
S
s(1)+ ~ N((KI,
(27)
k=1
Similarly, the distribution for the inner product similarity of negative pair S is
K
S
S ~ NN(K2, Kož).
(28)
k=1
Then the original contrastive loss for sample pair defined as:
exp(S+)
Lat - log
(29)
exp(S+) E, exp(S,)'
where S is the inner product similarity between a positive pair, and S represents the inner product
similarity between any other negative pair. Similarly, the partition contrastive loss is given by
1 E. (xp(KK(()) part
Let-part part = - -log
Ex exp K
(b)+ part
K Ej Ex exp( K Sprtt))
K
(30)
Σx exp K (()+t)
lolog
Σn exp(KS
part
(k)+
E3 E. exp K part.))
where Spart is the inner product similarity of the k-th partition between a positive pair, and Sport.j represents the inner product similarity of the k-th partition between negative pair.
Let
A exp(S*)
Ā E. (ep((K(())
(31)
exp(St) + Σ, Sj
B Ex exp(X KS s(R)+
E3 Ex exp(Ks(()) part..)
15

---

the comparison between the two losses is transformed to compare A with B B
Let
a 4
8 B
P1 E[exp(S*))
(32)
P2 = Elexp( K -5())+))
Q1 Elexp(S
Q2 E[exp(K S(()
and we implement the expectations to approximate a and 3:
Then we have
P3
Pt
a ~3
B P + (N
(33)
P2 + (N-
1)Q2
Therefore, R P
-N it means that whether the ratio between a and P. P2 11
( is greater depends on the comparison between P: and Q2
According the assumption that the inner product similarities of partitions follow the Gaussian distribution, the exponentials of the inner product similarities obey the log-normal distribution, and have
Elexp
exp(Km +
Ko. 2
P1 Elexp K ((h)+ exp(K K²0
(34)
Q Elexp(( = exp( K μ22
Ko²
Q2 E|exp(K - s )] cxp( K P2
K²o
2
We turn the comparison between Q142 and Q2Pi to the comparison between log & and log P. then we have
(2a"
Pi
P2
K - K2 (oî -02)
log
log
(35)
Q1
Q2
2
Since K K,K K2 O. According the assumption of - 2 0, we have
Pi
P2
log
log Q2
< 0.
(36)
This means that the partition contrastive loss the lower bound of the original contrastive loss, and the partition contrastive objective is an upper bound of mutual information than the original contrastive objective.
B
Experiment
B.1 Datasets
We supplement the concrete descriptions of datasets in the following. Besides, the statistics of the datasets for learning node-level representations are shown in Table and those for learning graph-level representations are shown in Table
Cora, CiteSeer, Pubmed, and DBLP are four most widely used citation network datasets. In these datasets, each node represents a scientific paper, and the node attributes correspond to specific keywords extracted from the paper. An edge between nodes indicates citation relationship.
Amazon Photo and Amazon Computers are co-purchase networks, where nodes represent products and edges indicate that two products frequently bought together. Node features derived using a bag-of words representation based on product descriptions.
NCT1 is molecular graph dataset used for predicting chemical compounds' activity against cancer cells. MUTAG is another molecular dataset, commonly used for mutagenicity prediction. DD also contains chemical compounds and is employed in general graph classification tasks.
16

---

COLLAB is a social network dataset used for collaborative filtering. REDDIT-BINARY (RDT- B) consists of Reddit threads and is used for binary community classificatio. IMDB-BINARY (IMDB-B) is dataset derived from movie collaboration networks, labeled for binary classification. REDDIT-MULTISK (RDT-M5K) a multi-class variant of the Reddit dataset, containing threads labeled into five categories.
Table 3: Statistics of different graph datasets for learning node-level representations.
Amazon
Amazon
Dataset
Cora
Citeseer
Pubmed
DBLP
Photo
Computers
Nodes
2,708
3,327
19,717
17.716
7,650
13,752
Edges 5,429
4.732
44,338
52,867
119,081
491,722
Features
1,433
3,703
500
1,639
745
767
Classes
7
6
3
4
8
10
Table 4: Statistics of different graph datasets for learning graph-level representations.
Data Type
Name
Graphs
Average Nodes
Average Edges
Classes
Biochemical NCI1
4,110
29.87
32.30
MUTAG
188
17.93
19.79
Molecules
DD
1.178
284.32
715.66
COLLAB
5,000
74.5
2457.78
REDDIT-BINAN
2,000
429.6
497.75
Social Networks
REDDIT-MULTI-SK
4,999
508.8
594.87
MID--BINARY
1,000
19.8
96.53
B.2 Hyperparameter Setting
The hyperparameter settings for node and graph classification are listed in Table respectively. For the hyperparameters used in the node-level datasets, Enc is the learning rate of the graph encoder in the pre-training phase, Epc Init Enc is the training epochs of the graph encoder in the pre-training phase, and Hid dim is the hidden dimension of the graph encoder. Similarly, Ir Lr_Dis, Inii__nttt______t________e Dis dis_ Dis act are the learning rate, training epochs, hidden dimension, and activation function for the discriminator the pre-training phase, respectively. Leaky represents LeakyRel.U activation function. Itr fr__nmmi_ is the number of iterations in the alternating training strategy. where the graph encoder is trained with Epc_ Le__FT___ncC___ epochs in each iteration and the discriminator is trained with Epc_FT FT Dis epochs in each iteration. Proj_din dim is the hidden dimension of the projector head, is the dimension the structure embedding, m is the number of feature elements in a partition, 7 is the threshold to distinguish the positive pairs and negative pairs for the discriminator, and the temperature coefficient. Pa1. Pf1, Pe2, and Pf2 are four hyperparameters for controlling the strength of graph augmentations. The hyperparameter settings in graph-level datasets similar to he node-level datasets, but we abandon the structure embeddings since it is not appropriate to node structure embeddings for constructing graph structure embeddings. Augmentation is the graph-level augmentation strategy adopted by the datasets, where 'dnodes' to drop nodes, and 'random' includes node dropping, edge perturbation, attribute masking, and subgraph sampling. All the augmentations adopt the default ratio as [39]
B.3 Ablation study
We perform ablation studies to assess the contributions of the exponential partitioned similarity ("Spart") and the semantics-consistency discriminator ("D") the CaliGCL framework, and the results are reported in Figure From the results, we could find that the exponential partitioned simi- larity and the discriminator could enhance the model performance. However, joint implementation of both methods effectively alleviate the biases in GCL and further improve the model performance.
17

---

Table 5: Hyperparameter setting for node-level datasets.
