Temporal Graph Thumbnail: Robust Representation Learning with Global Evolutionary Skeleton
Authors: Weining Shi
Zhisen Wen
Qinggang Zhang
Chentao Zhang
Zhihong Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Dynamic Graph Neural Network
Global Evolution
Von Neumann Entropy
Robust Representation Learning
Primary area: learning on graphs and other geometries & topologies
TL;DR: We introduce Temporal Graph Thumbnail (TGT), a robust representation method that captures temporal regularities via von Neumann graph entropy and feature mutual information to guide learning on noisy, rapidly evolving temporal graphs.

Abstract: Temporal graphs are commonly employed as conceptual models for capturing time-evolving interactions in real-world systems. Representation learning on such non-Euclidean data typically depends on aggregating information from neighbors, and the presence of temporal dynamics further complicates this process. However, neighbors often contain noisy information in practice, making the unreliable propagation of knowledge and may even lead to the model failure. Although existing methods employ adaptive spatiotemporal neighbor sampling strategies or temporal dependency modeling frameworks to enhance model robustness, their constrained sampling scope limits handling of severe noise and long-term dependencies. This limitation can be attributed to a fundamental cause: neglecting global evolution inherently overlooks the temporal regularities encoded in continuous dynamics. To address this, we propose the **T**emporal **G**raph **T**humbnail (**TGT**), encapsulating a temporal graph’s global evolutionary skeleton as a thumbnail to characterize temporal regularities and enhance model robustness. Specifically, we model the thumbnail by leveraging von Neumann graph entropy and node mutual information to extract essential evolutionary skeleton from the raw temporal graph, and subsequently use it to guide optimization for model learning. In addition to rigorous theoretical derivation, extensive experiments demonstrate that TGT achieves superior capability and robustness compared to baselines, particularly in rapidly evolving and noisy environments. The code is available at https://anonymous.4open.science/r/TGT-BDF2.

Published as a conference paper at ICLR 2026
TEMPORAL GRAPH THUMBNAIL: ROBUST REPRE- SENTATION LEARNING WITH GLOBAL EVOLUTIONARY SKELETON
Weining Shi', Zhisen Wen', Chentao Zhang', Qinggang Zhang", Zhihong Zhang**
School of Informatics, Xiamen University, Xiamen, Fujian, China
2 The Hong Kong Polytechnic University, Hong Kong, China
{weining, wenzhisen, 30920241154582]e @stu. xmu edu, en, zhihongexmu edu. en, ginggangg. zhange connect. poltu.ho
ABSTRACT
Temporal graphs commonly employed as conceptual models for capturing time-evolving interactions in real-world systems. Representation learning on such non-Euclidean data typically depends aggregating information from neighbors, and the presence of temporal dynamics further complicates this process. However, neighbors often contain noisy information practice, making the unreliable prop- agation of knowledge and may even lead to the model failure. Although existing methods employ adaptive spatiotemporal neighbor sampling strategies or tempo- ral dependency modeling frameworks to enhance model robustness, their con- strained sampling scope limits handling of severe noise and long-term dependen- cies. This limitation can be attributed to fundamental cause: neglecting global evolution inherently overlooks the temporal regularities encoded in continuous dynamics. To address this, we propose the Temporal Graph Thumbnail (TGT), encapsulating a temporal graph's 's global evolutionary skeleton as a t thumbnail characterize temporal regularities and enhance model robustness. Specifically, we model the thumbnai. i by leveraging von Neumann graph entropy and node mutual information extract essential evolutionary skeleton from the raw tem- poral graph, and subsequently use it to guide optimization for model learning. In addition rigorous theoretical derivation, extensive experiments demonstrate that TGT achieves superior capability and robustness compared to baselines, par- ticularly in rapidly evolving and noisy environments. The code available https:// thub. com //shhhhimt Schiwning/GG1
INTRODUCTION
Temporal graphs arise naturally in a wide range of real-world domains, including transportation
systems (Zhao et al.) 2019 Yu et al. 2018 Li al. 2018 Guo al. 2019), recommendation 2021 and social networks (Deng et al. 2019 Tang et al.)
systems (Xiang et al. 2010
Fan et al.
2009. Representation learning on non Bunlideld structures already difficult. the temporal variation of node features and graph topology makes it even more complex (Gravina & Bacciu 2024 Most studies on temporal graph representation extend the message aggregation and passing mechanisms (Gilmer et al.] 2017) from static graph learning to dynamic settings. Some studies address temporal dynamics by slicing graphs into discrete snapshots and analyzing them as time series (Seo et al. 2018: Li et al. 2018 Bai et al. 2021), while others model temporal changes in continuous, online fashion (Micheli Tortorella 2022 da Xu al. 2020 Rossi al. 2020).
However, real-world data are often noisy, posing substantial challenges to model robustness and generalization. For illustration, consider Q&A forum where user posts are modeled as nodes
"Corresponding author.

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Published as a conference paper at ICLR 2026
and user interactions edges. There are various sources of noise in this social network tempo- ral graph. including off-topic information in an answer, incorrect associations between answers and unrelated questions, and message disorder caused by network latency. Since context for representa- tion is captured by aggregating spatiotemporal neighbor information along the topology, redundant erroneous node features, spurious or obsolete edges, and incorrectly recorded timestamps can significantly degrade context quality, thus weakening the representation quality (Hou et al. 2020).
Several robust representation approaches have been developed combat noise in temporal graph learning. These studies can be broadly categorized into two main directions: a) neighbor sam- pling adaptation. Adjusting spatiotemporal neighbor sampling improves the reliability of aggre- gated messages by reducing the influence of noisy or unreliable neighbors. For instance, [Li et al.) (2024a) conceptualizes the neighbor update as sequence decision problem and employs reinforce- ment learning to address it effectively. Chen et al. (2018) models the propagation of noise a Markov chain and proposes a greedy algorithm to rewire edges for neighbor sampling. b) historical information assistance. By capturing consistent patterns in historical interactions, temporal mod- eling reduces sensitivity to transient noise or irregular events, thereby improving model robustness. Zhang et al. (2022) mitigates spatiotemporal distribution shifts by making full use of invariant his- torical pattems observed across sequences, while Yuan et al. (2024) derives consensual conditions for temporal information and devise information bottleneck to capture temporal correlations.
However, these approaches remain limited by their constrained sampling scope, restricting their ability to handle denser noises and capture long-term temporal dependencies. Specifically, [Lee al.) (2024) observed experimentally that models relying on neighbor or historical information struggled with long-term dependencies, leading to larger errors. Sannar et al. (2020) shows that when both neighbor and historical information become unreliable, the quality of the learned embeddings can significantly deteriorate. We attribute these limitations two main factors. a) lack of global evo- lution modeling. Existing methods adopt narrow scope of neighbor sampling, overlooking the useful information from global evolution. When local information insufficient to resist the noise interference, model performance degrades significantly. b) absence of effective constraints for de- noising. Excessive denoising can distort critical information, while insufficient processing leaves residual noises. In the absence of constraints, striking a balance between the two becomes difficult.
To this end, we introduce the Temporal Graph Thumbnail (TGT) framework, which characterizes the global evolutionary skeleton of temporal graphs a thumbnail and employs it as optimiza- tion constraint to guide effective compression and denoising of representations, improving both the capability and robustness of the model. Similar marize the video's s content, we extract a static graph, akin to a snapshot, to encapsulate the evolution- ary information of the temporal graph into its thumbnail 1. Specifically, TGT addresses the afore- mentioned challenges by addressing two key research questions: (i) How to model a t thumbnai i 1 from the temporal graph that effectively serves as its skeleton with the global evolution features, and (ii) how to design effective constraints based on the thumbnai i 1 to guide robust learning process.
video cover, which uses a single image to sum-
In our TGT framework, we design a thumbnail modeling approach that derives the conditional like- lihood estimation of the thumbnail 1 under the raw temporal graph. It characterizes structural evo- lution grounded in von Neumann graph entropy and captures node feature evolution using sequential node mutual information through the Donsker- Varadhan representation. Based on the thumbna 1, we further derive effective constraints for representation learning that approximates the mutual in- formation bounds between the data, tasks, and thumbnai 1, effectively balancing robustness and expressiveness. Our contributions can be summarized as follows:
We model the thumbnail for temporal graphs that captures their global evolution features through online computation of von Neumann entropy and alignment with node feature mutual information.
We formulate optimization constraints based the evolutionary information embedded in the thumbnail, guiding the learning process to enhance model capability and robustness.
We conduct comprehensive empirical evaluations across multiple benchmark datasets. The results demonstrate that TGT achieves statistically significant improvements in performance com- pared to baselines. The results highlight the effectiveness of our method characterized by substantial evolution constraint, as well as its robustness in handling systematic noise.
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2 PROBLEM STATEMENT
2.1 THUMBNAIL OF GRAPH SEQUENCE
We aim mine specific graph that captures the evolutionary skeleton of temporal graph, en- capsulating, the global evolution features of the graph sequence. Considering a temporal graph Ji=1 characterized by historical sequence of snapshots (notably, TGT focuses primar- ily on discrete-time dynamic graphs), our method models a static t thmbnaill Gr = (Voz. Eg, }. which is designed to distill and encapsulate the essential evolutionary information of the entire temporal process within g. The corresponding adjacency matrix M is composed of Mas = if (a, 3) E Egr, and Ma set of assignment matrices, denoted as S {S), S .s..... ..... 5"), which map V in G' to Vg,, via the function F. saa in S denotes the mapping from node in G to node in Gr at timestamp i.
otherwise. Based on sufficient temporal correlations, we construct
S = f if =a,
otherwise.
To estimate the evolution of the graph sequence 9, the posterior probability P Gr: S) can be formulated as a probability distribution function in aggregate (Han et al. 2015). Due to the tempo- ral relationship between snapshots, the joint distribution cannot be simply computed by multiplying independent distributions. Instead, we model the joint probability as a weighted conditional distri- bution over each snapshot and thumbnail, parameterized by learnable coefficients B.
Aut MasMo ]
(g Gr, = II [ (G,
II II
Σ
exe 1.
"E> BEV' BEVer
G'Eg
G'EG LaEV VEVey
where
== In
1 1
Pe
K2
Pe 1'll/oor
Bi
Po
(2)
P. is the probability of a matching error between observed graph vertices and the corresponding
vertices in thumbrail. while B the sampled probability computed time step based on the feature z of vertex a (Han et al. (2015) (more details for Eq. 2 appendix (A.)). The aforemen-
tioned conditional likelihood estimation is applicable to both directed and undirected graphs and
provides an effective representation of the temporal graph G, assuming the thumbnail is accurate.
2.2 REPRESENTATION LEARNING WITH THUMBNAIL
The problem we examine involves a temporal graph G. where the graph stream comprises snapshots G each timestamp t, up maximum timestamp T. Each snapshot G {V' E*} consists of vertices V = [or vertices V possess k-dimensional features represented by X E Baxk The adjacency matrix A' === { n is defined such that at = if (i,j) E E and afy = 0 otherwise. We aim to input the temporal graph and utilize the node features X1:T along with the topological structure A 1:T to derive the node embedding -z++ E = for the subsequent time step, thereby facilitating downstream tasks. We aim to develop the robust representation ZT++1 particularly under conditions where node and structural features are sparse, unreliable, or perturbed. To meet these requirements, the representation should adhere to the principle of minimal sufficiency (Tishby Zaslavsky 2015):
} of size n and edges E C V x V of size m at timestamp t.The
IBz arg min (ninini(((()])) 3I: Z)},
where B is the Lagrange multiplier to balance the two mutual information T. During optimization, the mutual information between Z and the target label is maximized to enhance the representa- tion capability of the model, while the mutual information between Z and the input data G is con- strained to improve the model's robustness. By instantiating the intermediate constraint Z through the thumbnail 1 (as shown in Eq. 8), effectively capture the evolution of topological depen- dencies in historical graph sequences, thereby improving representation quality and robustness for downstream tasks such as node classification and link prediction.
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3 TEMPORAL GRAPH THUMBNAIL
In this section, introduce Temporal Graph Thumbnail in detail. Our framework consists of two components: (i) an approach for modeling the thumbnail that effectively captures global evolutionary features, and (fi) the optimization constraints for representation learning guided by the thumbnail 1. Finally, we instantiate training and inference processes to illustrate the full pipeline.
3.1 THUMBNAIL MODELING FROM TEMPORAL GRAPH
Existing methods fail to capture global evolutionary features, primarily due the absence of an plicit model that characterizes the core evolutionary backbone underlying temporal graph sequences (Liu et al., 2023). This oversight weakens the temporal coherence between individual snapshots and the sequence as whole, rendering such methods less robust to real-world perturbations. To this end, network entropy has become widely adopted tool characterize the structural dynamics of com- plex systems (Anand et al. 2014). Among various forms of network entropy, von Neumann graph entropy (VNGE) exhibits distinetive mathematical properties that make well-suited for modeling the thumbnai 1 (The reasons for choosing VNGE are detailed in the appendix A.I.3).
Inspired by VNGE's ability to coherently track temporal topological changes (uaag et al. 2023a) and encode evolving structural features (Liu et al. 2022 Alstott et al. 2015), we characterize the structural evolution by analyzing the mutual information between the thumonnail Gr and the temporal graph G (G',..., ,..., G from the perspective of VNGE. We start by preparing the ap- proximation Hy N. adapted from Ye et al. (2014), to characterize structural evolution.
de d dout
HvN = 1
Σ
Σ (a,
|Vor
2r 1
dg dout
Jout
dout
(a,3)E Bor
where
den
Myas
dout
Mar.
VeVer
(4)
The set Firr, represents the asymmetric directed edges of Gr, where Fer, {u,, u) (u, v)
Fgr and (v,u) u) & ]. Given the intrinsic interdependence in the evolutionary process, snapshots are not independent. Thus, B' is learnable probability weight that quantifies the probabilistic contribution of the snapshot G¹, corresponding to the parameter K' and B as defined in Eq.
(proof appendix A.2 and A.3.I)
IA(Tr:) = HVN(9T) - Σ B'Hlnx(GrlGG"
(5)
G
Then, we use convert the Euclideen graph sequence into vertex embeddings of thumbnail. 1. We employ Donsker- Varadhan representation to derive an estimator for mutual infor- mation TD between and Gr (Belghazi et al. 2018) (proof in appendix (A.3.2).
