What Matters in Graph Class Incremental Learning? An Information Preservation Perspective
Authors: Jialu Li
Yu Wang
Pengfei Zhu
Wanyu Lin
Qinghua Hu
Venue: NeurIPS 2024 poster (2024)
Keywords: Graph neural network
Class incremental learning
Primary area: graph_neural_networks

Abstract: Graph class incremental learning (GCIL) requires the model to classify emerging nodes of new classes while remembering old classes. Existing methods are designed to preserve effective information of old models or graph data to alleviate forgetting, but there is no clear theoretical understanding of what matters in information preservation. In this paper, we consider that present practice suffers from high semantic and structural shifts assessed by two devised shift metrics. We provide insights into information preservation in GCIL and find that maintaining graph information can preserve information of old models in theory to calibrate node semantic and graph structure shifts. We correspond graph information into low-frequency local-global information and high-frequency information in spatial domain. Based on the analysis, we propose a framework, Graph Spatial Information Preservation (GSIP). Specifically, for low-frequency information preservation, the old node representations obtained by inputting replayed nodes into the old model are aligned with the outputs of the node and its neighbors in the new model, and then old and new outputs are globally matched after pooling. For high-frequency information preservation, the new node representations are encouraged to imitate the near-neighbor pair similarity of old node representations. GSIP achieves a 10\% increase in terms of the forgetting metric compared to prior methods on large-scale datasets. Our framework can also seamlessly integrate existing replay designs. The code is available through https://github.com/Jillian555/GSIP.

What Matters in Graph Class Incremental Learning? An Information Preservation Perspective
Jialu Li jialuli0tju. edu. edu. en
Yu Wang 1.23,* wang yuttju.uuu... eerttju.eduu edu. en
Pengfei Zhu
22,..2,3. 1.2,,
zhupengfei@tji edu cn
Wanyu Lin wan www.yuu..nnoollyu edu hk
Qinghua Hu2.2.3 huginghua@tju. edu, cn
CCollege of Intelligence and Computing, Tianjin University, Tianjin, China 'Engineering Research Center of City Intelligence and Digital Governance, Ministry of Education of the People's Republic of China, Tianjin, China Haihe Lab of ITAI, Tianjin, China
DDpartment Computing. The Hong Kong Polytechnic University, Hong Kong, China
Abstract
Graph class incremental learning (GCIL) requires the model to classify emerg- ing nodes of new classes while remembering old classes. Existing methods are designed to preserve effective information of old models graph data to allevi- ate forgetting, but there is no clear theoretical understanding of what matters information preservation. In this paper, consider that present practice suffers from high semantic and structural shifts assessed by two devised shift metrics. We provide insights into information preservation in GCIL and find that maintaining graph information can preserve information of old models in theory to calibrate node semantic and graph structure shifts. We correspond graph information into low-frequency local-global information and high-frequency information in spatial domain. Based on the analysis, propose framework, Graph Spatial Information Preservation (GSIP), Specifically, for fow-freuynccy information preservation, the old node representations obtained by inputting replayed nodes into the old model are aligned with the outputs of the node and its neighbors in the new model, and then old and new outputs are globally matched after pooling. For giggffreyyncc information preservation, the new node representations are encouraged to imitate the neig-ne pair similarity of old node representations. GSIP achieves 10% increase in terms of the forgetting metric compared to prior methods on large-scale datascts. Our framework can also seamlessly integrate existing replay designs. The code is available through
com/ Jil1ian555/ GSSIP
Introduction
In real-world applications, graph data is continuously generated. For instance, citation networks, new types of papers and their citations may constantly emerge, an ideal literature classifier needs to continuously distinguish literature in emerging research areas [ 2]. Therefore, it is critical for graph model incrementally integrate classes to as Graph class incremental learning (GCIL). However, this poses a major challenge known as catastrophic forgetting, where the model needs preserve previous information while continuously acquiring new information 3 4]]
extended graph, which is referred
"Equal contribution. CCorrepoondig author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Many approaches attempt to preserve information from previous models or graph data to prevent catastrophic forgetting GCIL, which can be divided into four groups. The parameter isolation methods entirely or partially preserve parameters of different tasks to protect model performance, such as dynamically incrementing feature extractors and prototypes 2]. Regularization methods, on the one hand, preserve important parameters, such as assessing parameter importance by considering loss and topology [II and maintaining orthogonality with parameters from previous tasks (5), on the other hand, preserve the absolute position of nodes in feature space output space, such as aligning the outputs of samples on old and new models [6. The replay methods preserve a few nodes or subgraphs to retrain the model to prevent forgetting, such saving representative nodes (7], selecting subgraphs according to node degree [8]. and compressing training graphs [9]. The hybrid methods combine different learning paradigms (i.e. the combination of replay and regularization methods), such as feature distillation after identifying critical nodes [10] and minimizing distribution disparity of selected nodes across new and prior models [1]]. These hybrid methods have demon- strated considerable potential and achieved state-of-the-ae results. Despite their effectiveness, the information preservation mechanism by existing methods remains unclear, making it challenging to develop effective solutions for GCIL. This motivates us to explore a fresh perspective: What matters in information preservation when learning from the old model to the new model for GCIL?
(a) Baseline
(b) Ours
(c) Target
Figure 1: Visualizations of semantic shift and structure shift.
We investigate the unique characteristics of catastrophic forgetting on graphs and find node semantic and graph structure shifts in GCIL. The visualization of node embeddings for new model of baseline (ERGNN 7 and our method on old classes of CoraFull dataset are exhibited in Figure
[ The structure learned by old model is selected as target instead of original topology due to noisy real structure [12]. Five nodes belonging to two old classes are randomly selected and connected (darker edges indicate more similar features). We detect distortion in the features of baseline (Figure [(a)) relative to those the target (Figure [[e]), especially nodes located within the black dotted box. The two categories can be well separated in the feature distribution of target but not in the baseline model and lead to false predictions (grey nodes in Figure I(a).. The topological correlation is also significantly changed, node #2 is in proximity to node #1 but distant from node #3. Surprisingly, within the representation space of baseline, node #2 appears to be moving closer to node #3 while simultaneously becoming more distant from node #1, which exacerbates catastrophic forgetting. Our method (Figure I(b) designs graph information preservation modules to mitigate shifts successfully.
In this paper, we inspect GCIL from the perspective of information preservation and theoretically find a key factor in redueing catastrophic forgetting risk with hybrid methods is preserving old graph information. We correspond graph information into low-frequency local-global information and high- frequency information in spatial domain. Subsequently, a Graph Spatial Information Preservation (GSIP) framework is proposed for calibrating semantic and structural shifts. In detail, the old and new representations of nodes are obtained after replay graph data is input to old and new models. The old representations of a node are locally aligned with new representations of node and its neighbors. Further, old and new representations are globally matched after mean pooling. Finally, new representations of nodes are encouraged to mimic neighbor distance similarities that appear in old representations.
The proposed GSIP can outperform existing replay designs by up to 10% in terms of the forgetting metric on large-scale datasets. It is easy to implement and can be easily adapted to information- preserving approaches to boost their performance. Experiments show that GSIP greatly improves

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over current information preserving methods under different experimental settings and calibrates node semantic and graph structure shifts. Our main contributions can be summarized as follows:
We provide theoretical insights into GCIL and find that preserving old graph information corresponding to low-frequency local-global and high-frequency information in spatial domain can calibrate semantic and structural shifts and reduce catastrophic forgetring risk.
We propose simple yet effective method that utilizes node representations on old and new models preserve node features, graph representations, and neighbor distances.
By combining with graph replay-based methods, our framework consistently achieves performance improvements across several benchmark datasets and shows the effectiveness of all the proposed components.
2 Related Work
2.1 Incremental Learning
Incremental learning requires the model to retain the capability of predicting old tasks while acquiring
information about new ones [3] 13
14 15 T6, 17 18]. Class incremental learning is not assigned
task ID and has greater training difficulty than task incremental learning [19 20]. Existing methods can be categorized into three groups. Parameter isolation methods dynamically adapt the model without restricting its structure and capacity, providing distinct parameters for each task [21 22 23 24, 25). Replay-based methods replay a subset of examples stored in previous tasks
generated using generative models mitigate forgetting [26] 27 28 29] Regularization-ban methods introduce an additional regularization term in the loss function to prevent modifications to 33
crucial parameters related to previous tasks 6 30
Traditional incremental learning methods for images or text lack topology learning, making challenging to achieve effective topology mining and information preservation. By contrast, we analyze the basics of preventing catastrophic forgetting in GCIL from information preservation and solve them in the spatial domain.
2.2 Graph Incremental Learning
Graph incremental learning focuses on handling streaming graph data, and numerous methods have been developed explicitly for graph data [34] [3] 36 37 38
39, 40 41). Topology-awar Weight Preserving (TWP) preserves key parameters and topology of previous tasks through regularization terms [1. Experience Replay Graph Neural Network (ERGNN) framework incorporates memory replay by storing representative nodes (7]. Sparsified Subgraph Memory (SSM) stores sampled sparse subgraphs in a memory repository to preserve structural information [8). Su et al. introduced regularization terms to mitigate catastrophic forgetting from structural drift [Л]. Zhang et al. redesigned the architecture into three-layer prototype that adaptively selects different parameter combinations for different tasks (2]. The Condense and Train (CaT) [9] framework compresses the graph into small but informative synthetic replay graph. Furthermore, two graph incremental learning benchmarks have recently been developed 42 43).
In comparison, GSIP combines graph information preservation to avoid catastrophic forgetting through frequency local-global and high-frequency information preservation.
3 Problem Analysis
Graph Class Incremental Learning (GCIL). GCIL addresses the problem of supervised node classification within the context of an expanding graph. Specifically, each G denotes a newly emerging subgraph within the overarching graph. A G consists of a node set Vt and an edge set Ex with its connectivity captured by adjacency matrix A E g=x=, where n is the number nodes. Each vertex is associated with node features X,, and target label Y, E [0,1)s where represents the total number of classes. At time t, the GCIL problem denoted as PGCIL is provided with a subgraph G {X', A'). The PECIL problem is formally defined with the following signature:
Paciu ( ,M- MM)
(I)
3

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where f is graph neural networks and M signifies an external memory capable of storing subset of training nodes or other useful graph data.
In the scenario of graph incremental learning, disparate tasks are mapped into distinct partitions of the graph. Once the learning for a specific task is completed, access to the corresponding data is restricted. Our objective is to learn a shared graph neural network model that distinguishes all classes from existing ones. Formally, we aim to minimize the loss caused by previously seen nodes time step T in CCIL, the statistical risk of catastrophic forgetting is defined as:
min Σ Xy [H (Y", .a (S (G';
(2)
where 0 indicates parameters of the model, H. represents cross-entropy loss, and denotes softmax activation function.
Replay-Based GCIL. Replay-based methods store replayed nodes or subgraphs in memory M by sampling. Catastrophic forgetting is solved by maintaining the historical distribution. These methods train a new model by minimizing loss of old task nodes on new model concerning the true labels. (M; the replay loss is calculated as:
Given node representations on new model Z (Z
Creplay
Σ H (Y (2,,01) (z)
(3)
M
SEM
Semantic Shift and Structural Shift. Due to mem- ory and privacy limitations, a large amount of old graph data cannot be accessed in graph incremental learning, which leads to material information of old models being gradually forgotten and seriously dam- ages new model performance on old classes. We de- sign two novel shift metrics measuring semantic and structural forgetting degrees when trained on novel classes to show that model divergence manifests in node-level semantics and graph-level structure as- pects. On CoraFull dataset, we conduct shift tests us- ing model representations Zold and Znew generated by classical replay method ERGNN (7). Specifically,
mmma
3000
6.0
Fpoch
Epozs
Figure 2: Semantic shift (left) and structural shift (right) between old and new models.
central kernel alignment [44] scheme is leveraged to compute Semantic Shift Score (SS5x):
IS(Z znow)
(n0 = tr tr(Zold
SSSx (2 2 nue
HS(Zold, 2n(n)= (n 1)2
HS( 2 (SS new Zunw
(4)
where C' is centering matrix Cn = In - 1/nl1T. In particular, Structural Shift Score (555s)
is derived by performing structure Á inference using feature cosine similarity, then computing differences between graph representations obtained by Anonymous Walk Embedding (AWE) [45
SSS (2 ( = - COS(AW
AW B(A)"")); )), A.j = 1 [COS(2,, 2,) > 6), (5)
where cosine similarity function COS b) = a Tb [[b]) is used to calculate feature similarity degree, and d is similarity threshold. Each task is trained for 200 epochs, and shift scores range from 0 (no shift) to 1 (completely different). We observe that SSSx and SSSA in Figure 2 gradually rise with the increase of epochs. Serious shifts are found in both node semantic and graph structure between old and new models as new classes are trained.
4 Graph Spatial Information Preservation
4.1 Graph Information Preservation
Model information preservation for GCIL can be defined as the mutual information of graph informa- tion across old and new models when considering the corresponding model parameters:
P Pgota_gme I grold, grow))
here, goold and @new are graph information on old and new models. We directly maximize mutual information between old and grew which inherits powerful encoding capability of gold to grew