Datasets
Cora
Citeseer
Pubmed
DBLP
Photo
Computers
Lr Enc
le-4
le-3
3e-4
2e-4
Sc-4
1e-3
Epc Inn Ene
300
300
8000
1000
6000
9000
Hid dim
[512,256]
[512.256]
[512,256]
[512,256]
[512,256]
[512.256]
Lr Dis
4e-4
8e-4
8e-3
8e-3
8e-3
8e-4
Dis dim
128
512
256
128
128
512
Dis_act
Leaky
Leaky
Leaky
Leaky
Leaky
Leaky
Epc_FT_ Fi_FTDDis
10
10
10
5
20
num
15
30
5
15
Epc FT Enc
15
15
15
15
5
15
Epc Init Dis
25
10
25
150
40
25
Proj dim
128
256
512
$12
512
$12
64
64
64
64
64
m
128
240
240
128
192
200
0.6
0.6
0.6
0.6
0.6
0.6
0.8
0.9
0.23
1.2
0.1
0.1
Pel
0.2
0.2
0.4
0.1
0.3
0.5
P/1
0.3
0.0
0.4
0.2
Pr2
0.4
0.4
0.1
0.1
0.5
0.5
Pf2
0.4
0.5
0.2
0.0
0.1
0.1
Table 6: Hyperparameter setting for graph-level datasets.
Datasets
MUTAG
IMDB-B
COLLAB
RDT-B
DD
NCT1
RDT-M5K
Ir Ene
2c-3
2e-3
Tc-3
3e-3
4e-3
3e-3
1e-3
Epc_ Enn
40
40
270
80
90
270
270
Hid dim
[128,128]
[128, 128]
[256,128] 128]
[128,128]
[256,256]
[128,128] 128]
[128,128] 128]
Epc FT Enc
15
15
15
15
15
15
Lr Dis
8e-3
8e-3
8e-3
8e-3
8e-3
8e-3
8e-3
Dis. dim
128
128
128
128
64
64
128
Dis act
Leaky
Leaky
Leaky
Leaky
Leaky
Leaky
Leaky
Epc_FT_DD
5
5
10
Itr_num
5
5
5
5
5
5
dim dim
128
128
256
128
256
256
256
Augmentation
dnodes
random
dnodes
dnodes
dnodes
dnodes
dnodes
m
128
128
128
256
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.8
0.05
1.2
0.1
0.05
0.04
Ga
Joh
wa
w!t
(a) CiteSecr
(b) MUTAG
IMDB-B
(d) RDT-B
Figure 4: Ablation study to analyze the exponential partitioned similarity Spart and the semantics- consistency discriminator D.
B.4 Quantitative analysis
We conduct quantitative analyses to evaluate the effectiveness of the exponential partitioned similarity measure and the semantics-consistency discriminator in CaliGCL.
To assess the partitioned similarity measure, we train the GNN encoder with both the general contrastive loss and the partitioned contrastive loss. We then compare the distribution of positively-
18

---

Alignment Distribution (%)
100
100
Alignment Distribution
80
880
10
10
0201 Peeitery Kineed ol Certnstte look.
Poctielly Alyned diesera Dvatavaivo Lesd
Nogativi Ailordd jonnec Crntractie Eoco
Sepprielyy Algyod ciinees Oontaottin Leis
Mruutndy-Akigee chassiimee Contrultine Lend
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Figure 5: Quantitative analysis on similarity estimation bias, where "aug" means the data with augmentations and "w/o aug" means the data without augmentations.
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aligned and negatively-alignen partitions in Figure a and (b) IMDB-B and RDT- datasets, as well as the average similarity among positive pairs, which is defined as the total similarity divided by the number of positive pairs in Figure (c and (d). The observed increases in the proportion of positively-aligned partitions and average similarity clearly demonstrate that the partitioned contrastive loss effectively strengthens positively-syneeristte features. Additionally, we compare the average similarity of positives with and without augmentations. Although random augmentations tend to reduce similarity among positives, the graph model using the exponential partitioned similarity still maintains good performance.
To evaluate the function of the semantics-consisteng discriminator, we compare its prediction predictions. Specifically, we count the number of sample pairs predicted positive pairs by the inner product and by the momantics-consistens pairs with predicted consistency label i = 1), and compute their ratios over the ground-truth positives, respectively. As shown Figure the discriminator yields more accurate predictions, supporting its role in effectively calibrating the contrastive loss.
accuracy against similarity-bart
discriminator (i.e.,
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(a) AFGRL
(b) GREET
(c) Bandana
(d) S2GAE
(c) iGCL
PIGCL
(g) SGRL
(h) CaliGCL.
Figure 7: t-SNE visualization of CaliGCI. and the comparative models Citeseer dataset.
B.5 Visualization
We provide t-SNE visualization comparing CaliGCL and other baselines to demonstrate the quality of learned representations. From the visualization results, we find the samples different classes in PiGCL and iGCL are mixed; the boundaries of different classes in GREET, S2GAE, Bandana, and SGRL are ambiguous without margins. In contrast, the boundaries of different classes CaliGCL separated with larger margins.
B.6 Model complexity
We analyze the time and space complexity of CaliGCL by decomposing key components: (1) Encoder forward pass. For an L-layer GNN such GCN GIN, the complexity of forward propagation typically (22)EE)d), where | the number of edges, d is the representation dimension. (2) Exponential partitioned similarity. For batch of N nodes, computing similarities with all other nodes has (KN22) complexity. (3) Discriminator. For N2 sample pairs, the total complexity is O Dd), where D is the dimension of the discriminator. Therefore, the complexity of CaliGCL O d KK + d).
B.7 Limitation
While the exponential partitioned similarity measure effectively enhances positively-synerristie features and suppresses negatively-synergisti ones, it remains a static mechanism that treats all partitions uniformly without explicitly distinguishing between aligned and misaligned components. promising future direction would to develop a learnable or attention brasee partition weighting strategy that automatically identify and emphasize positively-synergee features while down- weighting sreatively-synergginn ones in a data- manner.