TDv (G::g) = sup [r[[([V)] log EQ (e feF
(6)
We integrate I,(TT; 9), which characterizes the evolution of the topological structure, with (9T; 9). which reflects the evolution of node features, to maximize the mutual information between the thumbnail and the raw temporal graph. This objective guides the learning of the mapping F and the coupling weights B', as part of definition of the thumbnail in Eq. and
3.2 THUMRNAIL-GUIDER REPRESENTATION CONSTRAINTS
Existing representation methods limited in their ability model the global evolution inherent in temporal graphs, and they neglect the memory patterns and logical dependencies embedded within global temporal changes. With the guidance of the thumbnail, we formulate optimization straints that leverage global evolutionary information to enhance representation learning.
Based on local dependency assumption for static graphs, we extend it to temporal graphs.

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Assumption Given the relevant data from neighbors within certain distance range (k-hop) and time range (At) for vertex v in temporal graph G. v depends only on these spatiotemporal neighbors (U) and is independent of data from the rest of graph:
P( - Pe
Under this premise, we derive the variational bound of the mutual information. Constrained by the thumbnai il 1 Gr, we refine the temporal graph g the principle of minimal sufficiency (Tishby & Zaslavsky 2015::
GTIB arg (ongii(-(((-))) BI(G:
9r
Our objective is to employ the thumbnail Gr bottleneck constraint minimize redundancy in the original data while preserving sufficient evolutionary information to prevent distortion from excessive compression. Minimality is achieved by minimizing the upper bound of mutual informa- tion (G; Gr) to compress the redundancy of G. Sufficiency is achieved by maximizing the lower bound of mutual information ((Y; Gr) to capture essential features of G.
Upper Bound of I(g; Gr) Inspired by GIB (Wu et al. 2020), we extend the variational approx- imation to the mutual information between the thumbnail and the original data, termed the Thumbnail Bottleneck (TB) constraint. Specifically, Sgr Ser C [T.) is the index sequence of the temporal graph snapshot, is the temporal graph time span. To enable tractable optimization, we introduce variational distribution Q(-) to approximate the true posterior P(.).
((: I(g: hESerx U UfZ HesSoA) )
H TB
Σ IESOTA
TB STA
STx
P(Z Gr
|A,
2
P(Z GTX
(21-1)
where TB = E
log
TB ! STX
E
log
STA
Q(Z
Q
!))
10 and Z are Gr embeddings that capture temporal evolution information. This helps resist
perturbation attacks that violate inherent evolutionary patterns (proof in appendix A.3.3)
Lower Bound of I(Y; Gr)
The variational lower bound of [(X; Gr) follows the approximation:
P ( 9rx Z _ Ep(x)
E E P
(Z ( 9rx
(1)
I(Y: 1- E-1--EPy.gn. Brylgg
Q(Y)
Q(Y)
(10)
z0 and 20 are determined in Eq. I In practice, reparameterizing the probability distribution function P Prroves beneficial by exploiting prior data the conditional probability distribution. This approach eliminates learning mapping between Gr and Y, which requires instead the learning of an approximate probability distribution Q(Y) in a typically diver-dimensional vector space (proof in appendix A.3.4).
3.3 INSTANTIATION OF TRAINING AND INFERENCE
We instantiate the previously introduced probability distributions and elaborate on the detailed pipeline of the framework. We illustrate the process using link prediction as an example, with the full pipeline presented as pseudocode the appendix A.1.4
Thumbnail Modeling For the thumbnail modeling, we adopt the mutual information alignment method to design the loss function as follows:
Levolution = (1.(9r:)) LD (Gr;9)).
The two terms in the formula are composed Eq. and Eq. respectively We define in Eq. 5 graph snapshot within the temporal neighborhood [t At, fl. with the neighborhood size configurable as a hyperparameter. Furthermore, FF in Eq. is specified as neural network layers composed of learnable parameters.
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Instantiation of 9r) To instantiate the node feature term TB (1)
Sr defined in Eq. 9 we
Q(2 ) mixture of Gaussians, expressed as Σk=1 TA T. o² q.k.) Here, The Mg. and ogg
(2)) as $ Grx ppp of), and then: are learnable parameters. We set P(Z ( 2
m
( TB
Σ VEVA
( Grx Mp: Upp o
D(ugkk ogxk)
(12)
STX
For structural term TB (1) tion, Bernoulli(e), where Φ hyperparameter. Neighbor information is then aggregated by sam- pling from this Bernoulli distribution. The generative distribution IP P(Z)) GT A, GTA Bernoulli, with sampling probability Op computed from the node embeddings.
STa as defined in Eq. 9 we similarly set Q( (0) as Bernoulli distribu-
-(-1)) ) is similarly
1 1 ag
a + ap log
TB ( A
DDK (Bernoulli() || (ernoulli(o)) = (1 a log
(13)
1 -
Instantiation of T(Y; Instantiating Eq. 10 depends on the downstream task. For instance, in
link prediction the likelihood term P(Y Ø z is modeled as categorical distribution, where Q(Y) denotes the empirical distribution of ground truth links. During optimization, the final expected joint likelihood term converges toward unity, effectively canceling the constant offset.
N
(Y;
log Categorical Y
2 (xx =: -CCE(Wout Zor Y),
(14)
N
Wont denotes the weights of the downstream classifier. During optimization, this component refers to the calculation of cross-entropy loss, where N represents the number of training samples.
In summary, the mutual information loss function for bottleneck constraints imposed by the t chumbnail i 1 is as follows, where SA, Sx are the index sets satisfying Assumption [
LB = - 10-- + -( + B
TB
( STA + Σ
(1) Grx
(15)
LISA
tESx
The overall training objectives of the proposed model can be reformulated as follows:
L = LB + X Levatation:
(16)
where À is the hyperparameter of the Lagrange multiplier. We adopt the Graph Attention Network Veličkovié et al. 2018) as the backbone framework, meaning both F in Eq. 6 and P. in Eq. 2 are GaT-based encoders. The probability B in Eq. 2 derived by transforming node a's encoding at time through a feedforward layer with a softmax. Therefore, according to Eq. - we can predict global evolution using the thumbrail Gr, which is compressed by graph sequence G.
EXPERIMENTS
We employ link prediction as a downstream task to assess the capability and robustness of TGT com- pared established baselines. Specifically, aim address the following research questions: Q1 (Fundamental Capability): How does TGT compare with SOTA methods in terms of fundamental capability? Ans 1. Q2 (Robustness): How robust is TGT under various types of perturbations or
noise against other baselines? Ans 2. 2.2
2.3.
Q3 (Effectiveness of the Thumbnail): How ef-
fective is the thumbnail in modeling evolution' ? Ans 3.1. Why is von Neumann Graph Entropy
(VNGE) chosen for modeling thumbnail 1 Ans 3.2. Q4 (Contribution of Constraints): How do the constraints guided by the thumbnail contribute enhancing the model's robustness? Ans
4.1 EXPERIMENTAL SETUPS

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Table 2: Inductive link predication performance the Bitcoin, MathOverflow and MOOC datasets under clean settings.
Bitcoin
MathOverllow
MOOC
Model
AUC
AP
AUC
AP
AUC
AP
EvolveGCN
67.59+0.3
63.38+002
75.59++022
72.73=0.4
72.35+0.3
73.59++0.2
Dynamic
JODIE
74.47+0.3
75.50+0.4
67.06+1.2
66.32+0.6
73.19+0.7
71.78+0.9
GNNs
DyREP
70,43.1.5
69.79-1.44
63,50.10.5
63.37=0.6
81.36.0.11
78.35.10.3
(DGNNs)
TGN
69.361111
72.0910.7
64,5010..6
65.88.0.6
79.36:11.0
78.96.10.5
Robust
DIDA
73.57=0.3
71.27-0.4
74.37=0.4
75.24=0.3
89.84:0.55
88.49:004
Gener-
GIB+LSTM
7..79=0.5
69.73+004
77.52+0.3
75.03=0.7
92.34+0.3
93,29:+0.55
alized
DGIB
12.9911.3
73.24:0.6
80.2940.33
79.99=0.5
93.06:1011
93.1110.3
DGNNs
TGT(ours)
91.41=0.2
91.01=00.3
82.38+0066
81.17=0.4
95,42+0.44
94.56+00.7
Datasets We conduct experiments on the link predic- tion task on three widely-used datasets. MathOverflow forum question-answer relationship data (Paranjape et al. 2017); Bitcoin dataset is user transaction relation- ship data consisting of transaction records (Kumar et al.
Table 1: Details of datasets for experi- ments.
ManlUvennlow
MOUL
25,318
Edges
59, days Homogeneous
Tineyas
days
2016 2018); the MOOC dataset represents the actions taken by users on a popular MOOC platform (Kumar)
Type
Eislutoon
Premency
18.7
45
13724 97
et al. 2019). More detailed information is shown in Ta-
Arg. Stap. modes
7.054
21,683
7,047
Arg. Snap. edges
51,363
207.581
$1,749
ble 1 Evolution Frequency refers to how often the graph evolves per day. Avg. Snap.nodes/edges represents the
Seop. ypan
60 days
20 days
12 hours
average number of nodes/edges in each snapshot, where the snap time span refers to the temporal scale at which the graph is chronologically split. Datasets of varying scales and evolution frequen- cies covering most real-world tasks. Detailed descriptions of datasets are provided in appendix A.4
Baselines We selected baselines from dual perspectives to demonstrate the effectiveness of TGT. Among temporal graph representation methods, we selected four superior methods. TGN (Rossi ett al. 2020) is framework for representation learning on streaming temporal graphs: JODIE (Ku- mar et al. 2019h learns node interaction relationships using RNNs; DyREP (Trivedi al. 2019 codes nodes by modeling temporal point processes at dual scales; EvolveGCN (Pareja et al. 2020) captures graph sequence dynamics by evolving GCN parameters. Among robustness and bottleneck constraint, we selected three methods. DIDA (Zhang et al.) 2022) leverages robust and generalized prediction patterns on temporal graph representation; GIB (Wu et al., 2020) computes the informa-
tion bottleneck of graph snapshots, with LSTM adapted for dynamic scenes; DGIB Yuaa et al.,
2024 imposes history constraints on the information bottleneck to obtain representations.
Data Perturbation To verify TGT's robustness, , we adopted adversarial attacks on training data in multiple aspects. a) Feature Interference: To assess the robustness of TGT under untargeted attacks, we introduce random Gaussian noise to node features and apply varying perturbation in- tensities (controlled by noise amplitude). b) Structural Interference: For topological attacks, we employ the Nettack (Zügner et al.) 2018) to perform adversarial edge perturbations (e.g. deletion or negative sampling), specifically targeting substructures that maximize prediction loss in the proxy model. с) Temporal Interference: To evaluate TGT's stability and robust representation capabili- ties under temporal disruptions, we disrupt the chronological order of graph snapshots by randomly permuting them, simulating interference with the underlying temporal evolution dynamics (For more detailed information of attack settings, please refer to the appendix (
4.2 EFFECTIVENESS EXPERIMENT
To address Q1 we conducted comprehensive experimental analyses on three distinct datasets, and subsequently compared the results with established baselines to rigorously demonstrate the superior performance of TGT. Implementation details of TGT are provided in appendix [A.5] In the effec- tiveness experiment, we employed clean data setting without any data interference and utilized inductive link prediction as the downstream task for the evaluation of representations. Drawing upon the original data, we conducted negative sampling by selecting 10% of the total edges to facil- itate the training of the link prediction task. After repeating the experiments three times, the average and range of the results are presented in the Table 2
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Table 3: Robustness results (AUC) Bitcoin, MathOverflow and MOOC datasets with data pertur- bation at different levels.
Dutuset
Model
Clean
Feature Interference
Structure Interference
Temporal Interference
20%
107
EvalveG2CN
62.76110.3
36.7711022
54.2410.2
50.37.004
03.89_.0.3
05.4721..3
54.3710.4
J0DIE
74.47-0.3
00.28-0011
61.10+0.3
53.32+044
70.33-0.2
66.82+03
60.57-0.3
11.54
09:79-0.5
DyREP
70.43.0.5
64.911033
61.2510.33
56.63103
69.98.0.8
63.79.10.7
08.60.0.5
69 69.4310.88
65.84.0.5
57.3310.7
Bitcoin
DIDA
TGN
78.57+0.3
69:36:1.1
71.05+02
67.382:06
02.3120.4
68.45-0.3
64-20+02
59.06:133
70.93-0.2
06.6110.5
68 68.71+0.3
18:36
58.73=05
65.69-04
08.46:933
71.33+022
06.92:077
62.55-033
61.74
64.20+044
GIB-LSTM
70.79+0.3
60.26-033
63.73+0.55
54.37+033
68
65 05.09
62.95-02
09.15+0.3
67.17-06
6341+008
DGIB
1..9911.3
69.92.006
63.63..0.5
60.78.0077
70.13.0.3
65. 65.84100.6
59.13.004
71.27.0.55
68.44.077
62.53.0.55
TGT
91.42:0.2
89.43:0.5
80.23:20.4
90.02:04
89.83:0.6
85.00::0.5
80.95=0.7
90.7810.5
S8.14:0.6
05.64:0.3
EvalveGCCN
75.
67. 67.22=00.3 05.
61. 1.