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Proposition 1. The upper bound on graph information preservation can be estimated as:
-I -I 2new)
S 2| 2 1 = 2
(7)
we expect to maximize mutual information I (2old, 2nnw)), thus minimizing -I (2old, 2ncu) in estimation is needed.
Proposition T is proved in Appendix A.1 which suggests that graph information preservation is bounded with the square of Euclidean norm between old graph information .n'ad and new graph information n'oww
4.2 Spatial Property
Based the spatial properties of graphs, analyze the maintenance of graph information in spatial domain to capture complex spatial relationships between nodes and edges graphs.
Lemma 1. (Graph spatial information factorization (46)) The graph convolution between convolution kernel F and the signal x to obtain graph information is formulated as follows:
F 2 (F (F) x Fhh x
(8)
where F x and Fh x are low-/high- frequency graph information, F In + AD-
Ft = In - - AD D is diagonal degree matrix with D.,; Σ, Ar.. and A = A + In
represents adjacency matrix with self-loop. Two pieces of information in spatial domain can be derived follows:
F - E
3z
Fh X;
(9)
VW [
VINTIN
JEN.
JEN,
where N represents node neighbors.
According to Lemma there is an identity map that filters out graph information 2 with graph convolution, which provides effective solutions to correspond 2 gold and 2row spatial domain. For cach component i, low-frequency information preserving
A 2:12 is defined as:
(( Zgaid
2
Zohl
Znew
A 212
Zhen
(10)
VIN.INN
VIN.INN
JEA,
JEM,
where Zold and new denote obtained representations on old and new models. This implies that the low-frequency information preserving semantic gap between the sum of node features and their neighbor features of replay data on old and new models.
Sustained global connectivity is crucial to avert the erasure global seman- tic information inherited from the preceding model. As displayed in Figure 3 global semantic shift does exist. We extend the concept of first-hop neighbor nodes in the previous equation include the entire replay graph (i.e. multi-hop neighbors), which is denoted as:
0.5
555, [Global)
( Zood Σ
Zold
Zeew
A
Z
VIMII
VM
JEM
JEM
Opoch
(11)
Global se- mantic shift on old and new models.
Figure
Similarly, the generalized low-frequency information preserving is the gap between the sum node features and replay data features on old and new models. It is worth noting that Eq. (TO) provides semantic comparison from a local perspective, whereas Eq. compares from a global perspective.
For each component i, high-frequency information preserving A 2,h ||
is defined as:
Zold
Znew
A 2
2 =
7old
JEN. VINGINN
Znew
VINIIN
(12)
JEN.
5

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Hrihhfrreuenne information preserving captures the gap between the difference in node features and neighbor features on old and new models from topological space.
Motivated by the above concepts, we introduce the following definition:
Definition 1. (Graph spatial information preservation) A graph spatial information preservation
model mainly consists of three kinds of information preservation || AA2 ~ || -
A 2h (defined in Eq. 10
and
12
Gold
Old node representations
Old graph representation
Distance
Peoling
A old
Sample
puli
7 oGGa
g
L
Li
new
Z AAN
Posting
New node representations
New graph representation
time
Low-frequency
(b) High-frequency information preservation
Class
Classl
Clascl
information preservation
Figure 4: A high-level overview of GSIP framework. It consists of low-/high- frequency modules to preserve old information. The old and new node representations are used to calculate information preserving loss of node representations, graph representations, and neighbor distances.
4.3 Instantiations for Graph Spatial Information Preservation
The above analysis yields two crucial insights: (1) old model information preservation can be solved by preserving the learned graph information; (2) graph information preserving can correspond to low- frequency local-global information and high-frequency information from spatial domain to calibrate node semantic and graph structure shifts. Inspired by these two insights, we propose low-/high- frequency information preservation to adequately capture the old model's 's information. A high-level overview of GSIP framework is shown in Figure 4 The pseudo-code can be found in Appendix A.4
Low-Frequency Information Preservation. The node representations within the previous model are derived via iterative feature integration and neighborhood communication, so contains low- frequency graph information. The information of old model is aligned into the neighborhood of new model to better utilize oow-frequency information, which can be represented as:
L G= Σ -=Σ>>
Σ JEMJUNUUS
120ld 2
(13)
where Z is the output given by model. Low-frequency local information preserving loss uses Mean Squared Error (MSE) loss to locally match representations of nodes on old model Zold with representations of nodes and their neighbors on model new For replay methods that do not explicitly save neighbors, neighbor selection can be found in Appendix A.2 It is worth noting that since inputs become sparse when converted to probabilities, the softmax followed by Kullback Leibler (KL) divergence loss is not applied [47]
Preserving global information about low frequeeyy components aligns old model information whole and prevents catastrophic forgetting. Low- low-frequency global information preserving loss is introduced to minimize difference between global representations of old and new models, which is defined as:
4 | R(Zold) R(Z
(14)
where represents pooling method, which is computed by mean pooling R(2) 11/M Z, Similarly, MSE loss is used to calculate global representation gaps.

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HlihhFrrequeny Information Preservation. The high frequency part of spatial domain represents the difference between the features of nodes and neighbors. The updated model preserves old topology by incorporating prior local contextual information and then mitigates heterogeneous information propagation blockage caused by smoothness assumption. Specifically, for node ny NV denotes neighborhood node set and defines Sooldu,, as the similarity of selected node vector with adjacent nodes computed by old model:
exp (K (7,7ud 3ggh))
Sold (i) = (S Sold [Wa Sgld
(15)
Eyex, exp K Zppla 2g/h
where K(,,) represents kernel function that measures pairwise distances between each node and its neighbors the latent feature space, and clement-wise absolute values KC (Z,, 2y) 12, Z, is used. Then, we measure similarity distribution from new model S (U;, which is formed by:
exp (K (Z, -nean
gnew (oi,
Spew
Snew [i
Snew
(16)
EyeN, exp (KC (Znee Zzguu)
High-frequency information preserving is proposed to map neighborhood pairwise differences between old and new models in topologiaal space, information loss from old structure to new structure is more easily recognized with the help of KL divergence, which is denoted as follows:
Sold (vi,N)
Lu
Σ iEM
Sold (u,,Ni)lo log Snew Ni)
(17)
Model Learning. To combine different preserving losses, the final graph information preservation loss function is defined as:
Cgip = Li 3CG 7kn,
(18)
where 3 and y are loss weights.
Node classification loss is obtained by Lnc = 1/| EG H. (Y, a (f (G: Therefore, the overall model learning objective is the weighted sum of current node classification loss, replay loss, and graph information preserving loss:
L Li + EreplayCrepion + Cigiph
(19)
where Oreplay and agip are loss weights, and the value of Oreplay is relevant to the design of replayed method. More analysis about the preservation of other graph frequency information (i.e. mid-frequency information and high-frequency global information) is given in Appendix A.3
Experiments
5.1 Datasets and Setups
Datasets and Settings. We utilize five public datasets to evaluate the effectiveness of the proposed method in GCII, the statistics of datasets are reported in Appendix B. Three ways of dividing classes are used: one is divided unequally, and the other two are divided equally, with equal classes per task. The first dataset is CoraFull [8]. which has 70 classes, 30 classes are used as base classes for dividing unequally, then 20 classes are used as an increment, and we divide classes equally into 10 or 2 classes per task. Arxiv [49] and Reddit (50), both containing 40 classes, dividing unequally using 10 classes as base classes, then in increments of classes, and dividing equally with 10 or classes per task. Each dataset has tasks with 2 classes per task on Cora [5]] and Citeseer [51]. The
latest benchmark [42] is employed to implement ERGNN, along with CaT [9] is used to implement SSM and CaT, we follow their settings in graph class incremental learning. Our implementation and detailed settings are available in Appendix B.4 and B.5
Baselines and Metrics. We compare our method with the following baselines, including Finetuning, Joint, EWC (30,, GEM (3T] MAS (2), LwF 66. TWP M1, SSRM [II), and three replay-based based methods (i.e. ERGNN 77. SSM g. and CaT (9), where three graph replay methods apply our framework. Finetuning the lower bound baseline updating the model only with newly emerging graph data. Joint is the ideal upper bound and inputs contain all previous graph data. We choose two widely used metrics to evaluate the performance of the compared methods, including Average Performance (AP) and Average Forgetting (AF) [1].
7