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Reviews:
[{"review_id": "rwHD7XyWqy", "reviewer": "Reviewer_hzBf", "summary": "This paper proposes CaliGCL, a calibrated contrastive learning framework for graph representation learning. The method adjusts the negative sample distribution in contrastive loss using a soft calibration mechanism based on anchor hardness and positive similarity. Experiments on five benchmarks show that CaliGCL improves performance over various unsupervised and supervised GCL baselines. The authors also provide theoretical insights into calibration effects on similarity distributions.", "questions": "-Could the authors provide a sensitivity analysis regarding the calibration function components (e.g., anchor hardness and positive similarity)? It is unclear how each individually contributes to performance gains.\n\n-The theoretical section, particularly Equation (4) and the definition of entropy-based hardness, lacks clarity. Could the authors elaborate on the derivation and its role in model behavior?\n\n-How does CaliGCL scale in terms of time and memory complexity on large-scale graphs? An empirical or analytical comparison would be helpful to assess real-world applicability.\n\n-How does CaliGCL fundamentally differ from existing adaptive weighting or calibrated contrastive loss approaches in vision or graph domains? Please clarify the novel contribution beyond reweighting strategies.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths：\n\n-The paper targets a relevant and timely problem in GCL, namely the lack of semantic awareness in existing negative sampling strategies.\n\n-The proposed calibration mechanism is conceptually intuitive and can be incorporated into standard contrastive objectives with minimal modification.\n\n-Experimental results show consistent improvements on well-known benchmarks.\n\nWeaknesses：\n\n-Several parts of the theoretical analysis (e.g., Eq. 4, definitions of entropy and hardness) are not clearly explained, making the formal contribution hard to interpret for a broader audience.\n\n-The effect of individual components in the calibration strategy (e.g., hardness vs. positive similarity) is not sufficiently analyzed. This makes it hard to assess where the gains truly come from.\n\n-The paper does not discuss how well the method scales to large graphs or more realistic industrial settings, which limits its practical relevance.\n\n-While the calibration idea is useful, it can be seen as an extension of existing weighting schemes in contrastive learning, with limited theoretical or algorithmic innovation.", "was_revised": true, "final_justification": "The authors have addressed all of my concerns. After also reviewing the comments and perspectives from other reviewers, I have decided to raise my score to a 4.", "rebuttal": "***\nWe have carefully read your comments and made extensive efforts to understand your concerns, particularly the core issue regarding the definitions of “entropy” and “hardness”.\n\nAt the beginning, we think that the concepts of entropy and hardness you mentioned might not be directly applicable to our model, since our motivation is to calibrate biased similarity estimation caused by conflicting feature elements and address semantic inconsistency arising from data augmentations but doesn’t explicitly incorporate entropy-based reweighting or hardness-aware mining in the traditional sense. \n\nHowever, we also reconsider what deeper concern or theoretical ambiguity might have motivated your comment and why these concepts are brought up. Considering this, we attempt to infer your original intent and the underlying motivation behind your question, and suppose that “entropy” denotes the softmax-like contrastive loss, and “hardness” is the trade-off between a learning signal from hard negatives and the harm due to the correction of false negatives. Guided by this understanding, we provide the following response in hopes that it can address your concerns.\n\nIf we misunderstand your question or don't resolve the issues you intend to highlight, we would greatly appreciate any clarification in the discussion phase and try our best to provide a more targeted and clear response.\n***\n\n**W1 & Q2**: We would like to clarify that Eq. 4 does not involve this issue, and we assume that the equation you referred to might be Eq. 5, and our responses are based on Eq. 5. \n\n“Entropy” denotes the softmax-like contrastive loss, such as cross-entropy [1-2]. Given the fact that the contrastive loss is regarded as the cross-entropy, the definition of entropy is obtained by substituting Eq. 5 into Eq. 10 as\n\n$\\text{Entropy} = - \\log \\left( \\frac{\\sum_k \\exp \\left( K[h_u^{(k)}]^T h_v^{(k)} \\right) / \\tau}{\\sum_j \\sum_k \\exp \\left( K[h_u^{(k)}]^T h_{v_j}^{(k)} \\right) / \\tau + \\sum_{j \\neq i} \\sum_k \\exp \\left( K[h_u^{(k)}]^T h_{u_j}^{(k)} \\right) / \\tau} \\right)$\n\nwhere $K$ is the number of partitions, $\\tau$ is the temperature, $u_i$ and $v_i$ are the $i$-th samples in the augmented views $U$ and $V$, respectively. Here, $(u_i, v_i)$ is a positive pair, $(u_i, v_j)$ and $(u_i, u_j), j \\neq i$ are negative pairs.\n\n“Hardness” reflects the trade-off between learning from difficult negatives and avoiding false negatives [3-4]. In our unsupervised setting without labels, this difficulty arises from distinguishing positive and negative pairs. To address this, we introduce a learnable semantics-consistency discriminator that predicts whether a pair is positive or negative based on a threshold $\\eta$. We then define hardness as the absolute difference between the predicted consistency score and the threshold, rather than using an entropy-based formulation as\n\n$\\text{hardness} = 1 - |p_{v_i, v_j} - \\eta|$\n\nwhere $p_{v_i, v_j}$ is the predicted consistency score of pair $(v_i, v_j)$, and $\\eta$ is the threshold for judging whether the sample pairs are positive or negative.\n\nIn the theoretical section, Eq. (5) defines an enhanced contrastive similarity designed to reduce the influence of conflicting features within representation vectors. Traditional inner product similarity may suppress positively-synergistic signals due to cancellation from negatively-synergistic ones, even within positive pairs. To address this, Theorems 2 and 3 illustrate the behavioral benefits:\n\n1. Theorem 2 shows that the exponential partitioned similarity in Eq. (5) upper-bounds the standard exponential inner product. This allows positive pairs with aligned partitions to receive higher similarity scores, even when negatively-aligned partitions would otherwise suppress them, thereby enhancing contrastive discriminability.\n\n2. Theorem 3 shows that using partitioned similarity leads to a contrastive loss that provides a tighter lower bound on mutual information. This implies that our method better preserves semantic consistency and promotes the learning of more informative representations.\n\nThese theorems provide both mathematical support and behavioral interpretation for our model.\n\n[1] Representation Learning with Contrastive Predictive Coding, in Arxiv 2018.\n\n[2] A Simple Framework for Contrastive Learning of Visual Representations, in ICML 2020.