54. 1
06.68:005
63. 11:44
05.23=0.3
69. 39:333
67.90:003
30DIE
67.05:122
61.23-00.3 3
59. 0.3
51.92
64.71:004
59 13::0.3
53.19-0.3
66 34.3--
63.95-022
0..37-0.3
DyREPP
63. 03.50.00.5
59.13110.3
3
53.3210.6
54.1740.3
60.19.0.2
56.3910.2
03.26_00.3
61.711044
60.07.0022
53.3310.55
MathOverllon
DIDA
TGN
74.37.044
64.50+0.5
70.95.10.1
61.22-003
58 1.3
63.96101
44.144
59.49-0.2
47.17.033
23-00-
53,97-033
73.5710.22
40+-0.3
36-00-
23-23
GIB-LSIM
77.52.0.3
73.22
6307
05.83-0.3
37
63.21=08
75
71,14:0555
03
IXTER
20-20+0.3
73663-0-5
68
67
7447+00
70.43-0.3
79 66-66-1
77,75+003
04-34-0
TGT
82.3810.6
79.22..0.3
0.33
71.42102
80.01 0.3
77.741..33
75.93_.0.7
81.57.10.2
$0.71.00.5
76.37
EvalveGCN
72.35.10.3
56.23:104
62.3710.2
54.72.02
59.29.10.1 68.8310.1
66.98.10.3 02.31_0.2
62.57.1022
55 93
JIDIE
75.19:#0.7
63.15=002
55 36+012
53.71=04
09.15=044
61 21:42201.2
57.34-044
63.25-40.3
58.59=004
DyREP
81.36:1...1
74.60:102
66. 66.77..0.2
60.52.1.33
7..43:0.2
68 31.003
59.74-0.3
78.5610066
65 65.39.10.5
50.37.10.5
MOOC
DIDA
TGN
89 29-54+00.5
79. 19.36.1:1.0
79.73+0.1 1
73.632:04
7362-0.3
08.4120.3
00003400
64.01+0.1
84.8--022
75.3120.2
69 69:27::022
71 7147+003
01.33-0.3
61.03-044
86.5++0.2
78.2910.77
72.6820088
82,79+0.3
03
73:66+0.2
GIB-LSTM
33.3 92.3410.3
65.25.0.3
63.67
$6.05 10.
62.61104
74.37_0.3
89.8510.7
85.7610.4
69 68..7
LXGB
93.00:00.1
54.35=001
75.24
63.3220.3
07.
24
79.6--033
70.32-00.6
TGT
95.42-0.4
58.73-0.2
80.79-0.1
71.68-0.3
90,43-0.3
86.16-0.5
81882-0.
02.61+0.2
89.53-0.7
04.01-066
Ans 1. TGT consistently outperforms the baselines in fundamental representation capability across all three datasets, with its advantages particularly pronounced in scenarios with sparse neighbor in- formation. By analyzing the datasets and the experimental results, it is evident that TGT exhibits significant advantages in the Bitcoin dataset. Specifically, its AUC and AP values have improved by 16. 94% and 15. 51%, respectively, compared to the runner-up. This improvement suggests that TGT can capture data features more effectively and demonstrate superior generalization capabilities. On MathOverflow, where social network interaction noise is prevalent, Robust Generalized DGNNs significantly outperform DGNNs. Furthermore, on the MOOC dataset, characterized by frequent evolutionary changes, TGT displays a clear performance edge. These results demonstrate the ef- fectiveness of TGT's design, which leverages von Neumann entropy to constrain the information bottleneck associated with global structural evolution.
4.3 ROBUSTNESS EXPERIMENT
To address Q2 evaluate the robustness of the model through data perturbation. Following the experimental settings outlined in the preceding subsection, we intentionally injected anomalics into the training dataset and conducted series of experiments evaluate the model's robustness along three distinct dimensions: node features, topological structures, and temporal correlations. After repeating three times, the average and range of the results are shown in Table 3
Feature Interference Ans 2.1. TGT can effectively resist feature interference with different noise levels. Gaussian noise perturbation of node features significantly diminishes the performance of ordinary DGNNs. Comparatively, robust generalized DGNNs exhibited superior performance. Given limited node features in Bitcoin and MathOverflow datasets, DIDA, GIB+LSTM, and DGIB maintained efficacy, yet their robustness lagged significantly behind TGT on the MOOC dataset. TGT employs thumbnail to establish feature's 's constraint (Eq. thereby enhancing the model's robustness during optimization.
[2) and thoroughly filters noise,
Structure Interference Ans 2.2. TGT demonstrates strong robustness against structural-Ievel noise, maintaining stable performance under structural perturbations. Structural perturbations im- pact all baselines evenly, and structural noise is equally applied to neighboring samples across methods. Under interference, continuous DGNNs (c.g., TGN, DyREP) excel at online processing of abnormal edges, demonstrating enhanced robustness. TGT employs thumbnail to establish structure constraint (Eq. [3], effectively mitigating the interference of abnormal edges in crucial topological relationships between modeling nodes. As defined in Eq. 4 the von Neumann entropy captures richer topological information by inherently encoding structural dependencies, which pro- vides TGT with distinct advantage over existing SOTA robustness methods.
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200
400
EvcheSCN
TODEE
DyREP
300
65
TGN
GB.LLTMM
TGTJoum)
70-
60
62-
56
MathOwedlow:
Figure 1: The accuracy curves of the algorithms after being subjected to different degrees of tempo- ral interference on three datasets. By constraining the evolution process using von Neumann entropy, TGT is more resistant to temporal interference.
Temporal Interference Ans 2.3. TGT exhibits strong robustness against temporal interference, with consistently superior performance compared to baselines. Temporal perturbations significantly impact evolving patterns in temporal graphs. The robustness evaluation results across different datasets are presented in Fig. On the slowly evolving Bitcoin dataset, temporal interference minimally affects each method, with accuracy decline primarily attributed to feature and topology redundancy stemming from nodes and edges. the MathOverflow dataset with extensive evolution, redundant historical information impedes model accuracy improvement. In the rapidly evolving MOOC dataset, temporal noise presents a more formidable challenge to the robustness of the model. EvolveGCN, for evolution modeling, is highly sensitive to temporal interference. DIDA, designed for data distribution shifts, excels in anti-interference. TGT demonstrates competitive performance across three datasets. TGT utilizes Neumann entropy to model graph evolution trajectory(Eq, 5) enables effective representation generation despite severe temporal correlation disturbances.
4.4 ABLATION STUDY
we choose to use normalized node degree instead von Neumann entropy in Eq. the baseline of ablation W/O VNGE. We also added ablation control without thumbna W/O T, where the term (gr: 9) in Fq. 11 set to zero entirely. Ans 3.1. The thumbna i is es- sential for guiding the denoising process, removing this component leads substantial degradation in the model's fundamental accuracy and robustness. As shown in Fig, 2 the dataset shows obvious differentiation of ablation effects, which is helpful to analyze the effectiveness of thumbnail 1.
To address Q3
21.41
70-
Figure 2: Robust performance of TGT on temporal interference various datasets under different temporal evolution constraint ablations
With thumbnaill 1i's guidance, the model outperforms the setting that ignores evolution, whether using basic node degree information or the proposed VNGE. Ans 3.2. The VNGE-based evolution modeling significantly outperforms the one using other methods, demonstrating the effectiveness of our core innovation. It is confirmed by the detailed ablation in the appendix A.7 To address we conduct extensive ablation experiments to verify the roles of derived constraints (Eqs. 12 and 13 Ans 4. Each Thumbnail-brit based constraint contributes to the fundamental capabilities and robustness of the model, confirmed in appendix AA.] Furthermore, we conduct hyperparameter sensitivity experiments the Lagrangian coefficients. The results shown in the appendix A.9
Q4

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5 CONCLUSION
In this paper, we introduce a novel framework called Temporal Graph Thumbnail (TGT). By mod- eling the thumbnail from raw graph sequence based on von Neumann graph entropy and the mutual information of node features, we characterize the evolution skeleton of the temporal graphs to capture the global evolutionary information. Moreover, thumbnail serves as an interme- diary, establishing bottleneck constraint between the original data and the target task to enhance the model's s representation over critical information. Experiments validate TGT's superior robust- ness and generalization compared to other methods. Limitations and future work: Although TGT demonstrates stability on datasets with up to one million nodes, thumbnail modeling and real-time von Neumann entropy computation incur additional computational overhead, restricting its scalabil- ity to extremely large graphs. To address this, our future work will investigate pre-training strategies for thumbnail modeling to amortize processing time and hardware costs, while exploring the use of thumbnails as prompts in pre-trained temporal graph neural networks to improve model scalability.
ETHICS STATEMENT
We have reviewed the ICLR Code of Ethics and have ensured full compliance with its guidelines, particularly in terms of data privacy, transparency, and responsible computing practices. The datasets used in the experiments include those in A.4 and A.10 which are widely used.
REPRODUCIBILITY STATEMENT
To facilitate replication and extension of our work, we provide comprehensive resources: the code is given in Appendix A dataset and preprocessing details in Appendix AA.4 model hyperparameter configurations in Appendix AA.5 and complete derivations and theoretical proofs in Appendix AA.3 With the complete set of code, data, preprocessing details, hyperparameter settings, and theoretical derivations provided, the reproducibility our work is fully ensured.
ACKNOWLEDGMENT
The authors wish to thank the anonymous reviewers for their helpful comments. This work was partially funded by Xiamen Major Science and Technology Project (No. 3502220241028).
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A APPENDIX
A.1 FREQUENTLY ASKED QUESTIONS (FAQS)
A.1.1 CODE AVAILABILITY
To promote transparency and reproducibility, we have included the source code of TGT, along with training and evaluation scripts, an anonymous repository [ this repository in the Supplementary Material. The archive also contains the running logs and the hyperparamters for TGT (detailed TGT implementation
We have also included copy of
appendix (A.)). The noise generation methods we adopt are based on DeepRobust (Li et al. 2020 2] with detailed attacks implementation provided in appendix A.6 We warmly welcome other researchers to reproduce and extend our work.
A.1.2 WHAT ARE ADVANTAGES OF TGT?
TGT introduces three key advantages over existing methods for temporal graph representation, specifically addressing their fundamental limitations through a tailored and principled design.
Superior representation capability with thumbnail. Table demonstrttss that TGT consistently achieves superior representation capability across various data scenarios. Compared with other base- lines, TGT benefits from the additional global evolution information provided by the t thumbnail i,, leading to significantly improved representations. Notably, the advantage becomes even more pro- nounced in scenarios characterized by high randomness and frequent evolution.
DoepRobust:
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Enhanced robustness across various noise conditions. Table provides strong evidence that TGT exhibits robust performance against various types of noise. With the guidance the thumbnail constraints, TGT consistently outperforms baselines under different noise settings. In particular, TGT demonstrates significantly more robust performance under temporal perturbations.
Stapshot 6
6
To
4
To
Ts
Ta
Updstc
findaace
Thumbnail Sr
Figure 3: Visualization of the thumbnail modeling.
Solid and explainable theoretical foundations. We model the thumbnail by computing the von Neumann entropy and mutual information of sequential node features, followed by a princi- pled variational derivation of the representation optimization constraints. Logically complete and rigorous derivations are provided in appendix to validate the thcoretical soundness of the TGT. The relevant lemmas and assumptions well-established and broadly accepted in prior rescarch.
To further illustrate thumbnail clearly, we visualized the modeling process as shown the Fig. 3 The visualization clearly shows that the thumbnail preserves the persistent and fighfrequency structural patterns across time, while filtering out transient or noisy structures. Specially:
Node connectivity patterns that across snapshots retained with higher edge weights in the thumbnail, indicating that Gr successfully summarizes the stable evolutionary backbone.
Ephemeral low-consistency edges, which appear sporadically in individual snapshots, largely suppressed, demonstrating the denoising effect induced by the VNGE-basee constraint.
As shown in Fig. the thumbnail graph becomes progressively more coherent matching the theoretical goal of modeling the global evolution trajectory rather than any single temporal slice. This visualization also confirms, in human-interpretabl way, that the thumbnail effectively captures temporal regularities and structural evolution trends in the datasets, aligning with the conceptual description Method and Algorithm !
iterations,
A.1.3 WHY VON NEUMANN ENTROPY? HOW ABOUT OTHER GRAPH ENTROPIES TO MODEL THUMBNAIL?
The Neumann entropy (VNGE) plays crucial role in understanding the structural and topologi- cal complexity of temporal graphs, as it effectively captures the information content in a manner that aligns with the continuous evolution of time-varying networks (Passerini & Severini 2009). While other entropy measures, such as Gibbs entropy and Shannon entropy, can also capture structural evolution some extent (Bianconi] 2009). they are suitable for continuous computation. These methods typically rely on discrete snapshots, resulting in systemic information loss. More critically, these graph entropy measures do not satisfy the subadditivity property (De Domenico & Biamonte 2016), which poses challenge when designing local mutual information coupling (as seen in the second term Eq. 5} because direct weighted accumulation for aggregation is not feasible. This necessitates additional assumptions more complex modeling approaches. In contrast, VNGE can be computed online and, except in special cases (such as chain graphs or trivial graphs), satisfies the subaddirivity property (De Domenico & Biamonte 2016), significantly simplifying the process of capturing structural evolution. Given these considerations, we select VNGE over other graph entropies more concise approach for capturing reliable structural evolution.
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The robustness of TGT temporal order disruption stems from the VNGE-based constraint I2(r:: 9) (Eq. 5). which fundamentally captures the global structural distribution of snapshots rather than their temporal order.
Specifically, the VNGE defined over the normalized Laplacian ppectrum((pppendee A.3.II which depends solety the eigenvalue distribution of the graph. Thus, in constructing the temporal thumb- nail Gr, maximizing the mutual information between GT and the input sequence ç promotes align- ment of their spectral characteristics rather than merely aligning the temporal order of snapshots.
Since spectral quantities are permutation-invariinn with respect to node or snapshot order, the learned thumbnail inherently focuses statistical patterns of structural evolution that remain stable under temporal sequence shufiling (Braunstein et al. 2006). This means the model learns to retain infor- mation about how structural complexity evolves (c.g.. entropy growth, connectivity changes) rather than the specific temporal sequence of local variations. Therefore, the VNGE-based mutual in- formation regularization provides lemporal-order-agee agdde--ggg---ccccc bottleneck, enabling TGT to maintain robustness when snapshot order is permuted partially corrupted.
Furthermore, as supported by the results Table 5 TGT maintains consistent performance under temporal perturbation attacks, demonstrating that the VNGE constraint guides the model toward learning temporal invariant summary that mitigates the impact of sequence tampering.
A.1.4 COMPUTATIONAL COMPLEXITY ANALYSIS
Algorithm 1: Overall pipeline of TGT for link predication
Input: temporal graph 9 (G*) node features X X1:T+1, number of neighbors be sampled k; element-wise nonlinear rectifier T; Hyperparameters X, 3, A/;
Initialize: initialize all weights; set relative time encoding for each timestamp i: Z (0) for each timestamp t; Oufputtrepresennttiin Z Predicted label Y++1 of next timestamp link occurrences.
X
Sample (t) k-top closest spatiotemporal neighborhoods.
2: for layers / = ... and E yl:741 do for timestamp t in vange[[++]dd do
3:
4:
I,G9:g)) Eq.