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Table 1: Performance comparison CoraFull, Arxiv, and Reddit for GCIL setting. Results averaged among three trials. The best performing results (excluding Joint) are highlighted in bold.
CoraFoll
Arxiv
Reddit
Method
Unequally
Equally (10)
Equally (2)
Unequally
Equally (10)
Equally (2)
Unequally
Equally (10)
Equally (2)
APt
APt
APM
AFI
APt
APP
APt
APt
AFT
AP
AFT
APt
APt
APT
APT
AP1
Finetuning
23.95
-76.59
11.06
-85.77
2,70
-95 48
11.64
-70.41
5.41
-50.00
4,91
-87.61
14.66
-91 80
22,90
-94.42
5.83
-94.23
FWC
24.09
-75.78
11.15
-86.88
5.13
-93.08
11.93
68.97
14.83
-5.33
4.91
-87.88
13.79
-9.35
22.30
.9.27
9.66
-93.85
GEM
23.95
-76.05
11.23
-85.78
7.97
-90.00
11.61
-60.27
8.27
-44.22
4,92
-66.66
18.51
89.79
22.58
99.95
35.11
-65.67
MAS
24:20
-75.97
10.94
-82.37
4.43
-89.22
11.09
-66 76
12.32
-57999
5.29
-81.64
15:45
-0.50
25.54
0.01
5.98
-14.17
LwF
23.99
11.14
2.72
-95.08
11.93
-88.14
-9033
-9..99
-88.98
TWP
23.86
-75.74
11.01
-85.43
3.56
-94.66
11.93
-69.26
14.41
-6.55
4,90
.87.75
13.95
-96.17
21.22
-9644
9.34
-94.24
SSRM
63:62
-16.24
31.39
-60.61
3.22
-89.29
31.51
-45,12
26.61
-46.22
26.16
-61.24
78.40
-20.92
76.78
-23.16
83.96
-15.41
ERGNN
60.91
-19:47
24.39
-69.31
3.01
-94.34
31,18
-45.45
24.47
-19.11
24.70
-62.26
76.60
-23.22
75.22
-25
83.16
-16.21
GSIPP
67.22
110.91
71.15
11 11.37
44.79
-44.60
34,09
-32.59
33.88
-27.97
40.21
-28
96
90.82
66.05
89.59
2,03
93.03
55.50
Improve
6.31
8.56
46.
46.76
57.90
57.94
11.78
49.74
2.91
12.86
21.14
15:51
33.30
14:22
17.17
14.37
23.23
9.87
10.71
SSM
50.51
-10.56
62.90
-6.02
79.02
-4.24
63.48
-12.41
60.57
-10.09
63.91
-12.48
90.10
-5.83
86.91
-3.24
96.24
-1.64
GSIP
55.32
-2.50
63.86
0.08
79.3
0.70
63.36
-1.27
61.55 34
-6. 34
64,16
-8.87
90.74
-3.97
87.41
0.13
96.25
-0.65
Improve
4.81
0.96
0.29
-0.12
3.75
3.61
1,86
3,37
0.01
CaT
70.35
-5.26
76.35
-5.44
80.64
.4.31
71.66
8.33
70.16
.7.25
66.21
-12.73
96.39
-0.77
93.97
-1.31
97.64
.0499
+GSIP
71.06
-0.28
78.29
-1.25
81.10
2.68
71.52
-..76
70.57
-3.97
68.80
3.49
96.15
-0.23
94.23
0.21
97.55
1.04
Improve
0.51
1.94
0.46
40.14
0.41
3.28
16.22
0.54
1.52
-0.09
Joint
85.3
85,3
85.3
63.5
63.5
63.5
98.2
98.2
98.2
5.2 Performance Comparison
GSIP can improve the performance of existing replay-based information preservation methods. The effect of GCIL on five datasets is presented in Table T and Table 2 and the results with standard deviation are presented in Appendix C.T Joint does not provide AF due to its non-compliance with the incremental learning setting. The existing regular- ization term relies on the correlation between old and new classes leading to catastrophic forgetting, and some of them do not take topology into account. The hybrid method SSRM absorbs partial old infor- mation, resulting in extremely limited performance gains. GSIP consistently demonstrates significant improvements in AP and AF when combined with ex- isting replay methods. For example, ERGNN- GSIP improves both AP and AF by about 10% on Reddit. On CoraFull, SSM-GSIP under unequal partitioning situation improves AP and AF by 4.81 % and 8.06%, respectively. CaT-GSIP performs remarkably well on Arxiv, even surpassing the performance of Joint, which has a 16.22% increase in AF on a setting with an increment of 2. CaT experiences slight decrease in AP in some settings after using GSIP. This can be attributed to GSIP enabling better preservation of graph spatial information from old model, resulting in a lower forgetting rate. In addition, for ERGNN-GSIP, the AP and AF increase by 5.81 % and 9. 1.44% Cora and by 13.64% and 21.74% on Citeseer. On Cora and Citeseer datasets, the AF of SSM-GSIP improves by more than 5%, and CaT-GSIP achieves the highest performance in most cases, with the AP approaching the value of Joint.
Table 2: Performance comparison on Cora and Citeseer for GCIL setting. Results are av- eraged among three trials. The best perform- ing results (excluding Joint) are highlighted in bold.
Cora
Citescor
Method
Equally (2)
Equally (2)
AP1
AF1
AP1
AF1
Finetuning
32.58
-96.83
31.46
-77.66
32.58
-97.16
31.26
-78.22
GEM
32.70
-97.12
31.39
-77.70
MAS
31.84
-97.17
31.25
-76.67
32.58
-97.57
-78.29
TWP
32.58
-97.32
31.22
-78.14
SSRM
35.48
-70.01
51.91
-67.66
FRGNN
65.48
-46.09
47.65
-51.12
+GSIP
71.29
-36.95
61.29
-29.38
Improve
9.14
13.64
21.74
SSM
67.64
-19.78
60.99
-13.60
+GSIP
69.92
-11.82
61.86
-8.39
Improve
2.28
7.96
0.87
5.21
CaT
88.22
-440
75.08
-10.93
GSIP
89.60
1.84
77.02
-9.95
Improve 1
1.38
6.24
1.94
0.98
Joint
93.09
78.27
GSIP can consistently achieve excellent performance on old classes in different task IDs. The performance matrices of ERGNN on CoraFull before and after incorporating GSIP are shown in Figure 5(a)and Figure [()) It is difficult to remember the information of old classes before employing graph spatial information preservation. After implementing the GSIP scheme, the performance matrix demonstrates a deceleration in the forgetting process (i.e., the color of each column does not change much and the color deepens), which indicates that the catastrophic forgetting problem is mitigated due to the preserving of old graph information.
8

---

100
100
LRON
ERONN OSP
74
S3M
SM OSP
AF
20 13
150
123
20
400
(a) ERGNN
(b) ERGNNNGSIP
(c) Shift calibration
(d) The analysis of #M
Figure 5: (a)-(b) Performance matrices on CoraFull dataset. (c) Semantic and structural shift calibration of old and new models during increments. (d) Performance changes affected by #M on CoraFull dataset.
GSIP can offer information preservation capability to calibrate semantic shift and structural shift. Figure eecibiiss curves of shift scores during the incremental process for ERGNN on CoraFull. It can be noted that shift scores start at relatively high value, gradually decrease, and smooth out after graph spatial information is maintained, demonstrating that the low-frequency local-global information and high-fiequency information of old model are well captured, and semantic and structural shifts are nicely calibrated.
5.3 Ablation Study
We investigate the effectiveness of low- low-frequency local modules (LL), www-frequency global modules (LG), and high-frequency modules (H), the experimental results use ERGNN baseline (B). The results of ERGNN, SSM, and CaT with standard deviation are summarized in Appendix C.2 The
above components are added one by one to baseline for performance comparison. From Table 3 we observe that: (1) When LL is utilized, the model can easily learn the aggregation rules of nodes and neighbors from old model locally. AP (AF) improves by about 2% (6%) to 45% (55%) over the baseline demonstrating the superiority of LL. (2) Combining LG significantly improves performance, especially on Reddit. The reason may be that larger datasets have greater overall shifts during increments. (3) H also brings significant improvements, with AP (AF) improving by about 1.6% (0.9%) to 2.4% (7.4%) over B+LL+LG on Reddit. This indicates that the H module can extract more topology information for better performance.
Table 3: Ablation comparisons of graph spatial information preservation.
Reddit
Method
Uncqually
Equally (10)
Equally (2)
Uncqually
Equally (10)
Equally (2)
Unoqually
Equally (10)
Equally (2)
APt
API
AP1
API
APt
APt
AP-
AP
AP1
APT
AP1
AP1
AP1
API
B
60.91 -19.47 24.39 .9.31 301
.9.34
31.18
45.45
24.47
.9.11
24.70
62.26
76.60 -23.2
75.22
25522
83.16
-16.21
B+LL
65.79
-13.20
69.02 14.15
41.37
47.39
33.27
34.60
27.10
40.95
38.09
35.04 84.63
-12.09
84.26
-1.37
87.52
10.97
BiLLaGG
66.22
-12.78
69.77
.1313
41.84
46.73
34.00
.3.61
32,80
34.40
39.89
-2885
89.21
87.17
-948
91.34
-7.12
BALLALGH
67.22
-10.9
71.15
-1.37
44.79
44.60
34.09
33,59
33.8
-7.97
40.21
28.96
90.2
-6.05
89.59
-203
93,03
-5.50
5,4 Further Analysis
Hyper-Parameter Analysis. We analyze the impact of the number of storage nodes for each task #MM on performance. As depicted in Figure 5(d) it can be observed that the proposed method consistently outperforms the original method in terms of the -AF metric (the lower, the better), regardless of the value of #MM Interestingly, even with less memory, the proposed method still achieves better performance. CaT cannot be trained on 400 nodes due to Cuda memory constraints. We analyze the impact of loss weight Oigip on ERGNN, SSM, and CaT across CoraFull, Arxiv, and Reddit datasets with increments of 2 in Figure 6 For ERGNN, SSM, and CaT, aggi,.1 set to 1, [0.01, 0.01, 0.01], and (0.1, 0.01, 0.01] for three datasets. It can be observed that the performance change not as significant with the variation of a on SSM and CaT. However, different Ggip has a greater impact on performance with ERGNN-GSIP. The possible reason is that ERGNN selects representative nodes for replay, which may cause class imbalance and topology discarding. For ERGNN, SSM, and CaT, the optimal hyper-parameters agiip on three datasets are [50, 10, 1), [0.1, 0.1), and [1, 0.5, 0.5). Because of space limitation, we provide more curves about # in Appendix C.3 and hyper- parammerr analysis of loss weights 3 and y in Appendix C.4

---

23-
x30
Fya: X22
K50
1:30
100
X:300
4ju 200
(a) ERGNN-GSIP, AP
(b) SSM-GSIP, AP
(c) CaT-GSIP, AP
10-
20
1.20
1.:0
26-
Pae: 10
50
60-
App 300
Rhee 100
20
26
CoraFall
Andiv
Redde
Corafull
Araiv
Reddd
-20
Cor of
Ardie
Reddlt
(d) ERGNN-GSIN AF
SSM-GSIP. AF
(1) CaT-GSIP, AF
Figure 6: The analysis of agip in ERGNN, SSM, and CaT on CoraFull, Arxiv, and Reddit datasets.
(a) ERGNN, Task 1 ERGNN-GSIP, Task
(c) ERGNN, Task
(d) ERGNN-GSIP, Task
Figure 7: The visualization of node embeddings in Task 1 and Task 7 on CoraFull dataset.
To qualitatively demonstrate the effectiveness of our representations, we adopt t- SNE (52 to visualize the learned node embeddings. After learning the last task, Figure ((a) and Figure 7(b) show the results of the learned node embeddings in Task on CoraFull, while Figure 7cc) and Figure 7 demonstrate the results of the last task. We can clearly observe that GSIP possesses better representation ability by considering representations and classifying old and new classes well.
Visulization.
6 Conclusion
We contribute to the literature of GCHL by addressing the issue of information preservation from old model when adapting new classes. The key insight is that preserving graph information from spatial domain plays a vital role preserving information about old model, and subsequently calibrates semantic and structural shifts and reduces catastrophic forgetting risk. To accomplish this objective, we introduce a framework, GSIP, which utilizes the outputs of nodes in old model to diffuse the outputs of new model and its neighbors, then aligns the outputs of new model with old model after pooling. Finally, GSIP maintains kernel distance of neighbor pairs on both old and new models. The graph information is remembered by wwwffreqenncy local-global information preserving and high-frequency information preserving in feature and topological space. Evaluations over benchmark datasets show the superiority of GSIP in handling different dataset splitting cases. In the future, will investigate comprehensive analysis for the preservation of complicated graph signals.
Acknowledgement
This work was supported in part by the National Science and Technology Major Project under Grant 2022ZD01 16500, in part by the National Natural Science Foundation of China under Grants 62436002, 62476195, 61925602, U23B2049, and 62222608, in part by Tianjin Natural Science Funds for Distinguished Young Scholar under Grant 23JCIQIC00270, and in part by Zhejiang Provincial Natural Science Foundation China under Grant LD24F020004.
10

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14

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A Method
A.1 The Proof of Proposition 1
Proposition 1. The upper bound on graph information preservation can be estimated as:
-I ( g S ||2 -2 2 1| = |
(20)
we expect to maximize mutual information ( (rold; gnew), thus minimizing I estimation is needed.
-I (grold gnew) in
Proof. The mutual information of graph information on old model and new model 53 can be equivalent to:
I (2ood, 2new) = / d. d. p( g goold) aol
pl 2 )
Vnew pl (p (oodd 2odd)
E
p p 20ol) log
1d 2new
Melmaeee spl
grod (plde
gnem Vold
pl
gpold) (nnw
Σ
p 2man 2rold 2
pl2 pl
guem gold
p( n grldd) log Jppee
Σ
new oold
Σ
pl (
)p( 2rald g log(p( grold g)nw))
gnem Vold
Σ
pl (nnew g log grold)
T nneu Wold
E m H goold g gnnww
Σ
log pl 20old )p( grold)
gold
H. (grll)
H 2oold
2enew
H(2old)
+ goss grow [log p( grold ]]
+ R.ga gnew (log q( goold g]ew)) (
H( grold)
+ gg
[DxL(p( 2rld | (gneee) la( goold genew))
Eg Eguu g [log g( (rald) 2
(21)
where expectations over distribution p( ofd genww) H( (rld) has been removed since it is constant with respect to optimized parameters. The reason for the inequality is the non-negativity of entropy H and Kullback-L.eiber divergence DKL.
We expect to maximize mutual information I (2ro gnew), so we need minimize goold, genew) in the loss function, hence we can obtain:
(22) The conditional likelihood maximized fit the information of old model, and the new model receives compressed information needed to recover old model. A Gaussian distribution with mean μ and variance employed to estimate variational distribution q( @rold 2enew)) which is expressed as follows:
-I 2old, grnww) S EEg 2 [log q( 2rold 2 ne)
[
I ggoll, grnew)
log (golld genww)
(23)
[MI
(grotd
Mi (Fnew)))
2
log oy
+C,
20?
where C is constant. Mean squared error matching can seen as a specific instance of the above formula when ul ( genw and o 1, thus -I grold, grnww) I 2 2 g
D
15