\n\n[3] Understanding the Behaviour of Contrastive Loss, in CVPR 2021.\n\n[4] Debiased Contrastive Learning, in NeurIPS 2020.\n\n**W2 & Q1**: We conduct experiments to show the contributions of the anchor hardness or the positive similarity in the calibration strategy. To evaluate the anchor hardness, we remove hard negatives of anchors (hardness $\\ge$  0.9 and consistency score < $\\eta$) from the loss. Similarly, to evaluate the positive similarity, we remove all the positive pairs (consistency score  $\\ge \\eta$) but preserve the corresponding samples of anchors in the augmented views in the loss. The results are listed in Table 1. Additionally, we provide the performance gaps between CaliGCL and the models without either anchor hardness or positive similarity. Generally, larger gaps correspond to greater contributions of the components. From the results, we find that both the anchor hardness and positive similarity contribute to the performance gains, which validates the effectiveness of the discriminator to calibrate the false-negatives and false-positives. Besides, we observe that positive similarity contributes more in small datasets (Citeseer, Photo, and Computers) while the anchor hardness contributes more in relatively large datasets (Pubmed and DBLP).\n\nTable 1. Comparison of performance gains concerning anchor hardness and positive similarity.\n\n|Datasets|Cora|Citeseer|Photo|Computers|DBLP|Pubmed|\n|-|-|-|-|-|-|-|\n|w/o anchor hardness|85.76|73.77|93.46|90.64|84.42|86.17|\n|w/o positive similarity|85.79|73.61|93.44|90.50|84.65|86.26|\n|CaliGCL|85.87|74.13|93.72 |90.79|85.32|87.16|\n|Performance Gap (without anchor hardness/positive similarity)|**0.11**/0.08|0.36/**0.52**|0.26/**0.28**|0.15/**0.29**|**0.90**/0.67|**0.99**/0.90|\n\n**W3 & Q3**: We provide the comparative results on model performance, training time, and GPU memory across two large-scale graph datasets, namely Arxiv and Mag in Table 2. The results demonstrate that CaliGCL can be applied to the large-scale datasets. For the model performance, we could observe that CaliGCL has the best performance when compared with other graph models across the two large-scale datasets, which illustrates that the calibration strategy aids in refining the contrastive process and enhances the model performance. However, the training time and GPU memory are heavier than other models due to the discrimination process. Even so, the time and space overheads of the model are within several times that of other models, and overall, they are at the same level.\n\nTable 2. Performance and complexity comparison among the graph models across large-scale datasets.\n|Datasets|Arxiv|||Mag|||\n|-|-|-|-|-|-|-|\n||Acc (%)|Time (s)|GPU (M)|Acc (%)|Time (s)|GPU (M)|\n|AFGRL|56.14±0.14|1308.15| 6966.99|27.97±0.02|5795.03|8370.58|\n|NCLA|58.43±0.02|999.17|**2041.58**|25.29±0.01|3470.89|**2887.58**|\n|HomoGCL|62.53±0.48|**566.54**|12194.19|26.17±0.02|**3197.28**|18131.88|\n|S2GAE|60.88±0.00|1303.80|11455.05|23.66±0.00|4374.16|14795.73|\n|Bandana|66.78±0.00|655.88|9947.60|25.91±0.00|5003.85|15452.42|\n|SGRL|65.71±0.01|673.02|14027.77|26.68±0.33|7506.63|24060.54|\n|CaliGCL|**68.42±0.00**|1969.10|14853.70|**28.66±0.00**|11126.59|23586.28|\n\n\n**W4 & Q4**: We clarify that CaliGCL differs from existing adaptive weighting methods [1-3] in terms of motivation, methodology, and experiment results. Specifically, we highlight the following key contributions or innovations from theoretical and algorithmic perspectives:\n\n**Algorithmic Innovations:**\n\n1. Broader bias correction: While most existing GCL methods focus on false negatives [1–2], they often overlook false positive cases where augmentations disrupt true positive pairs. CaliGCL’s discriminator addresses both, enabling more comprehensive correction of contrastive supervision.\n\n2. Comprehensive calibration perspective: Previous adaptive methods typically rely on feature similarity [1] or homophily principle [2] for reweighting. Except for them, CaliGCL introduces a finer-grained partition-level view, allowing calibration at a more detailed level beyond feature similarity or local adjacency.\n\n3. Learnable semantics-consistency: Unlike prior methods that rely on fixed or heuristic similarity measures [1–3], CaliGCL uses a learnable discriminator to adaptively assess whether augmented pairs remain semantically consistent. This leads to more accurate supervision, especially in complex scenarios.\n\n4. Empirical superiority: Beyond theoretical contributions, CaliGCL consistently outperforms existing adaptive weighting methods across various datasets, demonstrating its effectiveness in addressing semantic shifts and feature conflicts for better representation learning.\n\n**Theoretical Innovations:**\n\nThe theoretical section provides a solid foundation for why our method generates more expressive representations than existing methods.\n\n1. Theorem 1 reveals the existence and effect of negatively-aligned partitions in positive pairs, supporting our motivation to suppress their influence.\n\n2. Theorem 2 proves that the proposed exponential partitioned similarity yields a more ideal similarity than the inner product, thereby improving feature discrimination.\n\n3. Theorem 3 further shows that the partition contrastive loss serves as a tighter upper bound on the mutual information lower bound, meaning that it preserves or improves representation quality than the standard contrastive loss.\n\n[1] HomoGCL: Rethinking Homophily in Graph Contrastive Learning, in KDD 2023.\n\n[2] Enhancing Contrastive Learning on Graphs with Node Similarity, in KDD 2024.\n\n[3] Debiased Contrastive Learning, in NeurIPS 2020."}, {"review_id": "93pa8vVWHa", "reviewer": "Reviewer_cbPN", "summary": "The paper aims to mitigate two problems of graph contrastive learning, \\emph{i.e.,} similarity estimation bias and semantic shift bias, which respectively causes truly positive pairs to appear less similar and leads to incorrect positive or negative assignments as well as training noises. Therefore, the authors developed CaliGCL that is a graph contrastive learning model for calibrating the biases by integrating an exponential partitioned similarity measure and a semantics-consistency discriminator. Extensive experiments are performed to demonstrate the effectiveness and superiority of the work.", "questions": "1.\tWhile pretraining the discriminator, whether the sample pairs confront the problem of random augmentations?\n\n2.\tThe paper mentions that “the partition contrastive objective is an upper bound of mutual information than the original contrastive objective”. Generally, the contrastive objective is the lower bound of mutual information. Please provide a more detailed discussion on this issue.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1.\tThis paper provides a deep insight into GCL and introduces two problems faced by existing works;\n\n2.