5:
(1-(1-1) t TZ Grx
Grx
W())
(.0))
+ ( (1-1)
(1-1)
1!!
(L---t)),(1-- W attention ac()
6:
Po.k
7:
(1(11)
A
Usevi fu E NE.AA(()) un u Bernoulli( 6.00
)):
E ((,,))AA..)) { (1-11))
8:
Z
x JEEV
Inv(CTi 9) WF (1) logexp(Ws W 2 20-000,,0-1)
9:
# Eq.
(()
10:
cvolution (9(::) IDV (Gr:9)
# Eq.
11:
end for
12:
GT + (
A W
13:
YT+1 Link lannk-predctor(cc (GT)
# Eq. (1) evatutionn
LIB f
Levolution
-IT
15:
C+LID + Levoluution
16: Update parameters by minimizing C and backkppoopagen
17:
end for
This section conducts computational complexity analysis and gives future work directions for improvement and optimization. The complete pipeline is summarized in Algorithm I with input feature dimension df and hidden dimension d'. The time complexity can be decomposed as following:
line 4: Computing the VNGE per Eq. costs O(EE,r |). During At, obtaining Ia requires:
O(At Egg l).
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line 5-8: Sampling & spatiotemporal neighbors and applying GAT feature-projection layers incurs O k- (1Vo> Id' + [)))
line 9: The final projection step costs O ( I d + IV ( dd"), where |V the num- ber of nodes in the previous At time of this temporal graph. Since the thumbnail's node set is no smaller than that of any single time window, we conservatively approximate the complexity as O VVr d'd". Putting these together for an L-layer architecture yields the time complexity of
o (L. + k- (1Var d + |Egr d)) + \V d
While TGT incurs extra cost for online VNGE computation, its total complexity is comparable to leading temporal GNN baselines. Furthermore, because the thumbnail contains no more nodes than the entire temporal graph and is typically much smaller, the effective constant in TGT's representation-leariig complexity is reduced.
Comparison of training time per epoch (s) with state-of-the-art baselines in robust temporal graph learning across multiple datasets.
Table 4:
Method
Bitcoin
MathOverflow
MOOC
tgbn-reddit
tgbn-genre
DIDA
23.46
37.77
8.91
27.65
17.80
DGIB
1.37
5.53
2.56
1.57
1.71
TGN
0.78
0.86
0.74
0.68
0.71
TGT
2.26
4.42
3.34
2.73
1.68
As shown Table TGT can efficiently handle datasets with node and edge counts on the order of hundreds of thousands. We conducted additional experiments measuring the per epoch training time of TGT compared with several baselines across five datasets (Bitcoin, MathOverflow, MOOC, igbn-genre, and tgbn-reddit (Huang al.) 2023b)). From the Table get:
Time overhead TGT is only moderately slower than lightweight temporal GNNs (e.g., TGN). On all datasets, TGT requires around 2-4 seconds per epoch, representing only a small overhead compared TGN (0.7-0.9s) despite the additional global evolutionary modeling, This overhead mainly comes from computing the approximated trace term Tr(i.?), which is efficiently computed using sparse matrix and the low-rank Laplacian estimator described in Appendix A.3.1
Although TGT incurs moderate additional cost due to the computation VNGE (which depends on node degrees in Eq. 4. the runtime remains close to lightweight TGNNs. MathOverflow appears slower primarily because of its larger per-snapshot node count, while igbn-reddit, despite larger average snapshots, benefits from lower-dimensional preprocessed degree matrix.
Nevertheless, TGT significantly more efficient than robustness-orientedd baselines (e.g., DIDA) and still remains competitive with IB methods (e.g., DGIB). This indicates that our thumbnail- guided constraints provide robustness without incurring excessive computational costs.
Memory footprint TGT remains comparable to baselines. Since TGT only stores:
static thumbnail graph Gr (small number of nodes)
low-rank Laplacian terms for VNGE approximation, the memory overhead remains negligible.
node features for each snapshot.
These components together introduce negligible GPU memory overhead. In addition, since snapshot construction is entirely flexible with respect to temporal granularity, practitioners may adjust the snapshot time scale according to available GPU memory, ensuring efficient resource utilization on different hardware configurations.
In all our experiments, including the large-scale TGB datasets, TGT fits comfortably within 24GB GPU memory (per RTX 3090 GPU), demonstrating that the method is scalable in practice.
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Although VNGE introduces additional computation compared with purely local aggregation mod- els, the overhead is modest in practice, and TGT achieves a favorable balance between robustness, global-evolution modeling, and computational efficiency. The additional results confirm that TGT is computationally practical even on large-scale datasets. For example, MathOverflow contains over 500,000 edges, and MOOC exceeds 400,000 (shown in Table []. Since the theoretical upper bound of TGT's computational complexity scales linearly with the number of edges across all snapshots, scaling larger datasets (e.g., datasets with millions of nodes) can be prohibitively time-consuming.
In our future work, we plan to explore pretraining-bee baseid approaches for thumbnail modeling, allow- ing the computational overhead to be offloaded to the data preprocessing stage and thereby improv- ing the scalability of the framework.
A.2 DETAILS FOR THUMBNAIL DEFINITION
The goal of the thumbnail 1 is to summarize holistic evolutionary process features from discrete snapshots in sequence. To this end, we define the structure of the thumbnail and the node correspondence between each snapshot in the sequence and the thumbnail, 1, thereby obtaining the posterior probability of the mex--times temporal graph state. Our derivation of the thumbnail il builds on widely recognized works (Luo & Hancock [2011 Han et al. 2015).
Specifically, consider a temporal graph & represented
sequence of length T: 9
(G',..., G"), where the i-th snapshot is G = EV) with node set V and edge set E We aim determine t thumbnail 1 GT (Va,, E. where Vør is the node set, Es, is the edge set, and the adjacency matrix M satisfies:
Mag = {1
if (a, B) E Egr, otherwise.
(17)
We first consider a single snapshot G For the adjacency matrix A of this snapshot
A = [1 (a. E F., 0 otherwise.
(18)
where a, 6 V and 3 E Vgr. For time step i, we define assignment matrix S with elements saa which denotes the correspondence between node a in snapshot G and node a in thumbnail:
{1 0 otherwise.
F(r)) =
(19)
aa
where FF denotes the mapping from snapshot Go to the thumbnail. In this paper, we instantiate F as GAT network layer with trainable parameters, which is trained using a mutual information alignment loss (see in Eq. II.. In practice, the assignment matrix we seek should maximize the conditional likelihood of the observed snapshots given the available thumbnail:
S' = arg max P (G* 19r,
(20)
S:
This is primarily based on the core idea that "the correspondence matches assigned to the nodes of the data graph are hidden variables which have arisen through noisy observation process" (Luo) Hancock, (000). Following standard methods for constructing likelihood functions of mixture distributions, we decompose the likelihood of the snapshot and sum over the corresponding nodes in the thumbnail:
P PC St. II Σ p (za Yaa, S*).
(21)
aEV*
Ta denotes the embedding of node in the snapshot G¹, while ya denotes the embedding of node in the thumbnail GT. P (Ia S*) denotes the probability of these two under the assignment matrix S This assumes that snapshot nodes are conditionally independent given the nodes of Gr. We define the model by specifying P (Ia Ya. St) through conditional probability:
P (I Ne. S')
P (5a Va, St)=
(22)
P P(y., St)
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After applying the definitions and properties of conditional probability and performing alge- braic rearrangements (Luo & Hancock, 2001)
Plaulthriin )P(v shi (((ba)
ILEV HEEE,
P (I (a)
P(zo Man)
P ((ra yo: Va: 5t)
P(auleso) P(wa)
πeev: Ilavvrr
P(ya)
(23)
v |
II II P P Warsbbl
P (ra ya)
BEV BEVET
Given that the snapshot node za conditionally independent of the thumbnail node Yo unless a correspondence sua exists, we get P(r. ya.) = P(r.). we can simplify the above equation to:
|'lx/Ver (24)
P (Ie yo. St)= B² II II BEV* BEVeT
P (zza (a. Sbo), where Ba
1 P (xa)
Modeling a stable thumbnail requires more stable node information, and nodes that appear briefly may contribute less to the thumbnail modeling. To address this, our TGT introduces weighting coefficient inversely proportional to the node frequency across snapshots within the time window, enabling tailored handling transient or non-persistent nodes throughout the sequence.
Specifically, we implement the design described in Eq. 24 Taking newly added nodes as an exam- ple, if Va new node, the probability P(x₂) the denominator of B small (as newly added nodes have low frequency of occurrence in the input snapshots). Its impact on P(z Me. $b2) (the thumbnail mapping function) amplified due the multiplicative relationship in Eq. 24 thereby enhancing the feature capture of transient or non-persistent nodes by the thumbnail.
To test for consissssst,,,y we make use of an indicator that verifies if snapshot nodes a, V correspond to valid edges in Gr. Formally:
[1 if (a. E' and (a, B) E Egr and F(b) = otherwise.
A Massh
(25)
When the edge (a. E E can be matched thumbnail edge (a, 8) Egg via the node correspondence F(a) and F(b) B, the indicator function takes the value 1; otherwise, it is O. To model the probabilistic relationship between snapshots and the thumbnail using Bernoulli distribution with error probability Pe. we define:
P.) Manhh pC 21-Aao
P (xa Ma. sig))
(26)
Substituting Eq. 26 Eq. 24 and then substituting the resulting expression into Eq. the likelihood function for single snapshot:
we derive
P(G' St, II aEVi aEVOr
Σ
K exp
1 Σ Σ
1 Massho
BEVI BEVOr
(27)
where μ In
1 Pe
Ki = p!
Bi
P
For the entire graph sequence g = (G (GI.....Gr GT)
we aggregate the likelihood functions of indi- vidual snapshots following the approach of Han et al. 20015. where = ".....,,777 denotes the
sequence of assignment matrices and then get Eq. 2
Ala NasMasSip ]
P [g) Sr = II [ (G Gr, ST.
II
Σ K exp
# I Σ BEVi BEVET
Oleg
ateg
Leei
V
1
P
v'I-VVrr Bi.
where
==
Ki = P.
P
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A.3 PROOF FOR EQUATIONS
A.3.1 APPROXIMATION OF HVN (EQ.
We initiate by providing improved version of the approximation method for the von Neumann entropy of directed graphs from Ye et al. (2014). Initially, we define the von Neumann entropy, which can be derived from the normalized Laplacian spectrum, outlined here:
A M
X V
HvN(G) = -Tr(Plog P) = Σ
log
(28)
where . llly denote the eigenvalues of the combinatorial Laplacian matrix, P represents the transition matrix with elements Pue. 1 if (u, e) is in the edge set E of graph G, and P.v. dus
otherwise. By scaling the normalized Laplacian matrix by the reciprocal of its trace, one obtains {1 5s density matrix given by - The eigenvalues of this density matrix are [VV [....... )m1) Con-
sequently, the von Neumann entropy of the density matrix associated with the normalized Laplacian matrix of the graph is defined as follows:
HvN(G) M
N [
log
(29)
The von Neumann entropy mentioned above depends on the computation of the spectrum of the normalized Laplacian, and hence, its computational complexity scales cubically with the number of nodes. The Taylor expansion for the expression In
A is given by:
[
2
3
4
2
In
1
1
(30)
If retain only the first term of the expansion and neglect the subsequent terms which contribute insignificantly, In [
3
can be approximated by (M - 1). Subsequently, the von Neumann en-
tropy Syx can be approximated by the quadratic entropy E, N (1 M yielding:
2 V
HVN(G)= = Σ
Σ N VV Σ X3.
(31)
V
Using the fact that Tr[i E, 24 the quadratic entropy can be rewritten as:
T
Tr
i2
HvN(G)-
(32)
[V
V
The normalized Laplacian matrix L has unit diagonal elements. For the trace of the normalized Laplacian matrix we have:
T+( [V]. Similarly, for the trace of the square of the normalized Laplacian, we have:
(33)
Tr 12 = Tr
12 ( 12 p 1/2 + 4 1/2 pT 1022TTT110 1/2) + 1 1/2 00002pp4-000 1/2 10//2/p/////// 12 11/1
1/2 p4 1/2
1/2 1T x 1/2 1/2p2 pt 4 1/2 p 1/2
+
1/2 pT 1/2 11/ 1/2 22/2)
[r]
Tr|P| - Tr |T]
Tr [P3] + Tt PΦ p
Tr
pT 4b PT
Tt
pr²)
=|V|+
1
TTr [p3] + Tr p pT
(34)
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where +b diag(o((), is the unique left eigenvector of P, and then: (details in Ye et al. 2014))
(1u)
d d
(35)
Φ(0)
To continue the development we first partition the edgeset E into two disjoint subsets Eg and E2, where Pi {[u, u), E and (1,,0) $ F), E2 {(u, ((u,)))( (u, E E and (v, E E). Then according to the definition of the transition matrix, we find:
Tr[Pa] = Σ E
Par
Σ
dost dout
WEV OEV
(u,e)E
(36)
p2 (u))
((u)
Tr|Po-
((v)
o(v)dout
WeV eEV
(u,D)E
Substitute Eq.
36 into Eq.
34
then Substitute Eq.
33 and Eq.
34 into Eq. get Eq. 4
HvN
Vor
2/ 1
dg dout
d dout?
oout
dour
(a, E
(a.0)
where
dat
Myns
yout
Σ
May-
TEVOT
TEVeT
Since the normalized Laplacian L. is sparse for most real-world graphs, Tr(E2) can be computed without eigenvalue decomposition by exploiting sparsity (Eq. 34 36):
Tr(L3) = Σ D2 = Σ J
2 uv
+
Σ &e
(37)
(u,))EFG
WE VeT
Both terms depend linearly the number of edges, leading O([Ea,]) complexity per snap- shot(Ye et al. 2014). During temporal aggregation, we process A/ snapshots, thus the total com- plexity of the evolutionary term becomes O(ât [E,r)). which matches the asymptotic bound stated in the paper. No strong assumptions are required, only the standard sparsity condition |E < |V, 12. which holds for most real-world dynamic graphs
Furthermore, in practice, we pre-normalize edge weights and reuse cached degree matrices across consecutive snapshots. This amortizes the per-snapsher cost and further reduces the effective run- time, as validated by our empirical runtime profiling in Table
A.3.2 DONSKER- VARADHAN ESTIMATOR FOR MUTUAL INFORMATION (EQ. 6
To prove Eq. 6 we need to show:
[EP[/(Vo) - log Eo [es(vo)))
((GT; sup JEF
which is equivalent proving:
(K || Q) Ep[T] - log log(Eole")) Epit;
P.t: log Σ Eitt
(38)
To find the extremum of Eq.