---

A.2 Neighbor Selection
Some replay-basee based methods do not explicitly save topology, so we treat nodes with high representation similarity and label correlation on old model as neighbors for each node when computing low- frequency local information preserving loss. Nodes whose feature similarity is greater than the given threshold and whose labels are the same can be neighbors, formulated as follows:
Ni Li (COS(Z)) 2g > 8) n (Y; Yj) M), M;
(24)
where the cosine similarity function (Z, 2,) = Z, 2 2 is used to calculate the degree of feature similarity, and 8 is similarity threshold.
We only use node feature similarity when calculating the loss of high-frequency information preserva- tion for neighbor selection. The absence of label correlation captures the distance between nodes on different classes, which is conducive to the model to distinguish different classes. The neighborhood selection process is expressed as follows:
M (j COs Zggd) > 8, E M),
(25)
where ó the similarity threshold.
A.3 The Analysis Other Graph Frequency Information Preservation
We perform analyses and experiments to assess the preservation of various aspects of graph frequency information, including mid- mid-frequency information and figh-frequency global information.
Mid-Frequency Information Preservation. According to the definition of mid- fid-frequency filtering graph convolutional networks [54] [53,, the mid- frequency convolution can expressed as:
= (In - 5ー AD (L, + - AD-
(26)
Through a similar analysis as mentioned above, mid- dii-fireyency information preservation defined as:
(( Zold
Zold
Zpen
| 2,
Znew
(27)
V W3
VIN
JEN?
JEN?
where N2 is the second-hop neighbors of nodes. The distinction between mid-/high- frequency information preservation lies in that mid hid-frequency signals calculate the differences between the node and its second-hop neighbors.
Frgg-Fre Global Information Preservation. The general formula for high-frequency global information preservation is represented as follows:
)
Zgdi
Zeeew
A 2
Zeel
Znew
(28)
VIMIMM
VMM
JEM
JEM
It implies that every node in replayed graph needs to calculate disparity with other nodes, Since graph neural network assumes that neighboring nodes have similar representations, the penalizing distance between nodes and their neighbors at multiple hops away is redundant and does not contribute to structural preservation. Moreover, the inclusion of this exhibits high time complexity (O(IM)2 k), where k denotes the dimensions of the hidden spaces).
imposes an optimization burden and
We add mid-freeqency information preservation (M) and high-frequenn global information preser- vation (HG) to GSIP in Table which can yield a slight performance improvement some cases. However, it does not lead better performance enhancements outstanding results. The possible reason is that the preservation of first-hop neighbors is sufficient to calibrate structural shift.
A.4 Algorithm
The proposed method is summarized in Algorithm |
16

---

Table 4: Performance comparison before and after adding other graph frequency information preser- vation on CoraFull dataset.
Unoqually
Equally (10)
Equally (2)
Method
APT
AFT
AP
AFT
APt
AFF
APT
AFT
APT
AF1
API
AFt
55.32
-2.50
67.22
-10.91
63.86
0.08
71.15
-11.37
79.31
0.70
44.79
-44660
GSIP
#0.75
#1.13
20.44
#0.62
#0.85
+0.76
+0.98
#0.74
+0.50
#0.25
#1.77
#1.67
55.28
-2.61
66.63
-10.07
63.89
0.20
69.96
-12.85
79.29
0.75
24.66
-68.48
+M/ HG
10.65
#1.04
20.49
+1.75
20.86
#0.68
20.85
20.94
40.67
40.52
#1.47
+1.35
Algorithm 1 Framework of GSIP
Input: At time step > 1: New input G. Memory M, Graph neural networks f. Labels Y, Learned
parameter Oold Max epochs U, Loss weights Orepiays Qgip
Output: Parameter (new which can mitigate catastrophic forgetting of preceding classes
1: Initialize gnew random
2: for u = 1 to U do
3: Lnc = (Y
4: Lreptay = Lee(VM [(M:0mew)
(MMM (M:
5:
Lgip = Ereg (MM;
6:
gnew c- arg min L fine CreplayCrepler + agip
7: end for
8: Add selected nodes to memory M
9: return Parameter (new
B Implementation Details
B.1 Datasets
As illustrated in 5 we utilize five public datasets to evaluate the effectiveness of our proposed method in graph class incremental learning. Three different ways for partitioning datasets are employed: one involves an unequal division, where more classes are designated as base classes to enhance model robustness, while the remaining classes are treated as novel classes; the other two ways involve an equal division, with an equal number of classes allocated per task. The first dataset CoraFull 443, which encompasses 70 classes. For the unequal division, 30 classes are used as base classes, and an additional 20 classes are selected as increments. Additionally, the classes are divided equally into either 10 or classes per task. Arxiv [49] and Reddit (50), both of which consist of 40 classes. In the unequal division, 10 classes are designated as base classes, with increments of classes. Similarly, the classes are evenly divided into either 10 classes per task. Each dataset has
3 tasks with 2 classes per task Cora [51] and Citeseer [5I]
Table 5: Statistics of datasets.
Datasets
CoraFull
Arxiv
Reddit
Cora
Citeseer
nodes
19,793 130,622
169,343 1,166,243
227.853 114.615,892
2,708
3,327
# edges
5,429 7
4,732
# class
70
40
40
6
task
3/7/35 /
7/4/ 20
7 //4/ 20
3
# base class
30/ 10/2
10 10/2
10/ 10/2
2
# novel class
20/1
5/ 10/2
5 10/2
2
B.2 Baselines
In this subsection, introduce the basclines in the main paper. These baselines follows:
Finetuning the lower bound haseline updating the model only with newly emerging graph
data.
Joint is the ideal upper bound and inputs contain all previous graph data.
17

---

Elastic Weight Consolidation (EWC) [30 quadratically penalizes model weights according their importance previous tasks.
Gradient Episodic Memory (GEM) 31 modifies gradients of the current task using gradi-
ents computed from stored graph data.
Memory Aware Synapses (MAS) [32 utilizes analysis of parameter prediction the
importance of parameters when adding regularization terms.
Learning without Forgetting (LwF)
6 utilizes information distillation reduce the dis-
crepancy between old and new models.
Topology-aware Weight Preserving (TWP) preserves the key parameters and topology of the previous task through regularization terms.
Structural Shift Risk Mitigation (SSRM) [ introduces regularization terms to mitigate catastrophic forgetting from structural drift.
Experience Replay Graph Neural Network (ERGNN) 7 framework incorporates memory replay by storing representative nodes.
Sparsified Subgraph Memory (SSM) tory to preserve structural information.
[ stores sampled sparse subgraphs in memory reposi-
The Condense and Train (CaT) 9 framework compresses the graph into small but
informative synthetic replay graph.
B.3
Metrics
We choose two widely used metrics to evaluate the performance of the compared methods, including Average Performance (AP) and Average Forgetting (AF) B1. When the model learns the latest task, all previous tasks are evaluated and a lower triangular performance matrix W = {Wer E w x formed, where Wte' is node classification accuracy on task after learning task t' S 1') and r is the total number of tasks. Average performance AP = 1/T Ei=1 Wer evaluates the average performance of model on previous task after learning from new task т. Average Forgetting AF 1/(r 1) Σi= (w++ - we,e) represents the average performance degradation on previous tasks after learning from task r.
B.4 Reproducibility
Our method is trained using fixed random seed to ensure the consistency and verifiability of results. We are committed to open-sourcing and sharing our code to promote academic collaboration and knowledge sharing, enabling other researchers to reproduce and validate experimental results.
Table 6: Incremental learning settings.
ERGNN-GSIP
d: 0.5, sampler: CM subgraph_sampler: sampler: random n_ encoders: 500, feat init: randomChoice, feat Ir: 0.001, hid dim: 512, hop:
SSM-GSIP CaT-GSIP
B,5
Detailed Settings
Our model is deployed in Py Torch with an NVIDIA RTX 3090 GPU and trained with 200 epochs for every task. We use Adam with weight decay for optimization. and the learning rate is set to 0.005. We two-layer GCN with hidden dimension 256 the backbone. All results are reported in means and standard deviations over 3 trials. The train- main-validation-tee splitting ratios are 60%, 20%, and 20% for all datasets. The saii-valiiatio--test split is achieved through random sampling, resulting in variations in performance across different rounds of random sampling. 8 set to 0.99 and the search space of 5 is (0.5, 0.6, 0.7, 0.8, 0.9}. Table is the hyper-parameter we adopt from
42 and (9.
C Experimental Results
C.1 Performance Comparison
18