\tThe proposed solution of exponential partitioned similarity measure is novel, which divides a sample representation vector into multiple partitions. Exponential function is adopted to emphasize the positive partitions and meanwhile it weakens the negative partitions. This method helps to mitigate the degraded similarity between positive samples caused by similarity estimation bias. The semantics-consistency discriminator also helps to distinguish semantically flipped sample pairs.\n\n3.\tExtensive experiments are performed to demonstrate the effectiveness and superiority of the work.\n\n\nWeaknesses:\n\n1.\tSome concepts should be further specified, like semantic shift bias and contrastive supervision. \n\n2.\tThe figures should be further improved. For example, it seems that the augmentations might possibly flip the nature of sample pairs, but Figure 1 only illustrates that positive pairs might become negative pairs. How about that the negative samples become positive pairs? Similarly, Figure 2 should include more details.\n\n3.\tThe design of discriminator is somehow unclear.", "was_revised": true, "final_justification": "The responses have basically addressed my questions, and I have no further comments and would like to maintain my previous rating.", "rebuttal": "**W1**: Semantic shift bias corresponds to the semantic relationships between sample pairs which are altered due to augmentations, such as the positive pairs becoming negative pairs. The contrastive supervision is the pre-defined supervision about how to realize the positive or negative assignments. In graph contrastive learning, the contrastive supervision is to assign corresponding nodes across views as positives while others as negatives. \n\n**W2**: Figure 1 shows that with augmentations, the ratio of positively-aligned partitions decreases, indicating that some positive pairs are no longer aligned and supporting the existence of semantic shifts. However, Figure 1 focuses on the condition that a part of positive pairs become negative pairs. To illustrate the condition that the negative pairs may become positive pairs, we count the negative pairs whose inner product similarities become positive after augmentation, and find that 99,742 negative pairs show semantic shifts. This illustrates the possibility that negative pairs turn into latent positive pairs. The relevant statistics will be added in Figure 1 in the refined version.\n\nFor Figure 2, we will supplement these semantics-bias pairs after the augmentation process in detail in the refined manuscript.\n\n**W3**: Since the augmentations may change the semantic consistency between sample pairs, we propose the discriminator designed for finding the semantic-consistent pairs for calibrating the contrastive process and preventing semantic shift bias. And it is designed as a four-layer MLP with activation functions. We provide the specific configurations across different datasets in Tables 1-2, where Dis_dim is the hidden dimension of the discriminator and Act is the activation function applied in the hidden layer. The last layer will add a Sigmoid function to output the consistency scores.\n\nTable 1. Design of the discriminator for node-level datasets.\n\n|Datasets|Cora|Citeseer|Pubmed|DBLP|Photo|Computers|\n|-|-|-|-|-|-|-|\n|Dis_dim|128|512|256|128|128|512|\n|Act|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|\n\nTable 2. Design of the discriminator for graph-level datasets.\n\n|Datasets|MUTAG|IMDB-B|COLLAB|RDT-B|DD|NCI1|RDT-M5K|\n|-|-|-|-|-|-|-|-|\n|Dis_dim|128|128|128|128|64|64|128|\n|Act|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|LeakyReLU|\n\n\n**Q1**: The sample pairs used for pre-training have not undergone random augmentations, since the random augmentations may change the semantic consistency of sample pairs. When using the biased samples to train the discriminator, it must restrict its reliability and effectiveness. In order to pick out reasonable positive and negative pairs, we only select the positive and negative pairs in the original graph based on the K-NN algorithm and the homogeneity principle.\n\n\n**Q2**: The same in your comment, both the partition contrastive objective and the original contrastive objective follow the InfoNCE objective form that is the lower bound of the mutual information. Theorem 3 is proposed for proving that the proposed partition contrastive objective provides a tighter upper bound than the contrastive objective, not an upper bound of mutual information. We provide discussions on this issue in the following:\nFormally, the InfoNCE objective for a positive pair $(h, h_+)$ over a batch of size $N$ is\n\n$\\mathcal{J}\\_{\\text{Info}} = \\log \\left( \\frac{\\exp(S(h, h_+)/\\tau)}{\\sum_h \\exp(S(h, h)/\\tau)} \\right)$\n\nIt corresponds to\n\n$I(h; h_+) \\geq \\log N + \\mathcal{J}_{\\text{Info}}.$\n\nTherefore, the InfoNCE objective is the lower bound of the mutual information. In the paper, the partitioned exponential similarity is defined as:\n\n$S_{\\text{part}}(h, h_+) = \\frac{1}{K} \\sum_k \\exp \\left( \\frac{K \\cdot \\left[ h^{(k)} \\right]^T \\left[ h_+^{(k)} \\right]}{\\tau} \\right),$\n\nwhere $h^{(k)}$ and $h_+^{(k)}$ are the $k$-th partitions of representations $h$ and $h_+$, respectively. The partitioned exponential similarity results in the partitioned contrastive objective as\n\n$\\mathcal{J}\\_{\\text{part}} = \\log \\left( \\frac{S_{\\text{part}}(h, h_+)}{\\sum_h S_{\\text{part}}(h, h_+)} \\right).$\n\nSince the objective obeys the InfoNCE formation, we also have\n\n$I(h; h_+) \\geq \\log N + \\mathcal{J}_{\\text{part}}.$\n\nHence, both the contrastive objective and the proposed objective are the lower bounds of the mutual information. However, Theorem 3 proves $\\mathcal{J}_{\\text{part}}$ is the upper bound of $\\mathcal{J}\\_{\\text{Info}}$, which leads to the following conclusion:\n\n$I(h; h_+) \\geq \\log N + \\mathcal{J}\\_{\\text{part}} \\geq \\log N + \\mathcal{J}\\_{\\text{Info}}.$\n\nIt means $\\mathcal{J}\\_{\\text{part}}$ is a more ideal objective than $\\mathcal{J}\\_{\\text{Info}}$, which induces a tighter lower bound on mutual information. Theorem 3 provides the inappropriate wording for misunderstandings. We will revise the wording of Theorem 3 as “The partition contrastive objective serves as a tighter lower bound on mutual information, and thus yields an upper-bounded estimate compared to the contrastive objective” in the refined version."}, {"review_id": "mGoJqKX3e9", "reviewer": "Reviewer_7U3j", "summary": "This paper proposes CaliGCL, a graph model for tackling two biases in graph contrastive learning: similarity estimation bias and semantic shift bias. To alleviate the similarity estimation bias, the authors raise an exponential partitioned similarity metric to emphasize positively-aligned partitions and suppress the negatively-aligned partitions. For eliminating the semantic shift bias, the authors apply a semantics-consistency discriminator to calibrate the sample assignments during the contrastive learning process. Empirical results, such as node classification, graph classification, and ablation study demonstrate the ability of CaliGCL to calibrate the biases when compared with other self-supervised graph contrastive and generative models.", "questions": "- What do “connected positives” mean in Figure 2? Do the connected positives mean the adjacent nodes? Besides, where are the encoders in Figure 2?