38
take the derivative with respect to b and set it equal to zero:
0 12 pit - log E; qieh) =
Øt;
Gjet
+ Pj Σ, Giet =0 0
(39)
- Σ Teet = ge
+ t, = log Pj + log Σ gict
qi
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Let log Σ qeet, and substitute t, into t;:
Σ pit, log Σ nc = 5 Piti log Σ
Each
Σ Pi
log
Pj
llg
gie
log
+a)
4j
Σ
Pt
log
Pi
log
Σ
c0h
qi
Pi
4i
di
(40)
Σ ps log
Pj
log Σ
Pj
+ +a
qi
qi
Σ Ps log
Pi qj
+ llog
= Σ pe log O
Pi DKI(plla).
Thus, the inequality Eq. becomes an equality at the extreme value. When substituting f(Vg) for T in Eq. 38 the equality holds.
A.3.3 UPPER BOUND OF I(g: Gr) (EQ.
9
For any groups of indices L, Sgr x Ser C [Z.] such that g 1. 760 (1 hess Ui HESA
and for any probabilistic distributions Q(Z ((2 ) Serx and Q(Z (1)
(1)
0r a ) E Sor
(1) (9; < 1(9; 12 JIESOTX u Ut HIESOR )
(1)
+
Σ
IB
GTX
Gr
IESorx
LESOrA
P(Z (1)
IA, Z
P(Z(()
(1)
12!-11
(
where
IBC STA
E
log
(1)
(1) IB
Grx
=E
log
9TX
(1)
GT
9r
2rx
By assumption 1
z(0)
2rrx Sr Htesx (Z
JLSSA ensures that the left side of the inequal- we define an order < on the of random
ity holds. As the second inequality, under Assumption variables (2
[
Grx JESSEy
U ( NESOTA as follows:
(1) 1. For distinct timestamps and I', 20) 6rx ATA 2 2 G
2. For given timestamp 1, z( (1) grx"
Subsequently, we define a sequence of sets based on this order.
1000 ST ( ((2a) 141 <1,12 <1,4 E Sørx la Serah.
GTA (1)) = (Z(1)) 7(1a)
11 <1,12 < 11,4 E Sorx la E Sera).
Grx
(1) We may decompose 1(9; 129 (200 JIESSOY U U12 G JreSery with respect to this order.
I(9: (2 JIESOTX U u1 JIESOTA) )=
Σ
(1) I(g:
9r ( + Σ
I(g; z
(H())
).
Srx Srx
LESOTA
IESorx
(41)
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Similar to Wu al. (2020), we get upper bounds for [(g; (
( and ((9; Z
|
(9:
2 Gr
| )S( 20-11 2r 14 $T
(
=I(Z A; z (H())
Gr A
(42)
21(2 A; 2(1)
A
=IB
(1) A - (L((P(2)) )[[((2
(Z (1)
0rA )) - IB
Gr
GTA
T(;; Z
>= (-11)
Grx
9rx
9rx "rA Grx STx
=1 (1-1)) 200) 211)
|H(
(
Grx GTA" STX STX
(43)
< <I 9Tx ' 19TX
(1-1)
z(1) z!1)
IB (1)
Grx
(I.(IP(
(1)
xxxx
5
IB ( 9rx
In summary, the proof of Eq. iss complete.
A.3.4 LOWER BOUND OF I(Y; GT) (EQ. 10
We apply the Nguyen, Wainright and Jordan's 's bound to prove the equation (Nguyen et al. 2010
Lemma Given any two X1 and X2 and any permutation invariant g. we have the varia- tional lower bound of (X1;X2))
T(X1, 2 E [x X2)] (p(X,))(Xx) [exp(g(X1, X2) 1]]]-
Iee P(Kilze :8ra) and obtain Eq.
We substitute g (Y,2 -1111 1+1++ log
10
Q(Y)
P(Y P
Grx:
701) (GrA)) EP(X)
Ep(gr) P(Y Grx 2or
2(1)
I(Y;9T) 2 1+ Ep(Y,
Q(Y)
Q(Y)
A.4 DETAILS FOR DATASETS
In this section, we provide detailed information on the three datasets used in the experiment. When selecting datasets, we consider multiple dimensions including time span, graph density, and evo- lution frequency. Since the TGT framework focuses on global evolutionary features, we prioritize datasets with high dynamic evolution frequencies in temporal graphs during our selection process. Additionally, under similar evolution frequencies and time granularity, we favor datasets with fewer nodes per snapshot. Such datasets exhibit greater randomness under the same perturbation ratio, enabling more effective validation of the model's robustness advantages in stochastic settings. Following these dataset selection criteria, we choose Bitcoin, MathOverflow, and MOOC for the main experiments and supplementary analytical ablation (see Table
Bitcoin Bitcoin is a cryptocurrency to trade anonymously over the web. Due to anonymity, there is counterpart risk, which has lead to the emergence of several exchanges where Bitcoin users rate the level of trust they have others. We integrate the data from the two trading platforms Bitcoin-OTC and Bitcoin-Alpha into time series dataset Bitcoin. Both these exchanges allow users to rate others on scale of -10 to +10 (excluding 0). According to guidelines, rating of -10 should be given to fraudsters while at the other end of the spectrum, +10 means "you trust him as you trust yourself". The other rating values have intermediate meanings. We assign two-dimensional features to nodes, which represent the trust and trustworthiness level, respectively, and sum up the rating of users.
MathOverflow MathOverflow is a temporal network of user interactions on the Stack Exchange website Math Overflow. Nodes represent users, and edges represent question-answer relationships. Three distinct types of interactions are modeled by directed edges (u, v, 1), where:
user answers user v's question at time I;
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user comments user e's question at time t;
user u comments on user e's answer at time t.
While constructing adjacency relations based on triplets, we assign six-dimensional features to nodes, statistically capturing the counts of: questions posted, answers provided, comments on ques- tions, questions commented on, comments answers, and answers commented on. We use these features to build a temporal graph for experimental purposes.
The MOOC dataset represents user behavior on a popular MOOC platform, modeled as directed temporal network. Nodes represent users and course activities (targets), while edges represent user actions on these targets. Each action annotated with attributes and a timestamp, where timestamps normalized to start from timestamp 0. The dataset is directed, temporal, and attributed. Additionally, each action includes a binary label indicating whether the user withdrew from the course after this action-i.e., whether this was the user's final interaction with the platform.
MOOC
For snapshot partitioning of a dataset, we follow these two principles.
Chronological order. We divide the dynamic graph strictly by chronological order to avoid any missing or overlapping temporal intervals. This ensures that temporal dependencies are preserved as faithfully as possible. Because different intervals naturally contain varying numbers of nodes and edges, the node/edge counts reported in Table reflect the average snapshot seale after segmenta- tion. Presenting the average scale helps provide an intuitive sense of computational complexity.
Handling extreme variations in temporal activity. In datasets with highly uneven temporal activity (c.g.. MathOverflow), directly segmenting by time may produce snapshots that are either extremely sparse or extremely dense. To address this, we follow chronological segmentation but enforce reasonable lower and upper bounds on snapshot size by merging overly small adjacent snapshots splitting overly large ones. This adjustment maintains strict temporal continuity (no overlap, no omission) while ensuring more stable computation and memory usage across snapshots.
A.5 DETAILS FOR IMPLEMENTATION
The complete process of TGT the link prediction task is shown algorithm T We implemented TGT utilizing GAT (Veličković et al.) 2018), incorporating 2-layered, 2-headed attention mecha- nism for facilitating message passing and aggregation processes. The hidden dimension is 64.
In experimental setup, we sampled neighbors with size of k 20 and constrained the spa- tiotemporal neighborhood to encompass At 5 snapshots. Neighbor sampling was conducted according to the Bernoulli distribution, guided by attention weights. The Adam optimizer was em- ployed, with an initial learning rate of 0.001, and the training process was iterated 1000 epochs with early stop. To ensure the robustness of our results, all experiments were conducted using three distinct random seeds: 0, 1, and 2. We reported the average performance and standard deviation derived from these repetitions. All the above experiments are conducted on two NVIDIA GeForce RTX 3090 GPUs with 24GB of memory each.
A.6 DETAILS FOR ADVERSARIAL ATTACK EXPERIMENT
Node Features To introduce perturbations to node features, we add Gaussian noise r where the interference amplitude r is set as the average of the maximum values of each node feature, and (O 1). Specifically, we randomly select 10%, 20%, and 50% of nodes in each dataset to simulate increasing levels of interference, gradually escalating the perturbation intensity.
Topology Structure To perturb the topological structure, we employ Nettack (Zügner et al. 2018). proxy-model-basee adversarial attack method that iteratively selects graph substructures with the highest impact on prediction loss as perturbation targets. This approach greedily identifies edges or node pairs whose modification most significantly affects the proxy model's gradient calculations, then applies perturbations such as edge deletion or negative sampling based on their impact on the optimization process. Specifically, Nettack simulates increasing perturbation severity by selecting the top 5%. 10%, and 20% of edges ranked by their influence on prediction loss, progressively intensifying structural disturbances to evaluate model robustness.
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Temporal Evolution To perturb the temporal evolution dynamics, we disrupt the modeling of underlying temporal dependencies by randomly permuting snapshots within the sequence. Specifi- cally, we randomly select and replace snapshot with another from different time step within the same sequence, thereby introducing inconsistencies in the chronological order. Across all datasets, apply such temporal perturbations 2, and times simulate increasing levels of disruption, allowing us to evaluate how model performance degrades with progressively stronger interference in the temporal continuity.
In addition to the untargeted interference mentioned above, we also conducted a detailed investiga- tion and selected five targeted attack methods interfere with our data, and used this to verify the robustness of TGT.
Vettack(Zügger et al., 2018): Nettack is pioneering white-box adversarial attack for graph neu- ral networks (GNNs), which perturbs graph structure and node features to misclassify target nodes while ensuring imperceptibility via preserving key graph properties; it adopts a simplified linearized GCN the surrogate model for computational efficiency, leverages gradient-guided greedy selec- tion to iteratively apply impactful modifications, and exhibits notable efficiency, generalizability across GNN architectures, and robustness to partial graph knowledge.
FGA (Chen et al., 2019): Fast Gradient Attack (FGA) is a white-box adversarial attack for net- work embedding. which perturbs graph structure via link rewiring to degrade the performance of downstream tasks; it leverages gradient information of pairwise nodes to select the most impactful node pairs, and exhibits strong generalizability across multiple network embedding methods as well as high efficiency with minimal link modifications.
SGA et al. )021): SGA is adversarial attack for large-scale graphs. which perturbs structure (edge additiions(eletionn)) to degrade downstream task performance; it adopts scalable strategies (e.g., sampling) for large-graph adaptability, leverages lightweight gradient-guided perturbation lection, and features strong scalability, high efficiency, and minimal imperceptible modifications.
Projected Randomized Block Coordinate Descent (PRBCD) is an adversarial attack for large-scale GNNs, which solely perturbs the adjacency matrix to degrade clas- sification performance; adopts randomized block coordinate descent with continuous relaxation of discrete 0,1 adjacency entries, leverages gradient optimization for efficient perturbation selection, and supports both evasion and poisoning attacks with strong scalability and low memory overhead.
PRBCD al. 2021:
GOttack(Ali) (Alom et al., 2025): Gottack is a universal adversarial attack for GNNs, which perturbs graph structure via topology-awar awwar graph orbits learning to boost misclassification rates; narrows the search space with orbit-based selection, and features high efficiency, strong rross-architecture generalizability, and subtle effective perturbations.
A.7 ADDITIONAL ABLATION EXPERIMENT ON THUMBNAIL BASED ON VNGE
To validate the efficacy of the thumbnaills 1's evolutionary constraints, which is implemented through the von Neumann Graph Entropy (VNGE), we conducted supplementary experiments on noise-perturbed datasets, serving as a complement to the results presented in Fig.
Table 5: Ablation study on holistic continuous evolutionary characteristics results (AUC) on datasets with data perturbation at different levels.
Dataset
Model
Clean
Feature Interference
Structure Interfeeence
Temporal Interferenee
10%
30%
5%
10%
27%
wio thumbnuil
7,.04:006
67.73 2
6188::0.3
56 12:003
69. 00.77
62887503
57.51:003
64.21:0.3
59.61:0044
55. 05.34-10.3
Bitcoin
VNGE
T
39.56..0.2
$6.9110..5
83.4510..3
30.3710..3
24.31.0.2
15.46.0..4
66.20_0.4
69.73_022
$3.26-1:0.
82.55.103
780410.3
79672903
68.37-0.2
72.42.0.3
94,7310.2
83.7910.2
81.5410.5
80.0110.3
74.37.0.33
71.0620.2
91.41:0.2
89.43:055
86.23:9.4
90.62-0.4
39.33-06
85.00:0.5
30.95-0.7
90.76:015
88.14:0.6
85.64:0,3
Bumbroil
16.11
24,344022
68.41-0.3
00:21-04
72.7040.5
68:299033
00:54-055
30.32-0.3
62.47-005
56, 86.07
MathOverllow
T
60.35:00.3
00.39:00.3
75.52.60.3
71.43+0.3
67.00
63.52-0.3
77.1340.3
77.43104
76.67:03
73.31
71.49:0.22
68.74:0.5
36.39-0144
18.61:10.1
06:12.03
33.30-0.3
69.83:1022
64.27-0.3
82.35-0.6
70.32--0.3
74.37-0.3
71.42-0.