---

Table 7: Performance comparison CoraFull, Arxiv, and Reddit for GCIL setting. Results averaged among three trials. The best performing results (excluding Joint) are highlighted in bold, and the standard deviations are shown in gray.
CoraFull
Andi
Keddit
Method
Uncqually
Equally (10)
Equally (2)
Uncqually
Equally (10)
Equally (2)
Unequally
Equally (10)
Equally (2)
API
APT
APC
AFI
API
APC
API
APT
AP1
AFT
AP1
AP1
API
API
AP?
AFT
AP
APC
Finetuning
23,95
-76.59
60.14
11.06
-8.77
10,28
2.70
-9548
11661
10.20
-70.41
10.76
82.03
5.1
-50.00
10.01
4.91
$0.41
-67.61
11.66
-9180
13.51
22.90
-91,12
11.74
10.78
5.83
-91.23 23
EWC
10.48
24,09
-75.78
20.61
20:25
11:15
20.32
21.99
5.13
21.94
20.15
11,93
-68.9
21.45
20.31
14.83
-57.33
81.16
20.01
4.91
#0.30
-87.5
20.27
13.29
-95.35
#1.54
=0.49
-95.27
+1.30
#2.03
23.07
GEM
+0.37
23.95
-76.05
+0..5
=0.29
11:23
-85.78
+0.07
+0.67
7.97
+0.67
-90.00
+0.38
11.61
-60.27
+5.38
=1.50
8.27
-44.4
+1 49
+0.08
4.92
+0.61
-86.66
+4.23
18:51
-89. 79
+6.06
=2.58
22.58
-93.93
+1.08
#4.31
35.11
-65.67
+4.57
MAS
24.20
-75.97
+0.97
10:94
-82.37
4,43
-89.22
11.43
11.09
-6..76
12:32
-57.99
5.29
-81.64
15:45
-0.50
25.54
0.01
5.98
-14.17
L.FF
-76.14
11.14
-85.67
2.72
11.93
70.66
14.69
-8.14
16.13
99.31
21.39
-93.29
7.59
98
10.63
10.55
20.34
10:63
10:25
20.05
10.20
10.74
11
11.15
10.01
20.09
15.17
13.99
10.69
10,60
10.51
10.74
TWP
10.17
23.86
-15.74
10.70
20.21
11.01
-85.43
10015
11.02
3.56
20.84
-94.56
10.08
11.93
10.24
669.26
30.54
14.41
-56.56
2:3
1..001
4,90
20.32
-87.75
13.95
10.90
11.11
-96.17
21:22
31.58
10.11
.96.11
13.46
9.34
23.78
-94.24
SSRM
63.62
-16.24
#2:25
31.39
-00.61
5.22
.89.29
31,51
-45.12
#0.11
36.61
-46.22
26.16
-61.24
78 40
-20.92
76.78
-23.16
#3304
83.96
-15.41
ERGNN
+1.43
-19.47
24.39
+0.73
-69.11
+0.20
+0.51
-94.34
31,18
41.75
-55.45
+2.67
24.47
-49.11
2:300
11.
-62.26 26
+0.75
-23.2
16.4
#11
75 75.20
+14.13
41.92
-16.21
ERGNN GSIP
67.22
-10.91
71.15
-11.37
64.79
-44.50
34.09
33.59
33.88
-27.97
40.21
-28.90
90.82
-6.05
89.55
-2.03
93.00
-5.00
(Ours)
+0.44
+1l.62
=1198
+0.33 74
+1.77
=1.007
+0.77
+1.37
=0.87
+1.63
+0.92
=11.94
=03 710
+1..68
-2.01
+4.62
+3.87
+4.11
SSM
+1.03
50,51
+0.94
-10.56
+0.50
62.90
-6.02
37
+0. 50
79.02
+0.23
-4.24
63.48
+0.78
+0.01
-12.41
+0.50
00..7
-10.09
19
+0.33
63.91
+0.55
-12.48
+1 16
90.10
+1.14
-5.83
+1.77
36.91
+0.56
-3.24
+0.24
96.24
+0.31
-1.64
SSM-GSIP
-2.50
63.86
0.08
-7.27
61.34
-6.34
-3.97
87.41
96.25
(Ours)
10,55
60.77
10:70
10.37
60.58
10:40
10.37
CaT
10.67
70.55
10:40
-5.26
20.41
76.35
10.58
-5 44
10.30
60.64
20.43
-.31
10.73
71.66
10.26
-8.33
20.14
70. 16
10.79
-7.25
10.12
66.21
20.09
-12.75
20.17
96.39
10:00
-0.77
93.97
+0.30
10.008
-1.31
30.09
97.64
20.01
.0.9
C.T-GSIP
71,06
-0.28
78.29
-1.25
81.10
2.68
71,52
-4.76
70.57
-3.97
68.80
3.49
96.15
-0.23
94.23
0.21
97.55
1.04
(Ours)
#0.54
+0.07
#0.11
50.25
#0.18
20.16
+0.64
+0.08
#0.14
50.51
#0.24
#0.39
60.30
#0.31
=0.28
+0.64
20.05
60.30
Joint
+0.1
85.3
85.3
+0.1
85.3
63.5
63.5
+0.3
63.5
98.2
98.2
The effect of GCIL on five datasets with standard deviation is presented in Table / and g GSIP improves existing information preservation meth- ods under different dataset splitting scenarios. The performance matrices of SSM and CaT on CoraFull before and after incorporating GSIP are shown in Figure [ and Figure 9 Remembering information from previous classes becomes chal- lenging without the implementation of the pro- posed GSIP. The performance matrix illustrates deceleration in the forgetting process after adopt- ing the GSIP scheme. This is evident by the minimal changes and deepening in color of each column, indicating mitigation of the catastrophic forgetting problem. This is achieved through the preservation of information from previous model.
Table 8: Performance comparison on Cora and Citeseer for GCIL setting. Results are averaged among three trials. The best performing results (excluding Joint) are highlighted in bold, and the standard deviations are shown in gray.
Cor
Citeseer
Method
Equally (2)
Equally
AP1
AFt
AP1
AFI
Finctuning
32.58
-96.83
31,46
-71.66
20.00
20.39
20.27
20.67
32.58
-97.16
31,26
-78.22
EWC
20.00
20.55
#0.15
+0.61
32,70
-97.12
31.39
-77.70
GEM
$0.19
40.16
#1.06
MAS
31.84
-97.17
31.25
-76.67
#0.10
#0.22
#0.55
+1.18
LW/
32.58
-97.57
31.44
-78.29
+0.00
+0.27
+0.00
#0.36
TWP
+0.00
32.58
+0.17
-97.32
+0.23
31.22
+0.22
C.2 Ablation Study
SSRM
#0.49
35.48
+1.12
-70.01
+4.59
51.91
#6.67
-6.66
Efficiency. As shown in Table 9 Table 10 and Table [T] the consequences of ablation experi- ments with standard deviation for FRGNN, SSM, and CaT as baselines (B) are presented. We in- vestigate the effectivvness of low-frequency lo- cal modules (LL), frequency global mod- ules (LG), and high-frequency modules (H). The above components are added one by one to base- lines for performance comparison. B is the baseline. B+LL indicates that it only uses low- frequency local preservation. B+LL+LG represents that it uses low- oww-freueeccy local-global preserva- tion. B+L.L+I.H+H is the full model, which uses low-frequency and high-frequency preservation. All the components are effective and can bring great benefits to the model in performance improvement.
ERGNN
$0.76
65.48
-46.09
#1.15
47.65
-51.12
+1.47
ERGNN-GSIP
71.29
-36.95
61.29
-99.38
(Ours)
20:91
1:.22
+2.28
67.64
-19.78
60.99
-13.60
SSM
23.14
18.10
#1.43
13.61
SSM-GSIP
69.92
-11.82
61.86
-8.39
(Ours)
11.46
16.74
22.23
13.82
88.22
-4.00
75.08
-10.93
CaT
20.49
11.04
20.35
11.07
CaT-GSIP
89.60
1.84
77.02
-9.95
(Ours)
#0.62
+3.89
#0.70
+1.15
93.09
78.27
Joint
#0.85
19

---

Taal
50
50
50
59
Tasas
Tasks
TAsas
Taaks
(a) CoraFull, SSM
(b) CoraFull, SSM-GSIP
(a) CoraFull, CaT
(b) CoraFull, CaT-GSIP
Figure 8: Performance matrices in SSM.
Figure 9: Performance matrices in CaT.
Table 9: Ablation comparisons of graph spatial information preserving strategy for ERGNN.
ConaFall
Arsiv
Reddit
Method
Unequally
Equally
Equally
Unequally
Equally
Equally (2)
Unequally
Equally (10)
Equally
AP1
AFI
-23.22
+1:12
11.43
10.30
60.73
10.51
10.83
22.67
43.03
60.75
65.77
111.67
134.13
s1.92
=2.66
B+LL
20.87
63.79
11.25
-13.20
10.11
69.02
20.05
-14.15
21.00
41.37
12.19
-47.30
11.48
33.27
11.29
-34.0
33:47
27.10
14.17
-00.95
11.02
38.00
11.67
-3504
$1.58
84.63
16.35
-12.09
11.12
84.26
11.96
-12.37
12.86
87.52
-10.9
$2.8
B+LL-LG
60.3
66.22
11.12
-12.78
10.37
69.77
60.20
-13.33
11.25
4184
11.16
-66.3
10.37
34,00
60.58
-33.61
12.71
32.80
13.17
-34.0
10.15
39,89
60.40
-2885
12:43
89.21
10.94
-6.97
12. 70
87.17
13.86
39.45
+3.58
91.34
=3.01
-7.12
BALL-LG-11
-10.1
44.79
44,60
34,09
33,59
33,88
-27.7
40.21
23.96
90.82
+6.05
-.03
93.03
-5.50
11.63
Table 10: Ablation comparisons of graph spatial information preserving strategy for SSM.
CoraPull
Artic
Reddi
Method
Unequally
Equally (10)
Equally (2)
Unequally
Equally (10)
Equally (2)
Unequally
Equally (10)
Equally (2)
AP1
AP1
AP1
AF1
AFt
API
AF1
AP
50.51
-12.41
-1.64
+0.50
+0.80
+1.77
B+LL
#0.76
54.75
#1.06
-5.45
=0.58
63.68
=0.60
-1.60
#0.65
79.30
+0.36
-0.49
#0.91
62.99
a0.18
-9.70
#0.48
60.13
+0.98
-7.04
#0.34
63.93
+0.58
.9.80
+0.48
90.55
+0.46
-406
$1.56
87.34
40.93
0.06
20.19
95.98
#0.30
.084
B+LL-LG
60.56
55.16
+1.21
63.67
=0.59
-0.02
79.31
63.40
-7.60
+1.02
61.17
-657
64.01
-9.22
90.70
-4.5
87.38
+0.93
0.14
+0:35
96.16
-035
B+LL-LG+Il
6.34
3.97
60.5
a1.13
20.85
20.76
20.50
60.25
21.13
20.82
20.77
40.70
40.37
20.3
20.44
30.40
11.00
10.91
20.37
20.64
Table 11: Ablation comparisons graph spatial information preserving strategy for CaT.
CoraFull
Artiv
Reddit
Mechod
Unequally
Equally
Equally (2)
Unequally
Equally (10)
Equally
Unequally
Equally
Equally (2)
APt
AP1
APt
API
AP1
AP1
AP1
API
AP1
AP1
AP1
AP1
AP1
API
B
=0.67
70.55
=0.40
-5.26
#0.41
76.35
=0.58
-5.44
+0.30
80.64
#0.43
-4.31
#0.73
71.66
=0.26
-8.33
#0.14
70.16
#0.79
-7.25
#0.12
66.21
+0.09
-12.73
+0.17
96.39
#0.40
-0.77
#0.30
95.97
+0.08
-1.31
40.09
97.64
#0.04
.0.49
B-LL
70.74
-1.32
+0.16
78.31
+0.15
-2.03
-0.89
71.42
+0.20
70.42
-4.00
+0.24
67.23
+0.44
-2.85
96.12
-0.29
93.23
+0.78
97.26
B-LLALG
20.42
=0.07
=0.27
20.33
20.19
=0.17
#0.92
71.41
40.04
10.15
40.59
20. 10.10
68.61
20.34
10.24
10.24
10.47
93.91
41.09
10.33
10.36
B-LLALGH
20.54
71.06
20.07
-0.28
20.11
78.29
40.25
-1.25
10.18
$1.10
2.68
10.64
71.52
10.08
4.76
10.14
70.57
-3.97
10.24
68.80
10.39
3.49
10.30
96.15
10.31
00.23
10.28
94.23
10.61
0.21
10.05
97.55
10.30
1.04
B-ll.
1.8-
B-LL+LG+N
B+LL+LG+N
3.2-
CI
B+LLL B-LLeGG-H
0.6-
100
200
200
Epoch
Epoch
Epoch
(a) The change of loss C.
(b) The change of loss Cp
(c) The change of loss Ca
Figure 10: Graph information preservation training losses of different variants with epochs on CoraFull dataset.
20

---

Information Pattern. Training losses of ERGNN-GSIP during incremental processes in the dataset inequality partition setting are illustrated. Specifically, we focus on the last two tasks and examine how the loss of different variants changes with increasing epochs. We make three observations: (1) Low-frequency local graph information is well preserved. Figure a0(a) measures the degree of feature similarity. It can be observed that losses of the first two variants start to rise around the 50th epoch. However, the GSIP loss remains much lower than the first two variants and converges quickly. (2) The learning of low-frequency information is ensured after global graph embedding similarity correction. can be seen that the loss of GSIP is slightly lower than the other two variants in Figure To(b] (3) GSIP can preserve high-frees high-frequency information and reduce the forgetting of topology. In Figure [O(c] we can see that the losses of the first two variants decrease and then increase at each increment. This demonstrates that node difference information is almost completely discarded in these variants.
C.3 PlyperPParameter Analysis of Memory Size
We analyze the impact of the number of storage nodes for each task #MM on performance of each task that has 2 classes. As depicted in Figure Figure [5] it can be observed that the proposed method consistently outperforms the original method terms of the AP (the higher, the better) and -AF metric (the lower, the better), regardless of the value of #.M CaT cannot be trained with 400 nodes on CoraFull due to Cuda memory constraints. Interestingly, despite having less memory, the proposed method demonstrates superior performance across three datasets. Furthermore, the performance remains relatively consistent when storing 200 or 400 nodes, indicating that our method does not incur higher storage costs. Notably, storing only 10 nodes per task yields performance comparable to storing 400 nodes on Reddit, highlighting the superiority of our approach.
80
AP (96
ERGNN CaT
ERGNN GSIP
GSIP CaT-CSIP
20
#
Figure 11: The change of AP affected by #M CoraFull dataset.
ERONN
ERONN
FRSWN CSP
30
30
380
200
400
10 50
110
200
400
Figure 12: The change of AP affected by #M on Figure 13: The change of AF affected by #M on Arxiv dataset.
Arxiv dataset.
AP1 (N4
62
JF (55)
02
ERONN:
23
20
TM
Figure 14: The change of AP affected by #M on Figure 15: The change of AF affected by #M on Reddit dataset.
Reddit dataset.
21