\n\n- Why do the authors use the Hadamard product to fuse representations $\\z_{v_i}$ and $\\z_{v_j}$ but not other fusion strategies like concatenation during the training process of the semantics-consistency discriminator? What is the advantage of the Hadamard product?\n\n- The paper claims that a semantics-consistency discriminator could realize the calibrated sample assignments. Can the authors provide experiment results on whether the discriminator has the ability to identify the positive or negative pairs during training?\n\n- Why don’t the authors sample the negatives and positives with similar sizes when training the discriminator? It can avoid the imbalance problem more efficiently than using a balanced softmax loss.", "limitations": "The authors have provided some plans for refining the limitations as future work, and the plans are reasonable.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n- This paper deals with the critical problems in graph contrastive learning, namely the similarity estimation bias and the semantic shift bias, which may cause biased representation learning.\n\n- This paper proposes a novel CaliGCL including the exponential partitioned similarity metric and the semantics-consistency discriminator to tackle the two biases and provides the theoretical evidence to analyze the proposed model.\n\n- Extensive empirical experiments on node classification and graph classification show that the proposed CaliGCL is superior to other comparative self-supervised graph models.\n\nWeaknesses:\n- The function of doubling the inner product of partitions in the proposed exponential partitioned similarity in Eq. (6) is not intuitive. As the paper claims, the exponential partitioned similarity can suppress the impact of negatively-aligned partitions by inducing their exponentials to approach zero while amplifying positively-aligned ones, it is unnecessary to double the inner product of partitions in Eq. (6). The authors need to clarify why the exponential partitioned similarity still implements the double inner product to achieve the amplification and the suppression.\n\n- Figure 3 only provides the mean performance of the proposed model, it would be better to supplement the standard deviation in Figure 3.\n\n- This paper needs visualization analysis to compare the proposed CaliGCL with other graph models, which better shows the embedding distribution and demonstrates the representation quality.", "was_revised": true, "final_justification": "Thanks for the authors' response, I have no more questions. I will keep my score.", "rebuttal": "**W1**: Eq. (6) corresponds to an example where a pair of positive samples $(h_v, h_v')$ has both negatively-aligned and positively-aligned partitions, and the inner product of the sample pair is the sum of the inner products of the partitions, namely\n\n$h_{v_i}^T h_{v_j} = [ h_{v_i}^{(1)}]^T [ h_{v_j}^{(1)}]+[h_{v_i}^{(2)}]^T [h_{v_j}^{(2)}].$\n\nAssume that $h_{v_i}^T h_{v_j} > 0$ and the first partition is negatively-aligned, namely $[h_{v_i}^{(1)}]^T [h_{v_j}^{(1)}] < 0$, it is easy to find that the inner product of the second partition has \n\n$[h_{v_i}^{(2)}]^T [h_{v_j}^{(2)}] > h_{v_i}^T h_{v_j}.$\n\nTherefore, as your comment, without doubling the inner product of partitions, it also holds the following conclusion\n\n$\\exp\\left( \\frac{[ h_{v_i}^{(1)}]^T [h_{v_j}^{(1)}]}{\\tau} \\right) \\to 0, \\quad \\text{and} \\quad \\exp\\left( \\frac{[h_{v_i}^{(2)}]^T [h_{v_j}^{(2)} ]}{\\tau} \\right) \\geq \\exp\\left( \\frac{h_{v_i}^T h_{v_j}}{\\tau} \\right).$\n\nHowever, this may not hold for the situation where the sample pairs have only positively-aligned or negatively-aligned partitions. For example, when the sample pair is divided into two positively-aligned partitions, we have\n\n$\\exp\\left( \\frac{[h_{v_i}^{(1)}]^T [h_{v_j}^{(1)}]}{\\tau} \\right) \\leq \\exp\\left( \\frac{h_{v_i}^T h_{v_j}}{\\tau} \\right), \\quad \\text{and} \\quad \\exp\\left( \\frac{[h_{v_i}^{(2)}]^T[h_{v_j}^{(2)}]}{\\tau} \\right) \\leq \\exp\\left( \\frac{h_{v_i}^T h_{v_j}}{\\tau} \\right).$\n\nIt is difficult to compare the values of the original exponential inner product and the sum of the exponential partitioned similarity for judging which is larger. To amplify the sum of the exponential partitioned similarity and make it larger, we follow Jensen’s inequality to leave the original exponential inner product less than or equal to the sum of the exponential partitioned similarity by multiplying the inner product with the number of blocks, and it corresponds to doubling the inner product in Eq. (6). This operator is necessary to generalize the conclusion to more common situations where the partitions are all positively-aligned or negatively-aligned, and we provide the corresponding proof in Theorem 2. Additionally, for Eq. (6), the double inner product has another merit to further facilitate the negatively-aligned partitions to approach zero more quickly while further amplifying the positively-aligned partitions.\n\n\n\n**W2**: We provide the classification results on IMDB-BINARY (IMDB-B) and REDDIT-BINARY (RDT-B) datasets with the means and standard deviations in Tables 1-2. In the refined version, we will include the shaded regions around the broken line in Figure 3 to indicate standard deviations.\n\nTable 1. Hyperparameter analysis for the temperature coefficient.\n\n|Datasets|Tau|Acc|\n|-|-|-|\n|IMDB-B|0.6|76.26±4.80|\n||0.7|75.68±5.39|\n||0.8|76.88±5.04|\n||0.9|75.68±5.33|\n||1.0|75.48±4.67|\n|RDT-B|1.0|89.44±2.89|\n||1.1|89.64±2.57|\n||1.2|90.29±2.69|\n||1.3|89.69±2.86|\n||1.4|89.69±2.81|\n\nTable 2. Hyperparameter analysis for the partition size.\n\n|Datasets|Partition Size|Acc|\n|-|-|-|\n|IMDB-BINARY|256|76.48±4.21|\n||128|76.88±5.04|\n||86|76.68±4.59|\n||64|75.87±3.17|\n||52|75.87±5.04|\n|RDT-B|256|89.84±2.92|\n||128|90.29±2.69|\n||86|90.00±2.52|\n||64|89.69±2.45|\n||52|89.49±2.51|\n\n**W3**: We provide a t-SNE visualization comparing CaliGCL and other baselines to demonstrate the quality of learned representations. From the visualization results, we find the samples of different classes in PiGCL and iGCL are mixed; the boundaries of different classes in GREET, S2GAE, Bandana, and SGRL are ambiguous without margins. In contrast, the boundaries of different classes in CaliGCL are separated with larger margins. Since we are unable to provide the Figures in the rebuttal, the visualization and the analysis will be included in the refined version.\n\n**Q1**: \"Connected positives\" refer to nodes that are connected with the anchor node in the topology, and they are the adjacent nodes. Based on the homogeneity principle, these nodes are always considered as positives. The encoder corresponds to the arrows between the input graphs and the feature information, such as feature information $\\textbf{H}_U$, $\\textbf{H}_V$, and $\\textbf{H}\\_\\mathcal{G}$ in Figure 2. We will supplement the missing details in the refined version.