80.01-0.3
77.74-0.3
15.93-0.7
81,57-0.2
76.37:0.4
shumbnonl
83.63:100
74.96:0033
0.5
72.11004
60:25:0:2
81.06::.4
00.7
00.78:003
85.30-002
72.78-10.4
60.23-0.3
54+03
81.17503
73.54-0.2
85.21+0.22
49
MOOC
W/ VNGE
92.03.10.2
86.73102
77.35.10.4
66.27_0.3
59.58.0044
$1.0105
77.3110.3
84.1910.22
79.11.0.3
70.59.10.4
TGT
95.42=00.4
88.73+0.2
90.79-0.1
71.68-0.3
90.43+03
86.16:0.5
S1.82-0.3
92.61=112
89.53-0.7
84.01 0..6
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To construct the ablation baseline w/o VNGE, selected normalized node degrees from Eq. as a substitute for the von Neumann entropy. We also added an ablation control w/o T, where the term (9r; 9) in Eq. [1 is set to zero entirely, and further ablation setting w/o thumbnail that completely discards the construction of thumbnail from Eq. 11
As shown in Table 5 the construction of the thumbnail 11 plays critical role in the model's rep- resentational capability and robustness. Even when structural information is excluded in the w/o ablation (where the thumbnail 1 determined solely by node features from Eq. 66, the model still improves its representation of temporal graphs using node semantics compared with w/o thumbnail.
In contrast naive alternatives for incorporating evolutionary constraints, such normalized node degree (w/o VNGE), the thumbrail modeled via mutual information based on VNGE (our pro- posed method) demonstrates significant advantages both representational capability and robust- ness. This confirms the effectiveness of the VNGE-based thumbnail in enhancing the model's ability eapture continuous evolutionary features while resisting perturbations.
A.8 ADDITIONAL RESULTS ON ABLATION STUDY
We conducted individual ablation studies the three purposed components investigate the spe- cific contribution of each to the experimental outcomes. When the data set has not been disturbed, the impact of each component on the experimental results shown in Fig.
BEL55
50.788
12233
90.35 79.63
AU029)
25141
80:37
51.17
Xibooin
MD0C
Figure 4: The figure shows the ablation experiment results three datasets. Where T represents the evolution constraint (Eq. [T]. A represents the structure mutual information (Eq. 13 and X represents the feature mutual information (Eq.
A.8.1 FEATURE MUTUAL INFORMATION
After setting the feature mutual information (Eq. 12 to 0, we repeated the robustness test of feature interference the three datasets. The results shown in Fig. 3 Without the feature mutual information, the model's robustness to feature interference drops significantly.
100
100
100
Iee
50.41
90
90
155.21
80
10000
11795
80.62
80
00.00
(1152
70
21.00
62177
12.6
docn
10%%
20%
5005
dims
10%5
20%
50%
dosc
10%
27%
30%
Figure 5: The figure shows the experimental results after the node features on the three data sets were perturbed to different degrees.
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A.8.2 STRUCTURE MUTCAL INFORMATION
With the structure mutual information (Eq. 13 set to zero, we conducted a repeated robustness assessment of structure interference across the three datasets, shown in Fig. 6 Compared with feature mutual information, structure mutual information has less impact on clean data, but its impact on the robustness of the model more obvious.
100
100
100
55.42
Ttar
90-
90
00143
85,00
80
13130
80
50.30
10144
$1.82
70.30
11:10
70
70
30,70
Chenn
9%
10%
20%
dhenn
5%
10%
20%
chenn
96
10%
20%
Figure 6: The figure shows the experimental results after the topology on the three data sets were perturbed different degrees.
A.8.3 ABLATION ON BACKBONE
In this section, mainly design experiments and discuss the impact of the encoder's backbone network configuration on TGT performance. The backbone corresponds to Lines 6-8 of Algorithm in Appendix A.1.4. where the encoder is applied. Although GAT is adopted in the main paper owing to its capacity to model node-specific contributions to the thumbnail through attention mechanisms, the TGT framework itself is backhone-agnostic.
Table The results under clean data and three types perturbations on Bitcoin dataset.
Feature Interference
Structure Interference
Temporal Interference
Backbone
Clean
10%
20%
50%
5%
10%
20%
2
GCN
90.86
88.93
86.89
81.73
89.66
85.27
78.13
89.24
86.05
82.75
GIN
91.44
90.24
84.52
79.87
87.72
85.33
81.02
89.90
88.31
84.13
GAT (Default) 91.41
89.43
86.23
80.62
89.83
85.00
80.95
90.78
88.14
85.64
To elaborate on this flexibility, have implemented two additional TGT variants with alternative encoders such as GCN (Kipi 2016) and GIN (Xu et al.) 2018). We repeat the experiments on the Bitcoin dataset by replacing the backbone and obtain the results shown in Table
These results demonstrate two points: (i) TGT's performance stable across different backbones. The changes in node encoder architecture have only minor influence clean-data accuracy and ro- bustness under perturbations. (ii) GAT performs slightly better, which expected because the atten- tion mechanism naturally aligns with the design of TGT. It allows different nodes in each snapshot contribute unequally to thumbnail, reflecting our motivation to capture heterogeneous importance.
In summary, while GAT is strong and intuitive choice for TGT, the framework does not rely specific backbone architecture and can be instantiated using standard GNNs such as GCN and GIN with comparable performance.
A.9 HYPERPARAMETER SENSITIVITY EXPERIMENTS OF TGT
We performed hyperparameter sensitivity analyzes the Lagrangian coefficients defined in Eq. and Eq. [6, examining their impact on robustness behavior and optimization of temporal evolution. We conduct experiments MOOC dataset, which has the most edges and the fastest evolution.
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The impact of Hyperparameter setting on thumbnail In this section, we primarily design periments and discuss the impact of hyperparameter settings on thumbnail modeling, and further provide suggestions the parameter selection methods for Lagrangian hyperparameters.
serves to specify the constraint strength exerted by the thumbnail over the encoding process. To clarify this relationship, have supplemented with experimental analyses the number of thumbnail nodes as a function of Different choices of À indirectly influence the thumbnail size, yet the thumbnail size remains close the average number of nodes (and edges) across snapshots. This, in turn, ensures the two terms are of the same order of magnitude during gradient computation, thus validating the rationality of confining A's value range to 10.
Table
Thumbnail statistics under different settings.
Dataset
Thumbnail
A
Snapshot Avg.
0.1
0.5
1
2
5
nodes
6,954
6,844
7,019
7,090
7,051
7,034
Bitcoin
edges
47,558
48,880
47,558
47,969
52,982
51,363
nodes
20.474
19,965
23,376
22,371
24.105
21.683
MathOverflow
edges
160,491
262,295
315,105
417,491
478,325
207,581
nodes 6,120
6,264
6,275
6,152
6,340
7,047
MOOC
edges
53,552
56,628
63,415
61,074
68,068
81,749
Regarding the terms balanced by (Eq. 15): the TB term computed via snapshot node sampling, while the downstream task mutual information is calculated using thumbnail nodes. Table conffrms that the two terms of the same order of magnitude, thus justifying the rationality of setting f's value range within 10.
Unfortunately, tuning these two hyperparameters is an empirical process, as changes in data sce- narios and specific requirements directly influence their optimal values. In practice, we recommend conducting small number of warm-up training runs for hyperparameter search, thereby achieving more reliable hyperparameter selection.
Specifically. during the pre-training phase, we estimate the noise level of the input sequence via the variance of von Neumann graph entropy across consecutive snapshots, denoted as Var (g')). This value is correlated with structural volatility, and we scale a proportionally to this variance:
Var(H(G')) Var (H(G)))
00
(44)
This guarantees that a is increased solely when the graph sequence undergoes unstable evolution. During the process of a's gradual increase, we select the value that delivers acceptable performance under the clean setting as the appropriate hyperparameter value.
Compression Balance Parameter of TGT The compression balance parameter B of thumbnail in Eq. [5] controls the trade-off between preserving relevant information and com- pressing redundant details the learned representation. We 0.1, 0.5, and 1, systematically increasing the compression strength to evaluate the robustness of the model under the compression constraints based on thumbnail.
values including 0.001, 0.01,
The robustness results for node feature and structure perturbations are presented in Table 8 When small, model performance degrades more significantly with increasing interference. As $ in- creases, while performance clean data slightly declines, the model's interference resistance im- proves substantially. Practically, setting 3 requires balancing robustness against the capability to represent clean data effectively, reflecting critical trade-off in the framework's 's design.
We performed the same experiment under temporal interference and observed consistent results, as shown in Fig. 7 demonstrating that the optimization objectives 7 B' , TB in Eq. 15 leverage the thumbnali Gr, which encodes rich continuous evolutionary features, as a compression con- straint. This approach prevents the disconnection between the temporal graph holistic evolutionary continuity and discrete snapshot features (e.g., nodes, topology), thereby enabling more coherent and robust model performance across perturbation. Finally set 0.1 in other experiment.
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Table 8: Robustness results (AUC) on MOOC with different 3 in TGT.
Feature Interference
Structure Interference
Clean
10%
20%
50%
5%
10%
20%
0.001
95.01
85.35
77.74
68.18
87.98
83.56
78.08
0.01
94.35
86.57
78.91
70.46
91.23
85.71
79.68
0.1
95.42
88.73
80.79
71.68
90.43
86.16
81.82
0.5
92.47
87.84
82.16
76.89
89.51
86.95
83.62
90.81
89.05
83.27
79.38
89.70
87.67
82.98
100
TCT
96.01
96.42
50:47
91.32
81.00
08.5399.7
01
05
A00%))
06.75
87.65
05.43
03:45
83.37
30.61
80
clean
11=1
11=52
1255
Figure The results after the temporal perturbation on the MOOC with different settings.
Evolutionary Constraint Parameter of TGT The evolution parameter a in Eq. 16 governs the strength of temporal continuity enforced by the thumbnail, guiding the model preserve coher- ent evolution patterns across time while filtering out noise and irrelevant transitions. We set to 0.1, 0.5, 1, 2, gradually enhance the strength of the continuity temporal constraint.
Table 9: Robustness results (AUC) MOOC datasets with different in TGT.
Feature Interference
Structure Interference
Clean
10%
20%
50%
5%
10%
20%
0.1
94.37
87.21
76.54
66.63
89.80
85.45
79.77
0.5
95.02
87.77
74.51
67.06
91.43
87.91
80.36
95.42
88.73
80.79
71.68
90.43
86.16
81.82
89.61
84.63
80.72
71.23
86.32
82.35
78.43
87.26
83.52
79.40
69.33
84.77
80.59
77.81
100
TCT
94.77.9.00255.4
0.1
52.61
[8:61
08.6490.01
89.53
AUO(%)
967116634
85.73
63.57
3400 83.47
83.40
81.34
03.334
80
79.34
T-
T242
0+6
Figure 8: The results after the temporal perturbation the MOOC with different settings.
Table Table@iindccater that under untargeted feature perturbations, the robustness improvement from con- straining the model with the von Neumann entropy, which captures topological evolution, rela-
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tively modest. By contrast, during structural attacks, increasing the parameter reduces the mag- nitude of performance degradation caused by perturbations. This trend is particularly evident the temporal perturbation experiment results visualized in Fig. | Notably, of node features becomes sparser, leading to degraded performance on clean data. After synthesiz- ing these experimental results, ultimately and clean-data representation.
increases, the encoding
1 as the optimal value to balance robustness
Window Size setting of TGT Regarding the time window size mentioned in Appendix A.5 we supplement the ablation experiment results on Bitcoin dataset. Interference settings are based on the most severe settings for each type of interference in the Table 3 (Feature 50%, structure 20%, temporal n=5).
Table 10:
The ablation experiment results (AUC) with different window sizes.
Interference
Window Size
Clean
Feature
Structure
Temporal
75.38
65.41
65.27
70.80
84.79
72.76
74.75
77.47
(baseline)
91.41
80.62
80.95
85.64
92.52
79.03
80.70
86.20
90.55
81.37
84.70
85.93
Notably, unduly small window sizes result in inadequate thumbnail modeling, thereby degrading representation quality. For window sizes larger than 5, the model's performance exhibits negligible fluctuations. This confirms that the neighborhood relationships determined by window size exert certain influence on the thumbnail. TGT exhibits strong adaptability to this hyperparameter.
A.10 ADDITIONAL EXPERIMENT SIMILAR WORKS
Numerous approaches have leveraged the information bottleneck to compress raw data for robust- ness (Wu et al. 2020 Sun et al. 2022 Yu et al. 2021b a). Some works have also attempted to adapt information bottleneck theory to temporal graphs for robustness (Yuan et al. 2024). In contrast to these related studies, which focus on achieving information compression, TGT prioritizes summariz- ing a reliable continuous evolutionary skeleton for temporal graph sequences as thumbnail. TGT is the first to integrate global evolution via the von Neumann entropy, characterizing the evolution- ary backbone of temporal graphs. By imposing compression constraints through the thumbna 1, it avoids excessive compression that causes information loss or insufficient compression that leaves noise redundancy, striking balance between essential feature retention and disturbance resilience.
To further establish TGT's superiority, we compare it against methods which perform information bottleneck temporal graphs and conduct supplementary experiments validate its distinct ad- vantages. We additionally selected two temporal datasets, Collab and Yelp. These datasets contain richer features but exhibit relatively lower evolution frequencies compared our primary choices.
Collab is an academic collaboration dataset containing papers published between 1990 and 2005. Nodes represent authors, while edges represent co-authorships. Each edge includes domain information such as "data " "database," "medical informatics," "theory," and "visualization," based on field of the co-authored publication. Node features are processed using word2vec Mikolov al. 2013), and edges are encoded via one-hot encoding.
Yelp a forum dataset containing business information, including attributes such as reviews, photos, check-ins, business hours, parking availability, and ambiance. We define users and busi- nesses as nodes and review actions as edges, processing node features with word2vec similarly to the Collab. We sample interactions from five major business categories, Pizza, American (New) Food, Coffee & Tea, Sushi Bars, and Fast Food, spanning from January 2019 to December 2020.
Our selected comparison baselines DGIB Yuaan al. 2024) and GIB (Wu et al. 2020 com- bined with an LSTM model, where GIB first generates discrete snapshot representations and the LSTM subsequently captures temporal dependencies. Experiments shows the superiority of TGT.
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Table 11: Robustness results (AUC) on Collab and Yclp datasets with different data perturbation.