---

C.4 Hyper-Paramest Analysis of Loss Weights
As shown in Figure T6 Figure 17 and Figure [8| the analysis for two hyper-parametent loss weights
3 and ERGNN, SSM, and CaT in terms of AF is conducted, and the results are in experimental
setting of increment 2. For ERGNN, 31 is set to [2c 5, le 1, le - 5] and "/1 is set to [2e
le 8, 1e 4] for different datasets. For SSM, 31 is set to [1. 1, 10] and "1 is set to [le 7, 1c 8,
le 3] for three datasets. For CaT, 31 is set to [le 1, 2, 2e - 4] and "i is set to [le 1, le 6.
2e 1] for three datasets. We notice that the model performance remains unaffected by changes in B
when y is set to 0 and by changes in when /3 reaches its optimal value.
81 81 x 10
x
81 x 50
X0
81 x 100
Vs x 100
CoraFull
Arxiv
Reddit
CoraFull
Anxiv
Reddit
Varying & with 0
(b) Varying
with /3
,3
Figure 16: The analysis of 8 and y in ERGNN-GSI CoraFull, Arxiv, and Reddit datasets.
A,
A x
Vi X50
A
100
AF (%)
AA (%)
CoraFull
Andiv
Reddit
CoraFull
Anxiv
Reddit
(a) Varying & with O
(b) Varying with /
3"
Figure 17: The analysis of /3 and ry in SSM-GSIP on CoraFull, Arxiv, and Reddit datasets.
10-
v2 10 10
8. 10
8. 100
Yn 100
AF (%)
AF T (%)
CoraFull
Anxiv
Reddit
CoraFull
Aniiv
Reddit
(a) Varying & with 0
(b) Varying with 8
3
Figure 18: The analysis of 33 and b in CaT-GSIP CoraFull, Arxiy, and Reddit datasets.
22