\n\n**Q2**: We use the Hadamard product to preserve fine-grained dimension-wise interactions between node representations. Though both the concatenation and the Hadamard product can preserve the dimension-wise information, the Hadamard product with dimension-wise multiplication emphasizes that the feature elements of a sample pair are closely correlated in corresponding dimensions, which is suited for relationship (similar or dissimilar) prediction. Besides, the Hadamard product doesn’t extend the feature dimensions when compared with the concatenation operator, which will save half the computation cost on calculating the consistency scores than the concatenation operator.\n\n**Q3**: The ability of the discriminator to realize the calibrated sample assignments during training mainly contributes to its identification for semantic consistency of sample pairs. To illustrate the identification ability of the discriminator, we have provided some experiment results to show that the discriminator has superior ability to predict semantic consistency of sample pairs, and the results are shown in Figure 6, where the blue histogram corresponds to the inner product and the pink histogram corresponds to the discriminator. \n\nIn the experiments, we use the inner product and the discriminator to judge the semantic consistency of sample pairs. Specifically, we assume the sample pairs with positive inner products are positive in the former method, otherwise negative, while the output of the discriminator can naturally give a judgement of semantic consistency of the sample pairs in the latter method. We compare the predictions with the true labels of the sample pairs to obtain the prediction accuracy, and the results are shown in Figure 6, where the ‘w/o aug’ and ‘aug’ represent the results of the prediction accuracy on the original data and the augmented data, respectively. Figure 6 illustrates that the discriminator has superior performance than the inner product on discriminating the semantic consistency of sample pairs under both the ‘w/o aug’ and ‘aug’ conditions, which demonstrates that the discriminator could further help the model to calibrate the contrastive learning process.\n\n**Q4**: The training for an ideal discriminator always needs numerous positive and negative sample pairs. Since we have no true labels to train the discriminator in an unsupervised setting, we attempt to find the reliable positives and negatives for training the discriminator. In our method, we only pick out the positives from the overlapped samples that are both adjacent and selected with the K-NN algorithm. Based on this strict condition, the positives are more reliable but always very scarce. However, the negative samples are easy to sample, so we can collect much more negative samples than positive samples, which leads to the imbalance problem between the positive and negative sample pairs. This means that though balanced sampling for positives and negatives is more efficient, it is difficult to conduct. To address the inevitable imbalance problem and prevent the encoder from giving consistent predictions as negative pairs, we adopt the balanced softmax loss function instead adjusts the sampling ratio between positives and negatives, offering a more reliable solution to the imbalance problem."}, {"review_id": "yPjJL53fMi", "reviewer": "Reviewer_8dv3", "summary": "This paper proposes a novel graph contrastive learning (GCL) model, CaliGCL, which targets at calibrating the similarity estimation bias and the semantic shift bias in the standard GCL framework. Considering this, on one hand, this paper raises an exponential partitioned similarity measure to emphasize the aligned partitions while suppressing conflicting partitions. Besides, the authors also provide theorems to explain the merit of the exponential partitioned similarity measure. On the other hand, this paper designs a semantics-consistency discriminator to assess the semantic relationship between sample pairs, which corrects the contrastive supervision from the semantic shift bias. The model ability is supported through comprehensive experiments on multiple graph datasets against advanced self-supervised graph representation models.", "questions": "(1) Does the model training converge? Since balancing the training between the encoder and the discriminator may be difficult, the authors should focus more on the loss convergence to ensure stable model training.\n(2) I am curious about whether the block number will cause the index flow? According to Equation 5, a large $K$ will amplify the inner product, which may cause index flow in practice.\n(3) The proposed CaliGCL is compared with some self-supervised graph models with learnable augmentations, which optimize the graph augmentations to improve the model performance. However, the paper still states that these models have the potential to distort semantic consistency between sample pairs. So why do these graph models with learnable augmentations still distort semantic consistency between sample pairs, and what are the advantages of CaliGCL when compared with these models?", "limitations": "The paper has clarified its limitation and the corresponding resolution, and the authors could refine their resolution to address the limitation.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n(1) Writing and organization of this paper are good and logical, and the motivation and the method are clear to understand and follow.\n(2) This paper deals with two underexplored bias problems, namely the similarity estimation bias and the semantic shift bias in GCL, and the perspective is interesting.\n(3) This paper proposes the exponential similarity measure and the semantics-consistency discriminator to mitigate the two bias problems, which make sense to calibrate the biases in GCL.\n(4) Experiments are comprehensive, including two downstream tasks across multiple public graph datasets to demonstrate the model superiority.\n\nWeaknesses:\n(1) How to calculate the normalized adjacency matrix $\\tilde{A}$ in Line 186 is missing. The authors need to give the specific mathematical formulation of it.\n(2) Lines 167-168 state that “the proposed exponential partitioned similarity could exceed the standard exponential inner product across all scenarios”. What do “all scenarios” mean? Why do the authors stress the “all scenarios”?\n(3) The authors state that shuffling the feature columns of representations before loss computation could ensure sufficient interactions among feature dimensions across partitions in Lines 220-221. Therefore, it should provide more experimental results to demonstrate the rationality of implementing “Shuffle”. \n(4) The authors need to provide a more comprehensive analysis on complexity when compared with other graph models.\n(5) The experiments in Figure 6 are conducted to evaluate the discriminator, but the legends are about the loss functions. There is a doubt as to which result was caused by applying the discriminator.", "was_revised": true, "final_justification": "All my concerns have been addressed and I raise the acceptance ratings.", "rebuttal": "**W1**: The normalized adjacency matrix $\\tilde{\\textbf{A}}$ is calculated as $\\tilde{\\textbf{A}} = \\hat{\\textbf{D}}^{-1/2} \\hat{\\textbf{A}} \\hat{\\textbf{D}}^{-1/2}$, and $\\hat{\\textbf{A}}=\\textbf{A}+\\textbf{I}_N$, $\\hat{\\textbf{D}}$ is a diagonal matrix, and its diagonal element is ${\\hat{\\textbf{D}}}\\_{ii}={\\sum\\_j} {\\hat{\\textbf{A}}\\_{ij}}$. This will be clarified in the refined version.