Datasel
Model
Clean
Temporal
20%
50%
10%
20%
2
x
GIB4LSTM
91 21.300.22
80.73e0.2
12.27=0.3
61.73=0.3
85,75e902
76:5==04
71.22+0.2
84,0==03
74.3440.5
60.255004
Collab
DG3B
92.17:022
08.05.0.3
73:72-003
64.18:066
87.47:00.1
80.73-0.2
74.43:10.3
63,32:02
80.73:10.2
59.46:0055
TGT
93.41:0.33
90.26:00.3
83.06.:03
74.92:0.3
91.19.04
85.07
78,72:0.4
84,4:003
$2.3110,2
73.11:003
CIB
77.52-04
71.71:002
62.63-0.2
54.76:0.5
75,41:002
72.56-044
65.31:0.2
74,02=03
68.3410.3
57.26+077
Yelp
76.88-022
71.54+0.4
67,54-0.5
62.980044
75.27-04
14-51-0.2
7345+0.3
75.39-0.3
72.11+0.3
65-22406
80.17.033
79.79.0.2
A.11 SUPPLEMENTARY EXPERIMENTS UNDER DIFFERENT DOWNSTREAM TASKS
To verify the robustness of the temporal node representations learned by TGT, we conduct additional experiments representative downstream tasks. Specifically, evaluate TGT node classi- fication and dynamic node property prediction. These experiments further demonstrate the model's robustness and its ability to generalize across diverse tasks.
A.11.1 SUPPLEMENTARY EXPERIMENT ON NODE CLASSIHICATION
To demonstrate that TGT produces general-purpose temporal node representations that transfer to other downstream tasks, we expanded our evaluation to a node classification task on both the Reddit dataset (Kumar al. 2019) and the MOOC dataset used in the main paper. The supplementary Reddit dataset is as follows:
Reddit: A subreddit-fccused sharing/diseussinn platform. We build the temporal dataset Red- dit from public sources (SNAP's 2046 subreddit monthly networks, Pushshift.io timestamped com- ments). is a directed temporal attributed network with subreddit hyperlinks (timestamps, sentiment polarity. post text attributes). featuring two core interactions: direct user replies and linear comment chains (less than 3 intermediates). Interactions have valence score (upvotes downvotes); node features (3D): activity (total comments), interaction quality (avg. valence), community participa- tion (joined subreddits), derived from historical stats. One-month snapshot: Ik active subreddits, 10k users, 672k+ interactions; post texts encoded as LIWC features; labels from ban records (366 labeled nodes); task: binary node classification (banned/unbanned).
Table 12: Node classification performance (AUC) under clean and perturbed settings on Reddit and MOOC datasets.
Interference
Dataset
Method
Clean
Feature
Structure
Temporal
EvolveGCN
56.4
48.6
51.7
51.8
JODIE
59.9
51.6
51.4
50.1
DyREP
63.2
54.5
57.6
52.4
TGN
67.2
53.1
58.1
53.5
Reddit
DIDA
58.2
49.0
54.3
49.7
GIB+LSTM
54.7
53.1
51.6
DGTB
55.7
54.6
$2.0
TGT (ours)
64.6
58.3
59.0
57.3
EvolveGCNN
67.3
$4.6
54,9
53.8
JODIE
68.5
53.7
56.5
55.4
DyREP
63.4
50.3
55.8
53,6
TGN
64.4
53.1
54.6
51.7
MOOC
DIDA
56.7
53.7
54.9
53.0
GIB-LSTM
59.1
51.2
54.7
52.3
61.3
54.7
55.2
TGT (ours)
66.4
60.3
62.6
61.4
The results reported in Table 12 each type of interference the Table (Feature 50%, structure 20%, temporal n=5)).
Interference settings based the most severe settings for
Under clean settings, TGT achieves competitive AUCs, comparable to or better than strong temporal baselines (e.g., TGN, DyREP). Under severe feature interference (50% Gaussian corruption), TGT retains substantially higher performance than most baselines (next best typically in the low-50s),
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demonstrating superior denoising ability. Under structural interference and temporal interference, TGT consistently outperforms alternatiees by meaningful margins.
These results show that the thumbnail provides robust evolution skeleton that helps not only link prediction but also node-level classification. Intuitively, by encoding persistent evolutionary skeletons, TGT yields embeddings that (i) suppress transient/noisy signals that hurt classification under corruption, and (ii) preserve long-term structural cues that are predictive for node labels.
A.11.2 SUPPLEMENTARY EXPERIMENT ON TARGETED ATTACKS WITH EXTRA BASELINES
To further measure the robustness of our TGT under more severe noise conditions, we conducted experiments on wider range of noise data. We implemented Nettack, FGA, and SGA using DeepRobus
f set the attack intensity n.perturbations to 20% of the average number of nodes, and implemented these five attack methods using the official code of PRBCU and GOttack (details of attack methods in Appendix A.6).
Table 13: Link prediction performance comparison under untargeted and targeted attacks.
Dalasel
Method
Clean
Untargeted Interference
Turgeted Attacks
Struel/20%) 5)
Temp-(n=55
Netluck
PGA
SGA
PRECD
GOllack
EvolveGiCN
67.59=00.
56.77=002
55 0.3
$4.37=0044
55. 99
47.07=066
52, 12.08:0.4
49. 49.70=007
51.08=02
JODIE
04.47:00.3
61. 10:033
60.57 5.3
58. 20:08.
54.3 04
55.24:0.3
58.00 SS.00:0.66
53.34:0.6
48. 38.54=033
DyRFP
70.43+0.5
61.25+0.3
0..
57.33+0077
53.46+0.7
54.09-0.7
53.71+1.1 55.71+0.3
49.05-0.5
TGN
69.36.11.1
62.3110.4
58.73_0.5
61.241088
52.17110
$4.711003
$1.210..8
50.70.0077
Bitcoin
DIDA
GIB-LLITM
70.79:10.3
73.57.10.3
3
63.73-20.55
62.90-0.2
60:60:04
63.41.088
64:29:004
60.19.0.55
62.2910.5
64.15:009
59.43:004
51.2710.3
63.43:0.6
61.991044
10.851100
57.89-077
60.50:044
DGiB
72.99+1.3
63.63-+0.5
59 59.13-0.4
62.53-006
62887004
63.04-003
62.53+0.7
60,55+014
58.81-03
Edgehank
62,27+0.77
54.38=0.9
$2.96=1.3
53.12=1.1
51.53+0.88
50,47=055
52 22:20:00.4
50,14+0.6
47.45-006
DYGFomee
87.9110.66
74.60:0.3
0.5
73.76
70.34:003
72.93:0077
71 64::0.7
69 69.6110.9
64.22:110
TPNer
01.10+1.3
17-0
13-13-044
77.67
60.27+1.1
70.10-1.3
0.-08-0.5
67.90+046
91.41.1022
86.23
80.95.0.7
85.6410.3
$2.11100.5
81.24.0.33
$0.32.0.9
78,2410.44
75.78
PaoivetCN
75.59+0.2
61.
55.23-0.3
56.63+0.5
53.16+0.4
50.13+06
52.33+0.7
52,53+0.7
50881006
JODIE
67.06.11.2
59.56
13.1010
35.37
54.04104
53.67..03
52.7210.5
51.1110,77
50.07.04
DyREP
63.30:10.5
6
53.26-003
53.33-10.5
52.73-+0.7
50 50:45:04
50.94:0.99
49.83:00
49 00
TGN
64.5010066
58.9610.3
53.97-003
55.25100.7
53.2 53.22.1055
$3.26.10.3
S3.2410.4
50.3110077
50.89.0.5
MathOverflow
GIB-LSTM
DIDA
77.52+0.3
74.37+00.4
69.38=06
68.63+0.1
63.21=08
67.00-04
65.4 65.44+015
6
58.4310.55
60.3++077
57.12:06
61 61.25-005
60.44-0.3
64-640088
54.11+0.5
56.41104
58. 10
05
DGBB
50.29+10.3
73.63-0.5
0.
70.24-005
61,22-077
6541:11
62.04:033
61:11110
60.33-04
Edgehunk
62.43+0.7
5500+00
11-03
53, 71-00-077
11443-06
50.99669
49602008
47.95-03
DYGFomee
$3.67.1..5
69.7
67.
67,81
6677199
53.01
60.711111
51,66119
00.7
[PNet
78.43+1.11
63.89:02
64.0--03
62. 62.86:503
54/22-14
51.19:066
57.78-0077
47.10=112
TGT
$2.38.10.6
74.37.0.
7.793-0.7
76.37
69.33.0.8
68.5910.4
66.5011055
61.17:14
EvolveGONN
72.35+0.3
62.37+022
52.31-022
55 55.93+0022
52 10+00%
40.51+033
S1 21-25
50,55+0.3
48.92-04
JODIE
73.1910.7
58.36
0..
58.59
56.95104
58.02.0.4
53.6910.3
52.2310.7
$1,69-03
DyREP
$1.36.0.11
66.77
59.37
17.1010
$5.42.0.3
$48310.4
52.6110.5
79.36=1.0
68.41:033
61.33-0.3
63.53-1111
79.00=077
61.65-033
9240077
54.77
5388=005
MOOC
GIB-L-TMM
DIDA
92.34 02.34+0.3
80.4+0055
65. 25-23.
73.62+0.3
74.37-0.3
69. 69.68+0077
73.66+002
66.04+033
63.26+004
64 64.76-0.5
68.58-044
64-03-
65,79+0.4
6311+-15
61,24+077
61.38-04
64.33-11
DGIB
93.06.10.1
75.24110.
79 69.
70.32.10.6
74.31.0.9
14.14.1.55
70.5110.7
68.6910A
66.03.077
Edgehhnkk
36.1410.7
6
62.31=1.1
61. 61.52+0099
60.71+0.5
5
60:32:044
60.78+055
59.64-03
DYGForee
91.1311.2
77.2410.5
75.55_0.3
10.5
69.07
70.07
67.37.0.7
66,28115
63.41208
TPNer
96445.0.7
73.16=13
77.25-0.4
74.17-0.
73.66-+0.7
12.96=13
67
65.7340.3
64.73-03
TGT
95.42=0.4
80.79:0011
81.2-003
84.01:0066
$0.33:9.3
78.11:0.4
01.38:0.7
77.65:0.6
78.60:0.4
In addition, we supplemented three more advanced baselines that outperform in clean data setting, thus more comprehensively validating the superiority of TGT. Edgebank (Poursafaci et al. 2022) is nmeooiizttionbasee baseline for dynamic link prediction, which leverages temporal edge reoccur- rence patterns to enhance evaluation rigor and exposes the flaw of easy negative sampling in existing protocols, featuring strong performance across settings. DYGFormer (Yu et al. 2023) is a dynamic graph learning framework integrating a novel architecture and a unified library, enabling efficient modeling of temporal dependencies and achieving strong generalization across diverse dynamic graph tasks. TPNet (Lu et al. 2024) is a temporal link prediction model that boosts performance via temporal walk matrix projection, enabling effective modeling of temporal dependencies.
To verify the reliability of our TGT, we generated the noisy datasets to conduct inductive link pre- diction experiments. The results in [ consistenly show that TGT achieves the highest per- formance under every targeted attack across all datasets. For instance, Bitcoin, TGT reaches 82.11 under Nettack and 78.24 under PRBCD, whereas strong basclines such as DGIB and DYG- Former fall 5-15 points lower. Similar trends hold on MathOverflow, where TGT maintains 70.46
'DeepRobust: http:s/ss/gtaaaaoooo//////////// oPPpop/ttarrr///tabooooo/////ssr Satpttttsss_________________ _g____at_ccalll___ca______a
'GOuack: ht httppp////gittbb..oo///aaa/eeeee
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under Nettack, substantially outperforming the next-best baselines. These results demonstrate that TGT does not merely resist mild perturbations but remains robust under attacks that specifically tar- get influential nodes edges associated with link prediction, indicating that its thuddeail-guided global evolution modeling prevents overreliance on vulnerable local structures.
Combined with its strong performance under untargeted feature, structural, and temporal interfer- ence, these findings provide direct and comprehensive evidence that TGT delivers outperforming robustness even against the most advanced targeted attack strategies.
A.11.3 SUPPLEMENTARY EXPERIMENT ON NODE PROPERTY PREDICTION
To further validate TGT's robustness and generalization ability across additional downstream tasks, we conducted supplementary experiments following thorough research: we selected the large-scale igbn-genre and redditeddit datasets(details in Table 14) from the TGB (Huang et al. 2023b for node property prediction, and adopted TGB's standardized task metrics. This design effectively supplements the verification of TGT'ss cross-task adaptability while demonstrating its superiority.
Table 14: Details of TGB datasets for experiment on node property prediction.
Dataset
nodes
edges
Steps
snap. nodes
snap. edge
tgbn-genre 1,505
17,858,395 27,174.118
133,758 21,889,537
1307 6843
78,345 440,603
tgbn-reddit
11,766
Tgbn-genre dataset is a bipartite weighted user-music genre interaction network (based listening behavior). Nodes are users and genres; edges denote time-stamped user-genre listens, with weights as song-genre percentage contributions. It is built by cross-referencing LastFM- song-listens (Ik users, 1-month song listens) and Million-Song Dataset (song genre weights). Re- tained genres meet two criteria: more than 10% song weight and Ik dataset occurrences. Genre names cross-checked eliminate typos.
Tgbn-genre:
Tgbn-reddit: Tgbn-reddit dataset is a user-subreddit interaction network (users and subreddits as nodes, edges indicating time-stamped user posts on subreddits) spanning 2005-2019, with the prediction task of ranking the subreddits a user will interact with most in the next week.
Node property prediction performance (NDCG@ 10) under clean and perturbed settings on Tgbn-genre and Tgbn-reddit.
Table 15:
Untarget Interference
Target Attacks
Dataset
Method
Clean
Feat.
Struct.
Temp.
Nettack
FGA
SGA
PRBCD
GOttack
EvolveGCN
0.343
0.321
0.316
0.305
0.296
0.253
0.287
0.277
0.262
DyREP
0.351
0.317
0.301
0.288
0.240
0.267
0.237
0.207
0.192
TGN
0.367
0.323
0.311
0.325
0.259
0.274
0.236
0.246
0.219
Tgbn-genre
DGIB
0.332
0.311
0.307
0.303
0.277
0,278
0.263
0.303
0.293
DYGFormer
0.365
0.306
0.291
0.280
0.247
0.231
0.275
0.252
0.209
TGT
0.363
0.348
0.354
0.345
0.342
0.314
0.319
0.307
0,313
EvolveGCN
0.310
0.291
0.276
0.297
0.201
0,210
0.207
0.197
0.182
DyREP
0.312
0.288
0.278
0.263
0.231
0.227
0.210
0.183
0.193
TGN
0.315
0.285
0.277
0.271
0.206
0,217
0.224
0.201
0.196
Tgbn-reddit
DGIB
0.304
0.269
0.265
0.261
0.223
0.227
0.216
0.211
0.187
DYGFormer
0.316
0.279
0.271
0.234
0,200
0.204
0.198
0.161
0.183
TGT
0.313
0.289
0,280
0.278
0.243
0.246
0.231
0.221
0.210
We processed the data using the same perturbation methods employed in the experiments detailed in Appendix 11.2 with NDCG@ 10 adopted as the evaluation metric. This setup aims to verify whether the methods' predictions of class importance align with the ordering of the ground truth.