---

C.5 Visualization
To qualitatively demonstrate the effectiveness of representations, we utilize t-SNE 52 to visualize the node embeddings of ERGNN and ERGNN-GSIE After learning the last task, Figure Figure disbydssply display the results the learned node embeddings in Task on Reddit, while Figure 19(c) and Figure 19dd) illustrate the results of the last task. GSIP exhibits superior representation ability, effectively considering representations and accurately classifying old and new classes.
and
(a) ERGNN, Task 1 (b) ERGNN-GSIP, Task
(c) ERGNN, Task 7
(d) ERGNN-GSIP, Task
Figure 19: The visualization of node embeddings from ERGNN and ERGNN-GSIP in Task 1 and Task 7 of Reddit dataset.
C.6 Time Complexity Analysis
The time complexity of framework mainly comes from three aspects: (1) node classification O(V]K** + [E]k + [Vik), (2) replay scheme
and (3) graph spatial information preservation O [k + k). The total time complexity is O(V |M]k |V]K+ |M|k + |& E k + k), where and M are subgraph vertices and memory. & and &M edge of subgraph vertices and memory, then k and k denote the dimensions of inputs and hidden spaces. The time complexity increases linearly compared with baselines. The running time on CoraFull presented Table I2 For ERGNN, [V] > IM| leads to phenomenon that the higher the number of tasks, the shorter training time. On the contrary, more tasks result in longer training time due the properties of compressed graphs in SSM and CaT.
Table 12: Running time (s) of each epoch under three dataset partitioning cases CoraFull dataset.
Method
Unequally
Equally (10)
Equally (2)
ERGNN
0.8073
0.3694
0.1660
+GSIP
1.1913
0.4672
0.3792
SSM
0.0110
0.0279
0.0410
+GSIP
0.0113
0.0312
0.0494
CaT
0.0106
0.0189
0.0488
+GSIP
0.0147
0.0235
0.0524
D Discussion
D.I Linitation
The primary limitation of GSIP lies in its focus on replayed designs and the lack of connection other methods. Also, the paper does not examine other GCIL. settings on the graph, such as the lack of clear task boundary, which would be interesting direction explore in the future.
D.2 Broader Impact
Considering the broader implications of our work, we posit that the proposed framework for infor- mation preservation of the old graph model will support the development of systems in open environment based on machine learning. However, accessing old data may raise privacy concerns, and the dynamic updating of systems could inadvertently marginalize under-represented groups, potentially have negative impact on outcomes and interfere with fair decision-making.
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Reviews:
[{"review_id": "oYEVk0ABNm", "reviewer": "Reviewer_Za5b", "summary": "The paper focuses on the challenge of graph class incremental learning (GCIL), where a model must classify new nodes of emerging classes while retaining knowledge of previously learned classes. The primary issue of GCIL is identified as catastrophic forgetting, where new learning overwrites old knowledge. To address this, the author introduce the concept of information preservation, suggesting that preserving graph information can help mitigate semantic and structural shifts that cause forgetting. The paper proposes the Graph Spatial Information Preservation (GSIP) framework, which preserves both low-frequency (local-global) and high-frequency information in the graph's spatial domain. GSIP aligns old and new node representations, ensuring old graph information is retained. Experimental results show that GSIP significantly reduces forgetting by up to 10% on large datasets compared to existing methods.The framework is shown to be effective across various benchmark datasets.", "questions": "- Why the method only consider the low and high frequency information in the graph’s spatial domain? Except for these two frequency, there are still other frequency information worth to be captured, e.g., medium, and more to capture complicated signals.\n- Why for the loss function (Equation 19), there is only one scaling factor \\alpha for the second loss term?", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The proposed method GSIP preserves both low-frequency and high frequency information of a graph, which can alleivate the catastrophic forgetting issue of GCIL.\n- The proposed method is proved to be effective across various benchmark datasets.", "weaknesses": "- The design of the method, which only capture low and high frequency information is not comprehensive. There are more complicated signals to capture, e.g., many spectral GNNs are designed for that.\n- The method is only evaluated on 3 datasets, corafull, arxiv and reddit. It would be great to be evaluated on more datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your constructive comments!\n>**Q1: The design of the method, which only capture low and high frequency information is not comprehensive. Why does the method only consider the low-/high- frequency information in the graph’s spatial domain?**\n\nA1: Considering the preservation of low-/high- frequency information is essential for addressing the challenge of catastrophic forgetting. We conduct analysis and experiments and conclude that high-frequency information preservation sufficiently calibrates structural shift.\n\n\n1. **The preservation of low-/high- frequency information has effectively calibrated feature and structural shifts, yielding satisfactory results.**\n\n* Figure 5(c) on Page 8 shows that the semantic and topological shifts are well-calibrated with the final semantic shift score and structural shift score almost reduced to 0. \n* Table 1 on Page 8 shows that catastrophic forgetting is greatly alleviated after using GSIP, the forgetting rate on ERGNN-GSIP is reduced by up to 57.94\\%.\n\n2. **The analysis and experiments of mid-frequency information preservation.** \n* **Method analysis.**\nWe conduct a brief analysis of the preservation of mid-frequency information. According to the definition of mid-frequency filtering graph convolutional networks [1][2], the mid-frequency convolution can be expressed as:\n$\\mathcal{F}^m={({I} _n-\\widetilde{{D}}^{-\\frac{1}{2}} \\widetilde{{A}} \\widetilde{{D}}^{-\\frac{1}{2}})}{({I} _n+\\widetilde{{D}}^{-\\frac{1}{2}} \\widetilde{{A}} \\widetilde{{D}}^{-\\frac{1}{2}})}$. Through a series of analysis, mid-frequency information preserving is defined as:\n$$\n\\left \\| \\Delta{\\mathbf{z}^{m} _i}\\right\\| _2^2 = \\left\\|\\left({z} _i^{old}-\\sum _{j \\in \\mathcal{N}^2 _i} \\frac{{z} _j^{old}}{\\sqrt{\\left|\\mathcal{N} _i\\right|\\left|\\mathcal{N} _j\\right|}}\\right)- \\left({{z} _i^{new}-\\sum _{j \\in \\mathcal{N}^2 _i} \\frac{{z} _j^{new}}{\\sqrt{\\left|\\mathcal{N} _i\\right|\\left|\\mathcal{N} _j\\right|}}\n}\\right) \\right \\| _2^2,\n$$ where $\\mathcal{N}^2$ is the second-order neighbors of nodes. The distinction between mid-/high- frequency information preservation lies in that mid-frequency signals calculate the differences between the node and its second-order neighbors. \n\n* **Experimental result.**\nWe have added mid-frequency information preservation **(M)** in method, which could yield a slight performance improvement in some cases. However, it does not lead to better performance enhancements or outstanding results. The possible reason is that the preservation of first-order neighbors is sufficient to calibrate structural shift effectively. In the future, we will investigate comprehensive analysis and instantiations for the preservation of complicated signals.\n\n| | |CoraFull| |Arxiv| |Reddit| |\n|:----|:----|:----|:----|:----|:----|:----|:----|\n| | |AP|AF|AP|AF|AP|AF|\n|Unequally|GSIP| **55.32±0.75**| **-2.50±1.13**|63.86±0.85|0.08±0.76|**79.31±0.50**|0.70±0.25|\n| |+M|55.28±0.65|-2.61±1.04| **63.89±0.86**| **0.20±0.68**|79.29±0.67| **0.75±0.52**|\n|Equally (10)|GSIP|63.36±1.13| -7.27±0.82| **61.34±0.77**| **-6.34±0.70**| **64.16±0.37**| **-8.87±0.58**|\n| |+M|**63.48±1.16**|**-7.21±1.06**| 61.28±0.71|-6.50±0.79| 64.08±0.12| -9.41±0.71|\n|Equally (2)|GSIP|**90.74±0.44**| **-3.97±0.40**| **87.41±1.60**|0.13±0.91| **96.25±0.37**| **-0.65±0.64**|\n| |+M|90.65±0.29|-4.01±0.40| 87.34±1.69|**0.18±0.92**|96.11±0.33|-0.82±0.44|\n\n\n\n[1] Jincheng Huang et al. Robust Mid-Pass Filtering Graph Convolutional Networks. WWW 2023.\n\n[2] Haitong Luo et al. Spectral-Based Graph Neural Networks for Complementary Item Recommendation. AAAI 2024.\n\n>**Q2: The method is only evaluated on 3 datasets, Corafull, Arxiv, and Reddit. It would be great to be evaluated on more datasets.**\n\nA2: **We have supplemented results from the newly added Cora and Citeseer datasets.** As shown in Table R1 of PDF file, each dataset has 3 tasks with two categories per task. GSIP achieves significant performance improvements, validating the effectiveness of GSIP. For ERGNN-GSIP, the AP and AF increase by 5.81\\% and 9.14\\% on Cora, and by 13.64\\% and 21.74\\% on Citeseer. The AF of SSM-GSIP improves by more than 5\\% on two datasets. CaT-GSIP achieves the highest performance in most cases, with the AP approaching the value of Joint.\n\nAs for implementation details, we use the same experimental setup as three existing datasets, the experiment includes three hyper-parameters $\\alpha_{gip}$, $\\beta$, and $\\gamma$. On Cora, for ERGNN, SSM, and CaT, the hyper-parameters are [5e-3, 2e1, 2], [1e-1, 1, 5e-2], and [1e-2, 1e3, 1e2] respectively. On Citeseer, for ERGNN, SSM, and CaT, the hyper-parameters are [5e-2, 5e3, 1e1], [1e-3, 1e2, 5], and [1e-3, 1, 1e-3] respectively. The number of budgets $\\|\\mathcal{M}\\|$ is 10\\% of the number of budgets for the existing datasets.\n\n\n>**Q3: Why for the loss function (Equation 19), there is only one scaling factor $\\alpha$ for the second loss term?**\n\nA3: **We have added a new hyper-parameter $\\alpha_{gip}$ to adjust the weight of $\\mathcal{L} _{gip}$** and Equation (19) is modified to $\\mathcal{L}=\\mathcal{L} _{nc}+\\alpha _{replay}\\mathcal{L} _{replay}+\\alpha _{gip}\\mathcal{L} _{gip}$.     \n\n**We analyze the impact of $\\alpha_{gip}$ on AP and AF of ERGNN, SSM, and CaT across three datasets with increments of 2 in Figure R2 of PDF file.** For ERGNN/SSM/CaT, $\\alpha_{gip,1}$ are set to [1, 1, 0.1], [0.01, 0.01, 0.01], and [0.1, 0.01, 0.01] for three datasets. It can be observed that the performance change is not as significant with the variation of $\\alpha_{gip}$ on SSM-GSIP and CaT-GSIP. However, different $\\alpha_{gip}$ have a greater impact on performance with ERGNN-GSIP. The possible reason is that ERGNN selects representative nodes for replay, which may cause class imbalance and discard structure. For ERGNN, SSM, and CaT, the optimal hyper-parameters $\\alpha_{gip}$ on three datasets are [50, 10, 1], [0.1, 0.1, 0.1], and [1, 0.5, 0.5].", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xpKEDtJeib", "reviewer": "Reviewer_9jHn", "summary": "This paper proposes an innovative framework named graph spatial information preservation (GSIP), which alleviates catastrophic forgetting in graph class incremental learning (GCIL), by preserving low-frequency local-global information and high-frequency information in both the feature space and the topological space.", "questions": "Question 1: From line 50 to line 52, how to quantitatively understand “a larger distortion”? According to Figure 1, does “a larger distortion” indicate that the black dotted ellipse in Figure 1 (a) has a longer major axis than that in Figure 1 (c)? If so, since the five nodes are randomly selected and connected, then what about other red nodes having the same class but not included within the black dotted ellipse?  \n\n\n \n\nQuestion 2: Do the parameter isolation methods, the replay methods, and the regulation methods mentioned in the Introduction section (from line 28 to line 37), have the same meaning as those mentioned in the Related Work section (from line 83 to line 89)?  \n\n\n \n\nQuestion 3: From line 216 to line 217, based on Equation 14, since the mean pooling has been applied, what is the motivation and reasoning to introduce the mean squared error (MSE) loss to evaluate global representation gaps?", "limitations": "The authors have not adequately addressed the limitations and the potential negative societal impact of their work.  \n\nSuggestion for improvement: Data privacy should be discussed.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Strength 1: This paper has good originality to identify a unique challenge in GCIL task, which is the lack of theoretical understanding of information preservation.  \n\n \n\nStrength 2: This paper provides detailed mathematical derivations to explain how catastrophic forgetting can be alleviated by preserving graph information in the spatial domain.  \n\n \n\nStrength 3: This paper conducts comprehensive experiments, along with ablation studies, hyper-parameter tuning analyses, and case studies, to demonstrate the effectiveness of the proposed framework.", "weaknesses": "Weakness 1: The generalizability regarding Figure 1 could be further improved.  \n\nWeakness 2: The writing in terms of clarity and conciseness could be further improved.  \n\nWeakness 3: It is a bit confusing to take another average (MSE) immediately after an average (mean pooling). The reasoning for Equation 14 could be better clarified with more details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable suggestions!\n\n>**Q1: From line 50 to line 52, how to quantitatively understand “a larger distortion”? According to Figure 1, does “a larger distortion” indicate that the black dotted ellipse in Figure 1 (a) has a longer major axis than that in Figure 1 (c)? If so, since the five nodes are randomly selected and connected, then what about other red nodes having the same class but not included within the black dotted ellipse?**\n\nA1: We need to clarify that larger distortion does not mean that the major axis is longer. We present a deeper understanding of larger distortion, generalizability regarding Figure 1, and quantitative results.\n\n\n1. **The understanding of larger distortion.** We believe that distortion indicates that the feature distribution on the new model deviates from the feature distribution of the old model. The two categories can be well separated in the feature distribution of target, but not in the baseline model and lead to incorrect predictions and catastrophic forgetting. \n\n2. **Generalizability regarding Figure 1.** We have included a new figure to further illustrate the generalizability, it can be seen from Figure R1 of PDF file that the distribution of nodes predicted incorrectly (grey nodes in the figure) also includes some nodes outside the black dotted ellipse. We have supplemented another black dotted ellipse to mark incorrectly predicted nodes to increase clarity.\n\n\n3. **Quantitative analysis.** We use the structural shift score defined in Equation (4) on Page 4 to quantitatively understand the degree of distortion. The overall structural shift score is 0.1359, the structural shift score for nodes predicted correctly is 0.1431, and the structural shift score for nodes predicted incorrectly is 0.1646. \n\n>**Q2: Do the parameter isolation methods, the replay methods, and the regulation methods mentioned in the Introduction section (from line 28 to line 37), have the same meaning as those mentioned in the Related Work section (from line 83 to line 89)?**\n\nA2: **Although both are classifications of existing methods and use the same classification names, there are still some differences in terms of the perspective of review and the level of detail.** In the Introduction section, we summarize existing methods from the perspective of information preservation as maintaining the previous model or graph data. We provide a detailed introduction to the form of information preservation for each type of method. The Related Work section introduces the definition and relevant papers for each type of method according to the existing classification.\n\n\n>**Q3: From line 216 to line 217, based on Equation 14, since the mean pooling has been applied, what is the motivation and reasoning to introduce the mean squared error (MSE) loss to evaluate global representation gaps?**\n\nA3: Our motivation for using another MSE loss stems from the instantiation of low-frequency global information preservation. MSE loss is employed to measure the differences in global features obtained after pooling. It is more stable in application and has demonstrate the effectiveness of this term in ablation studies.\n\n\n1. **We have indeed stated that Equation (14) is an instantiation of the low-frequency global information preservation in Equation (11).** As we stated in Lines 184-187 on Page 5, the conclusion drawn from Equation (11) is that to maintain low-frequency global information, it is necessary to reduce the difference in representation of the full replay graph formed by nodes and their multi-order neighbors between new and old models. The two terms of Equation (11) represent the representations of the replay graph on the model, and we instantiate each term through mean-pooling. Additionally, the full graph representation is also meaningful, as it represents the overall graph information contained in each category dimension. \n\n2. **The reasons for using MSE loss to measure the distribution difference between the representations of old replay graph and new replay graph can be encapsulated in three aspects:** \n* The MSE loss is more stable compared to other distribution difference measure functions [1]. \n* The use of MSE loss in Equation (14) does not duplicate that of MSE loss in Equation (13). Equation (13) is an instantiation of the low-frequency local information formed by Equation (10). Both work together to maintain low-frequency graph information. \n* From the ablation study in Table 2, after applying low-frequency local information preservation (Equation 13) on the Reddit dataset, ERGNN-GSIP achieved AP and AF of 84.63\\% and -12.09\\%. Upon incorporating low-frequency global information preservation (Equation 14), the AP and AF increased significantly to 89.21\\% and -6.97\\%, representing an improvement of up to 4\\%. This confirms the effectiveness of utilizing another MSE loss.\n\n[1] Chaitanya K Joshi et al. On representation knowledge distillation for graph neural networks. TNNLS 2022.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0MQzIcxdga", "reviewer": "Reviewer_BwJ9", "summary": "This paper studies the graph class incremental learning problem, and specially focuses on theoretically investigating what matters in preserving the information from the old classes. \n\nThe authors theoretically demonstrate that maintaining the graph information can preserve information of the old model, such that the node semantic and graph structure shift can be overcome. Based on this finding, the authors split the graph information into low- and high frequency parts, and designed alignment based techniques to preserve these two types of information.\n\nThe proposed GSIP can be integrated with different baseline methods, and the empirical results on three datasets demonstrated the effectiveness of GSIP.", "questions": "1. What is the rationale behind the definition of the structural shift score? Does this implies the difference between each node and its neighbors?\n\n2. The forgetting is demonstrated through the node semantic shift and the structural shift, why is the following analysis conducted from the low- and high-frequency perspectives?", "limitations": "The authors have discussed limitations regarding the method design, positioning of the work against other baselines, and empirical settings.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. The proposed work includes both theoretical foundations and empirical improvements.\n\n2. The semantic shift and structural shift are quantified and demonstrated on CoraFull dataset, which makes the mechanism of the forgetting more tangible.", "weaknesses": "1. The writing requires improvements. The abstract and introduction do not provide a clear overview of the work, since there are multiple unclear terms like graph information and local-global parts. I would recommend the authors to revise this part. At least these terms should be briefly introduced in Introduction before being used.\n\n2. The notations are not consistent. In 4.1, the bold Z-old and Z-new seem to be same as the Z-old and Z-new above in Section 3, but are in different fonts.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your positive comments!\n\n>**Q1: The writing requires improvements. The abstract and introduction do not provide a clear overview of the work, since there are multiple unclear terms like graph information and local-global parts. I would recommend the authors to revise this part. At least these terms should be briefly introduced in the Introduction before being used.**\n\nA1: Thanks for your comments. We will provide a brief introduction to some terms before using them, such as defining \"graph information\" as \"Information in the graph data containing node semantic information and graph topological information\", and changing \"local-global parts\" to \"local-global information\".\n\n\n>**Q2: The notations are not consistent. In 4.1, the bold Z-old and Z-new seem to be the same as the Z-old and Z-new above in Section 3 but are in different fonts.**\n\nA2: Thank you for the comments. We will modify the font format of bold Z-old and Z-new in Section 4.1 for easier distinction. They have different meanings, the bold Z-old and Z-new in Section 4.1 represent old graph information and new graph information, and Z-old and Z-new in Section 3 represent the node representations of the old model and the new models. \n\n\n>**Q3: What is the rationale behind the definition of the structural shift score? Does this imply the difference between each node and its neighbors?**\n\nA3: We utilize the structural shift score to quantitatively measure the degree of forgetting of the topological structure learned by the new model compared to that learned by the old model. We have to emphasize that the structural shift score implies the gap between the differences between nodes and their neighbors in the new and old models, rather than merely measuring the differences between nodes and their neighbors. \n\n**The rationale behind the definition of structural shift score comes from two aspects:**\n\n1. **Structural shift score is an effective way to quantitatively measure catastrophic forgetting from topological structure when learning new tasks.** If the new model can effectively mimic the topological structure of the old model, then forgetting can be mitigated. Firstly, we utilize node representations from old and new models to infer topological structure, and similar features suggest potential edges connecting nodes. Subsequently, we employ the Anonymous Walk (AWE) [1], which reflects topological information, to obtain representations of inferred old graph and new graph. Finally, we measure the difference in graph structure representations using cosine similarity as structural shift score, and the greater the difference, the higher the score. The real topological structure is not targeted due to potential noise [2]. \n\n2. **The structure shift score is used as a metric to evaluate structural shift.** As shown in Figure 2 on Page 2, structural shift does indeed exist during the incremental process, and the structural shift becomes increasingly larger as new tasks are learned. As shown in Figure 5(c) on Page 8, the structural shift is reduced to almost near 0, indicating that our method has well-calibrated the structural shift.\n\n[1] Sergey Ivanov et al. Anonymous Walk Embeddings. ICML 2018.\n\n[2] Seungyoon Choi et al. DSLR: diversity enhancement and structure learning for rehearsal-based graph continual learning. WWW 2024.\n\n>**Q4: The forgetting is demonstrated through the node semantic shift and the structural shift, why is the following analysis conducted from the low- and high-frequency perspectives?**\n\nA4: Graph information encompasses both feature information and topological information. We utilize the analysis of graph information preservation and the decomposition of graph information to map the preservation of graph information to the spatial domain and instantiate it. Low-frequency information preservation mitigates feature shift, while high-frequency information preservation alleviates topological shift, which has been validated through experimental verification.\n\n1. **Low-frequency information corresponds to feature information.** Low-frequency information represents the sum of nodes and their neighbors, which is similar to the process of message passing and information aggregation in graph neural networks to obtain higher-order node features. Preserving low-frequency information is equivalent to aligning higher-order features formed by general GNN and thus effectively mitigating node semantic shift.\n\n2. **High-frequency information corresponds to topological information.** High-frequency information represents the differences between nodes and their neighbors, maintaining the graph structure information through the support of the old model. The distance between a node and its neighbor is a measure of whether an edge exists. The closer the distance between the node and its neighbors, it means that there is a high probability that there is an edge, and vice versa. \nThe old model implies the graph information of the old data because it has seen the old graph data during training. We utilize the similarity between nodes and their neighboring nodes from the old model's output and then optimize the new model to imitate the structural similarity distribution of the old model. High-frequency information preservation fortunately addresses the challenge of topological shift.  \n\n3. **Experimental verification.** As can be seen from Figure 5(c) on Page 8, the semantic shift score and the structure shift score are almost close to 0, thus semantic and topological shifts are well calibrated through the preservation of low-frequency and high-frequency information.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HK5jBW6xxf", "reviewer": "Reviewer_T3gk", "summary": "This paper studies graph class incremental learning (GCIL), which requires the model to classify emerging nodes of new classes while remembering old classes. The paper provides a theoretical analysis of GCIL and finds that preserving old graph information that corresponds to local-global low-frequency and high-frequency components in the spatial domain can calibrate semantic and structural shifts to reduce catastrophic forgetting risk. Then, the paper proposes a method to utilise node representations on old and new models to preserve node features, graphs, and neighbor distances.", "questions": "Please see the above weakness.", "limitations": "The paper presents several limitations: The primary limitation of GSIP lies in its focus on replayed designs and the lack of connection to other methods. Also, the paper does not examine other GCIL settings on the graph, such as the lack of a clear task boundary, which would be an interesting direction to explore in the future.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper studies graph class incremental learning (GCIL), addressing the problem of supervised node classification within the context of an expanding graph. The motivation is interesting, and the challenge really exists in the real-world graph tasks.\n2. The paper gives a theoretical analysis of GCIL and divides the framework into low and high-frequency modules to preserve old information.\n3. The paper evaluates the proposed method on three public datasets and the better results compared with baselines show the effectiveness of the proposed method.", "weaknesses": "1. The motivation for preserving global low-frequency information about the graph is uncleared. Why only separate local and global components for low-frequency and not separate local and global components for high-frequency? More theoretical analysis and studies are needed.\n2. Why is the experimental setting for each dataset's tasks configured as described in the paper? Is it following previous research works? It is highly recommended to provide results for different task settings.\n3. It is highly recommended to add the implementation details, especially the hyper-parameter for each dataset in the paper.\n4. More hyper-parameter experiments about the method are needed. For example, the parameter sensitivity analysis about the \\alpha.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your positive comments!\n\n>**Q1: Why only separate local and global components for low-frequency and not separate local and global components for high-frequency?**\n\nA1: The rationale for maintaining low-frequency global information (LG) without high-frequency global information (HG) is that the design of HG is redundant, incurs high computational costs, and does not yield performance improvements.\n\n1. **Redundant method design and higher time complexity**. \n* We need to clarify that LG is meaningful as it represents overall graph information contained in each category dimension. It can be instantiated using graph pooling and does not need to waste much time. \n\n* The general formula for HG preservation is\n$$\\left\\|\\Delta{\\mathbf{z}^{\\widehat{h}}_ i}\\right\\|_ 2^2 = \\left\\|\\left({z}_ i^{old} - \\sum_ {j \\in \\mathcal{M}} \\frac{{z}_ j^{old}}{\\sqrt{\\left|\\mathcal{M}\\right|^2}}\\right) - \\left({z}_ i^{new} - \\sum_ {j \\in \\mathcal{M}} \\frac{{z}_ j^{new}}{\\sqrt{\\left|\\mathcal{M}\\right|^2}}\\right)\\right\\|_ 2^2.$$ HG implies that every node in replayed graph needs to calculate disparity with other nodes. Since GNN assumes that neighboring nodes have similar representations, penalizing distance between nodes and their neighbors at multiple hops away is redundant and does not contribute to structural preservation. Moreover, the inclusion of this term imposes an optimization burden and exhibits high time complexity ($O(|\\mathcal{M}|^2 \\cdot {k})$, ${k}$ is dimensions of hidden spaces).\n\n\n2. **Lower experimental result.**\n* According to ablation experiment in Table 2, the preservation of LG in ERGNN-GSIP on CoraFull result in an increase of 0.75\\% in AP and 1.02\\% in AF. On Reddit, the preservation of LG in ERGNN-GSIP led to an increase in AP by 4.58\\%, 2.91\\%, and 3.82\\%, and in AF by 5.12\\%, 2.92\\%, and 3.85\\% across three task divisions. \n\n* We conduct experiments on CoraFull and find that the addition of HG preservation loss does not lead to performance improvement.\n\n| |Unequally| |Equally (10)| |Equally (2)| |\n|:----|:----|:----|:----|:----|:----|:----|\n| |AP|AF|AP|AF|AP|AF|\n|GSIP|**67.22±0.44** |-10.91±0.62|**71.15±0.98**|**-11.37±0.74**|**44.79±1.77**|**-44.60±1.67**|\n|+HG|66.63±0.49|**-10.07±1.75**|69.96±0.85|-12.85±0.94|24.66±±1.47|-68.48±1.35|\n\n\n>**Q2: Why is the experimental setting for each dataset's tasks configured as described in the paper?**\n\nA2: We adopt standard configuration from prior studies [1][2][3][4], which is the current mainstream setup, and integrate two new task setups. Additional configurations have contributed to a reduction in variance. Subsequently, we conduct tests on two new datasets within the general setup.\n\n1. **Task settings as previous works + New task configuration.** We clarify that existing works are set with 2 classes per task. We have followed the setup of previous works [1][2][3][4], corresponding to our setup of Equally (2). We introduce two new settings: Equally (10) and Unequally. \n\n2. **New task configuration helps to reduce variance.** We believe that  new task configurations can bring a certain level of robustness. As shown in Table 5 on Page 16, On Reddit, the variances of AP and AF for Equally (2) reach 3.87 and 4.11 for ERGNN-GSIP, while for Unequally, the variances of AP and AF are only 0.70 and 0.68, with a reduction of more than 3.\n\n3. **New experimental results.** As shown in Table R1 of PDF file, we have added results on the Cora and Citeseer datasets with an increment of 2 and a task number of 3, validating the excellence of our method.\n\n[1] Xikun Zhang et al. CGLB: Benchmark tasks for continual graph learning. NeurIPS 2022.\n\n[2] Yilun Liu et al. Cat: Balanced continual graph learning with graph condensation. ICDM 2023. \n\n[3] Junwei Su et al. Towards robust graph incremental learning on evolving graphs. ICML 2023.\n\n[4] Zeyang Zhang et al. Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet. ICML 2024.\n\n\n>**Q3: It is highly recommended to add the implementation details, especially the hyper-parameter for each dataset in the paper.**\n\nA3: **We summarize experimental setup and hyper-parameters used for each dataset.**\n\n1. **Experimental setup.** Each task undergoes a training regimen of 200 epochs. For optimization, we employ the Adam algorithm with weight decay, setting the learning rate to a value of 0.005. The model's architecture is anchored by a two-layer Graph Convolutional Network (GCN) featuring a hidden dimension of 256. The datasets are partitioned into train-validation-test splits with ratios of 60\\%, 20\\%, and 20\\%, respectively. \n\n2. **Hyper-parameter.** The settings of three scaling factors $\\alpha_{gip}$ (the weights of $\\mathcal{L}_{gip}$), $\\beta$, and $\\gamma$ on each dataset are shown in Table R2 of PDF file, where U, E(10), and E(2) are three task partitioning modes, the first/middle/last three rows are ERGNN, SSM, and CaT.\n\n\n>**Q4: More hyper-parameter experiments about the method are needed. For example, the parameter sensitivity analysis about the $\\alpha$.**\n\nA4:  As shown in Figure R3 of PDF file, we conduct the analysis for $\\alpha$ (has changed to $\\alpha_{replay}$) with ERGNN, SSM, and CaT on three datasets. \n\n$\\alpha_{replay,1}$ is set to $\\alpha_{replay,o} \\times 0.1$ for three methods, the hyper-parameters we used are all $\\alpha_{replay,1} \\times 10$ (i.e., $\\alpha_{replay,o}$, dark blue bars in Figure R3). $\\alpha_{replay}$ is the hyper-parameter designed in the baseline and $\\alpha_{replay,o}$ is the original setting. \n\nAlthough this setting may not be optimal for performance, we still followed the baseline's setting for a **fair comparison**. In ERGNN, the greater the loss weight, the better the performance, possibly because ERGNN focuses too much on these independent replay nodes. In SSM and CaT, except when $\\alpha_{replay,1}$ is used, performance hardly changes with an increase in the loss weight, indicating that $\\alpha_{replay}$ is not sensitive.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We extend our gratitude for the valuable feedback and insightful suggestions provided by all the reviewers. We have diligently addressed the questions and suggestions raised during the official review process and have provided comprehensive response to the reviewers in the corresponding rebuttals. 