\n\n**W2**: “all scenarios” mean all the possible combinations of feature partitions, namely all positively-aligned partitions, all negatively-aligned partitions, and mix of both positively-aligned and negatively-aligned partitions. Lines 167-168 mean that under all possible combinations of feature partitions, exponential partitioned similarity yields similarity greater than or equal to global exponential inner product, where Lines 162-166 only give a simple example to clarify that the proposed exponential partitioned similarity has the ability to amplify the positively-aligned partitions while suppressing the negatively-aligned partitions in the mixed scenario, which cannot generalize to the scenarios where the representation vectors have only positively-aligned or negatively-aligned partitions. And we stress the “all scenarios” to illustrate that through Theorem 2, the proposed exponential partitioned similarity can not only amplify the positively-aligned partitions and suppress the negatively-aligned partitions in the mixed scenario, but can be generalized to common scenarios.\n\n**W3**: Feature shuffling ensures interactions across dimensions and avoids amplifying or suppressing fixed partitions during the training process. To illustrate the function of “Shuffling”, we provide the results of the model with and without shuffling in Tables 1-2. From the results, we could find that the model without shuffling has inferior performance when compared with that with shuffling. It is because that without shuffling, only the fixed partitions are amplified or suppressed during the whole training process. In contrast, the shuffling operator benefits the interactions across feature dimensions and ensures the model performance.\n\nTable 1. Model performance without “Shuffling” for node-level datasets.\n\n|Datasets|Cora|Citeseer|Pubmed|DBLP|Photo|Computers|\n|-|-|-|-|-|-|-|\n|w/o Shuffling|85.58±0.76|73.05±0.71|86.15±0.23|84.50±0.13|92.55±0.21|90.50±0.32|\n|w/ Shuffling|85.87±0.62|74.13±0.54|87.16±0.19|85.32±0.13|93.72±0.24|90.79±0.28|\n\nTable 2. Model performance without “Shuffling” for graph-level datasets.\n\n|Datasets|NCI1|DD|MUTAG|COLLAB|RDT-B|RDT-M5K|IMDB-B|\n|-|-|-|-|-|-|-|-|\n|w/o Shuffling|77.2±1.9|76.7±3.8|90.3±5.3|77.1±1.6|89.5±1.4|56.1±1.6|75.3±5.1|\n|w/ Shuffling|80.8±2.0|80.4±2.5|90.4±5.8|77.2±1.7|90.3±2.7|56.5±1.5|76.9±5.0|\n\n**W4**: To provide a comprehensive comparison among the graph models, we first define $N$ as the number of nodes, $d$ as the dimension of the representations, $D$ as the hidden layer dimension of the discriminator, $|\\mathcal{E}|$ as the number of edges,  as the number of sampling adjacent nodes, $C$ as the number of clusters, $V$ as the number of augmented views, $k$ as the number of sampling negatives, $K$ is the number of partitions, and $T$ as the iteration of K-means algorithm. Based on these symbols, we provide the complexity of the comparative models in Table 3. From the results, we find that our model has $\\mathcal{O}{(N^2)}$ complexity and though it is more computationally intensive than the graph generative models, it is comparable with the graph contrastive model, which illustrates the practicability of the proposed model.\n\nTable 3. Complexity comparison between graph models.\n|Model|Complexity|\n|-|-|\n|AFGRL|$\\mathcal{O}((2L\\|\\mathcal{E}\\|+TNC+N)d+N)$|\n|SUGRL|$\\mathcal{O}((L\\|\\mathcal{E}\\|+L+Nm+Nk)d)$|\n|NCLA|$\\mathcal{O}((\\|\\mathcal{E}\\|+L\\|\\mathcal{E}\\|+N^2)Vd)$|\n|GREE|$\\mathcal{O}(N^{d_s}+(2L\\|\\mathcal{E}\\|+N^2+N^2D)d)$|\n|HomoGCL|$\\mathcal{O}((NCT+2L\\|\\mathcal{E}\\|+N^2)d)$|\n|S2GAE|$\\mathcal{O}((1+d)L\\|\\mathcal{E}\\|d)$|\n|iGCL|$\\mathcal{O}((2L\\|\\mathcal{E}\\|+N^2V+N^2)d)$|\n|PiGCL|$\\mathcal{O}(2L\\|\\mathcal{E}\\|d+N^2d+N\\log N)$|\n|Bandana|$\\mathcal{O}((L+d+Ld)\\|\\mathcal{E}\\|d+\\|\\mathcal{E}\\|)$|\n|SGRL| $\\mathcal{O}((2L\\|\\mathcal{E}\\|+\\|\\mathcal{E}\\|+N)d)$|\n|CaliGC|$\\mathcal{O}((2L\\|\\mathcal{E}\\|+N^2+N^2D)d)$|\n\n**W5**: It is inappropriate to adopt the loss functions as the legends to cause misunderstandings. In Figure 6, the legend for the blue histogram is the result of the inner product and the legend for the pink histogram is the result of the discriminator. In the refined version, we will replace “General Contrastive Loss” with “Inner Product” and replace “Partitioned Contrastive Loss” with “Discriminator” for clarity.\n\n**Q1**: We have recorded the losses of alternate training strategies (including losses of both encoder and discriminator) during the fine-tuning process to clarify that the model training gradually converges. Besides, both the losses of the encoder and the discriminator become stable in the training process. Since we are unable to provide Figures in the rebuttal, the loss plots will be included in the refined version.\n\n**Q2**: Though $K$ amplifies the magnitude of exponential partitioned similarity, it doesn’t cause index flow in practice. When calculating the contrastive loss with the representations, they are always normalized, which means the inner product of a sample pair is limited in $[-1,1]$. Therefore, the training process will cause index flow unless $K$ is very large. However, according to the experiments, we find $K$ is always smaller than five. Additionally, another hyperparameter, namely the temperature coefficient $\\tau$, could also adjust $K$ to avoid the index flow problem.\n\n**Q3**: Though with learnable augmentations, the graph models [1-2] generally optimize the strength of graph augmentations (e.g., the ratio of edge removal), or the combination of adopted augmentation strategies. However, the input data is still modified randomly but with a more appropriate ratio or augmentation combination. Without explicit constraints, they may still destroy meaningful structures or features. In contrast, CaliGCL uses a discriminator to explicitly check semantic consistency between sample pairs and attempts to calibrate the semantics-biased sample pairs to correct the contrastive learning process directly, which further reduces the possibility of semantic shift caused by the random augmentations.\n\n[1]Graph contrastive learning with personalized augmentation, in TKDE 2024.\n\n[2] Adversarial Graph Augmentation to Improve Graph Contrastive Learning, in NeurIPS 2021."}]

Author rebuttal:

Meta-review: This paper introduces a calibrated contrastive learning framework for graph representation learning. CaliGCL adjusts the negative sample distribution with a soft calibration mechanism based on anchor hardness and positive similarity. Both theoretical and empirical results are provided to demonstrate the effectiveness. The writing clarity is argued by multiple reviewers, and has been alleviated after the rebuttal. Other minor issues, such as more in-depth analysis of experimental results or the exploration on large-scale graphs, are also suggested for improvement. All four reviewers give positive scores to this submission. Considering the scope of this paper, I will recommend acceptance as a poster.
Decision: Accept (poster)