We are delighted to be recognized for our efforts in this research. We would like to thank Reviewer Za5b for acknowledging the motivation and effectiveness of our model in avoiding catastrophic forgetting on graph class incremental learning. We also extend our appreciation to Reviewer 9jHn for their strong recognition of our novelty, theoretical contributions, and comprehensive experiments. Furthermore, we appreciate Reviewer BwJ9's acknowledgment of our theoretical foundations, empirical improvements, and quantitative analysis. Additionally, we are grateful to Reviewer T3gk for their strong endorsement of the motivation, theoretical contributions, and effectiveness of our work.

We have conducted analysis and experiments on mid-frequency information preservation, validating our results on new datasets and analyzing the weights of graph information preservation loss. 
We have advanced an understanding of larger distortion and performed a quantitative analysis. We have clarified the differences between the overview of recent papers in the introduction and related work sections and elucidated the motivation and reasons for using MSE loss to measure the discrepancy in global representations. 
We have explained certain terms before their use and modified some notations. The rationale behind the definition of the structural shift score has been expounded. We have elucidated the reasons for modeling from low-/high- frequency information preservation. 
We present the reasons for not employing high-frequency global information from the perspective of method design and experimental outcomes, explain the existing task division, list detailed experimental details and hyperparameters, and conclude with an analysis of the replay loss weights.

**We also provide a rebuttal PDF attached below in this global response section to exhibit additional results.** We have supplemented Table R1 with results from the newly added Cora and Citeseer datasets. Figure R1 illustrates the visualization of node representation on node feature shift and its generalizability. Table R2 shows the hyperparameter settings used in the experiments. In Figure R2 and Figure R3, we analyze the effect of loss weights of graph information preservation loss and replay loss on performance.

Meta-review: This paper proposes a graph spatial information preservation (GSIP) framework to alleviate catastrophic forgetting in graph class incremental learning (GCIL), with a focus on theoretically investigating what matters in preserving the information from the old classes. The authors have done a good job in the theoretical analysis in the topic of graph class incremental learning, which appears to be a nice addition to the community. The proposed method has also been shown to be effective across various benchmark datasets. Overall, the reviews are positive, and I recommend accepting the paper. However, there are some comments on the clarity of the presentation, and I encourage the authors to carefully revise the final version.
Decision: Accept (poster)