Specifically, while strong baselines such as TGN, DGIB, and DYGFormer suffer noticeable degradation under perturbations, TGT maintains the highest NDCG@ 10 across all untargeted and targeted attacks on TGBN-genre (0.354/0.345/0.344 un- der structural, temporal, and Nettack attacks respectively, far above TGN's 0.31 1/0.325/0.259). Re-
The results are shown in Table 15
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garding node property prediction robustness, both TGBN results show that the thumbnail-guided global evolution modeling provides stable ranking consistency even under aggressive adversar- TGBN-reddit, TGT achieves 0.243 under Nettack, outperforming
ial conditions. For example, all baselines by significant margins (the best baseline is approximately 0.231).
These observations confirm that TGT's core mechanism, using VNGE-constraingg thumbnail to encode long-term structural regularities, yields node representations that are not only scalable (con- firmed in Table 4 but also substantially more resistant perturbations.
A.12 RELATED WORK
Temporal graph representation learning
In temporal graph representation learning, representations are derived from spatiotemporal neighbor
info via message passing and aggregation based on topology. For instances, some approaches utilize recurrent neural networks (RNNs) to learn dynamic embeddings in temporal interactions (Kumar
et al. 2019: Pareja et al. 2020 Han et al., 2020). Alternatively. other methods rely attention mechanism to update neighbors Rossii et al. 2020; Trivedi et al.) 2019). Furthermore, numerous scholarly investigations have focused on model's 's robustness and generalizability, [Zhang et. (2022)
integrates a disentangled spatiotemporal attention network with intervention and regularization to improve representation quality under distribution shifts. Zhu al. (2019) boosts robustness by in- jecting random noise. There are also some attempts to fuzzy the adjacency matrix (Wu et al. 2019 Entezari et al.) 2020). Other methods also consider the evolutionary trajectory of temporal graph in representation learning. Shamsi et al. (2024)combin combinnig topological data analysis with recurrent neural networks to effectively predict the evolution. Wang et al. (225) models both temporal in- teraction trajectories and semantic edge types in continuous-time dynamic heterogeneous GNN, enabling richer representations of complex evolving graphs. Yao et al. (2022) builds a hierarchical spatial graph via a quadtree and uses graph attention to model long-term dependencies in trajec- tories, enabling efficient and accurate similarity computation, especially for long trajectories. Li et al. (2024b) introduces mutual evolution framework with a predictive module and an unlearning module co-training to improve unlearning efficacy and efficiency in graph neural networks while preserving prediction performance. There also methods based on generative models that at- tempt to conduct more precise representation learning in various aspects, such as knowledge graphs (Zhang et al. 2024a by and retrieval augmented generation (Xiang et 2025 Zhang et al., 2025). The above works illustrate the importance of modeling the evolution and demonstrate the value of this research question. However, none of these methods mentioned above focus on robustness; when data is disturbed, the modeling of evolutionary trajectories may be inaccurate. To address the lack of theoretical support for generalizability and robustness in previous research, we derive an solution to enhance model interpretability.
Von Neumann entropy in graph evolution
For a quantum system with density matrix P. the von Neumann entropy (VNGE) is SyN - -Tr(plogp). p describes mixed state system consisting of pure quantum states (b)) each with probability p.,, defined as p Σ Pa (tby). where T is the number of pure states. Sy. plays crucial role in the quantum measurement process, and can calculate the expectation of the mea- surable quantity. Braunstein interpreted the scaled regularized Laplacian as a density operator, en- abling VNGE for graph characterization (Braunstein al.) 2006). Subsequent research revealed that its measurement is pivotal in comprehending network system structure and topological complexity (De Domenico et al. (2015). For instance, aids depicting quantum statistics in topological net- works (Passerini & Severini 2009) and assessing graph irregularity (Passerini & Severini 2008). Liu et al. (2018) used VNGE to study dynamic genomes universal patterns, extending its use in dynamic cases. Then Huang al. (2023a) designed an attention network considering VNGE for evolutionary feature extraction. We combine VGNE to set cvolutionary constraints so that the latent Hinge graph can summarize more coherent temporal change characteristics.
Information bottleneck in graph
Alemi et al. (2016) applies variational estimation and integrates the information bottleneck theory (IB) (Tishby et al. 2000) to enhance robustness and generalizability in deep representation learning.
Wu et al. (2020) formulated hypothesis for transforming graphs into IID data, and subsequently
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Published as a conference paper at ICLR 2026
derives an IB optimization tailored to discrete structures. Methods such as Seo et al. (2024) and Sun iet al. (2022) each offer distinct solutions for extracting the IB objectives from static graph data. Yu
al. (2021b 2022 2021a have also demonstrated the remarkable performance of IB in discovering
key information and inducing important substructures, Yuan et al. (024) first attempts to extend IB
temporal graphs. Chen al. (2024) investigated the application of the IB in the extrapolation of temporal knowledge graphs. However, these methods overlook graph evolution, as IB constraints are limited middle latent variables. By leveraging the VNGE to characterize continuous evolutionary features, our TGT can access richer evolutionary information, which significantly enhances its abil- ity to resist interference under perturbed conditions. Furthermore, we integrate temporal evolution via VGNE to capture entire sequence. The results are presented in Table 11
A.13 LIMITATIONS AND FUTURE WORK
The main limitation of TGT is the scalable problem caused by the additional introduction of thumbnail characterization. Since the theoretical upper bound of TGT's computational com- plexity scales linearly with the number of nodes and edges across all graph snapshots (as shown in appendix A.1.4), scaling larger datasets (e.g.. datasets with millions of nodes) can be prohibitively time-consuming, While TGT demonstrates strong robustness under wide range of feature, struc- ture, and temporal perturbations, well as various targeted attacks, we acknowledge that our re- liance on a thumbnail-barn attacks specifically crafted to target the thumbnail. For example, an adversary could attack (i) the thumbnail structure via ontology-level perturbation, or (ii) the bottleneck constraint by designing gradient-aligned manipulations to weaken the VNGE-based regularization. These are meaningful and challenging scenarios that we agree deserve deeper investigation.
architectural design may introduce new vulnerabilitties under adaptive
In our future work, we plan to explore pretraining- ing the computational overhead to be offloaded to the pre-training stage and thereby improving the scalability. We will also explore the usage thumbnail 1 as a prompt, aiming to further improve the model's capability and enable the injection of external prior knowledge.
approaches for thumbnail modeling, allow-
A.14 BOARDER IMPACT
This paper aims to advance the field of graph learning by proposing robust representation method for temporal graphs. Our TGT framework inherently resists noise during representation learning, offering robustness against the ubiquitous real-world data noise and potential adversarial attacks such as data poisoning. This capability has positive social implications by enhancing the reliability and trustworthiness of machine learning applications in critical domains. Furthermore, we anticipate that our work will not pose direct social or ethical negative impacts.
A.15 THE USAGE OF LLM
LLM was employed exclusively for writing assistance, including grammar correction, spelling ver- ification, and text polishing, without contributing in any other way to our work.
36


Reviews:
[{"review_id": "yjaVvcZf35", "reviewer": "Reviewer_ER5F", "summary": "This paper introduces the Temporal Graph Thumbnail framework designed to achieve robust representation learning on dynamic graphs. TGT addresses limitations in local aggregation methods by modeling the temporal graph's global evolutionary skeleton as a static thumbnail. The thumbnail is constructed by maximizing the mutual information with the raw sequence, leveraging von Neumann Graph Entropy to capture structural evolution coherently. The $G_T$ is then used as a bottleneck constraint within the Information Bottleneck principle to guide effective denoising and compression. Extensive experiments demonstrate TGT's superior performance and resilience across various adversarial perturbations, particularly against temporal interference.", "questions": "1.\tPlease elaborate on the mechanism by which the static node set $V_{GT}$ is selected or learned from the dynamically evolving node sets $\\bigcup_i V^i$. How does the mapping function $\\mathcal{F}$ and the Bernoulli sampling (L778) handle nodes that appear and disappear over time, ensuring that persistent, core evolutionary information is captured?\n\n2.\tThe complexity analysis states $L_{evolution}$ is $O(\\Delta t \\cdot |E_{GT}|)$. Since the VNGE approximation requires computing $Tr[\\tilde{L}^2]$ (L958), what structural assumptions or approximation techniques are used to achieve this near-linear complexity?\n\n3.\tSince $L_{evolution}$ maximizes $I(\\mathcal{G}_T; \\mathcal{G})$ and the IB term minimizes $I(\\mathcal{G}; \\mathcal{G}_T)$, what theoretical guidance is used for setting the Lagrange multipliers $\\lambda$ and $\\beta$ to ensure the optimization converges to a non-trivial solution that balances fidelity and compression?\n\n4.\tGiven TGT’s superior robustness against random snapshot permutations, please provide a theoretical explanation for how the VNGE-based constraint $I_s(\\mathcal{G}_T; \\mathcal{G})$ (Eq. 5) ensures the learning process is robust to chronological disorder? Does the VNGE effectively provide a temporal invariant summary that suppresses the impact of sequence tampering?\n\n5.\tGiven the severe sensitivity of clean data performance to $\\alpha$ (Table 7), what noise-aware tuning methodology would you recommend to practitioners for selecting an appropriate value for $\\alpha$ when the ground truth noise characteristics of the target application are unknown?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1.\tThe introduction of the \"temporal graph thumbnail\" concept, which summarizes the global evolutionary skeleton, is a novel approach for capturing long-term temporal dependencies and regularities often missed by localized aggregation methods.\n\n2.\tThe use of von Neumann Graph Entropy is well-justified due to its unique properties, providing a solid and principled mechanism for characterizing structural dynamics within the Information Bottleneck framework.\n\n3.\tThe empirical section is rigorous, testing robustness against three distinct types of noise at varying intensities. The results convincingly establish TGT's state-of-the-art resilience, particularly against chronological disorder.", "weaknesses": "1.\tThe mechanism by which the mapping function accommodates transient or non-persistent nodes across the entire sequence remains unclear, impacting the core interpretability.\n\n2.\tThe total loss (Eq. 16) includes two terms. I think the difference objective function between fidelity and compression is not theoretically resolved, requiring a clearer explanation of the roles of hyperparameters in balancing these constraints.\n\n3.\tThe authors claim a near-linear complexity $O(\\Delta t \\cdot |E_{GT}|)$ for the $L_{evolution}$ calculation. But the approximated VNGE computation involves spectral quantities like $Tr[\\tilde{L}^2]$, which typically necessitates non-linear complexity in $|V_{GT}|$. So I will concern about scalability for large graphs.\n\n4.\tThe sensitivity analysis for the evolutionary constraint parameter $\\alpha$ (Table 7) shows a dramatic drop in clean data performance as $\\alpha$ increases. This high sensitivity makes the parameter practically difficult to tune effectively without prior knowledge of the target data's inherent noise level.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "ofj15ctC5S", "reviewer": "Reviewer_RLpw", "summary": "The paper proposes a framework for learning representations from temporal graphs. The method constructs a thumbnail, which is defined to be a compact static summary graph, and captures the global evolutionary skeleton of a dynamic graph sequence using von Neumann graph entropy and node mutual information. This thumbnail serves as both a model constraint and a denoising guide, and enforces a balance between information compression and representational sufficiency via mutual information bottlenecks. The paper integrates this into a GAT-based architecture and evaluats it on Bitcoin, MathOverflow and MOOC datasets. The results show improvements in both capability and robustness against feature, structural and temporal perturbations compared to other dynamic graph neural networks and variants.", "questions": "- The paper acknowledges that real-time computation of von Neumann graph entropy introduces overhead and limits scalability to large graphs. Is this significant and have the authors run quantitative experiments or have insight to offer on runtime or memory usage compared to other temporal GNN baselines (e.g., TGN or DGIB)?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper introduces the TGT method that summarises a temporal graph’s global evolutionary skeleton as a static graph distilled from the sequence, using von Neumann graph entropy for structure and node mutual information for features. This shifts robust temporal GNNs away from purely local neighbor/history modeling toward an explicit global evolution model.\n\n- Beyond modelling the thumbnail, the method introduces thumbnail-guided mutual-information constraints that act as a bottleneck between raw data and task outputs. This couples global evolution with robust representation learning in a nice training objective.\n\t​\n- Experiments are strong and cover the three standard benchmarks of Bitcoin, MathOverflow and MOOC, with diverse evolution frequencies and evaluation spans inductive link prediction under clean data in Table 2 and robustness under feature, structure and temporal perturbations at multiple intensities.\n\n- The paper defines the temporal-graph setting, the thumbnail and assignment matrices and the training components. Research questions guide the narrative and figures/tables are tied to those questions.", "weaknesses": "- The contribution remains confined to improving node embeddings for downstream tasks such as link prediction. While the method introduces an interesting modelling layer (the thumbnail), it does not extend this to broader temporal-graph reasoning or applications (e.g., forecasting, anomaly detection, or event prediction) to demonstrate its significance. \n\n- All experiments evaluate link prediction only. While this is a common benchmark, it represents a narrow test of temporal representation robustness. Node classification or dynamic community detection would better demonstrate whether the thumbnail’s global-evolution modelling benefits.\n\n- The ablations test the removal of VNGE and the thumbnail, but there is little exploration of the effects of hyperparameters (e.g., window size, Beta, etc) beyond a brief mention in the appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "PfzWkEABgA", "reviewer": "Reviewer_QMEp", "summary": "The paper studies modeling of discrete-time temporal graphs and facilitates tools from information theory to construct a novel framework called TGT, that simultaneously do temporal graph property prediction as well as temporal information compression in the sense of von Neumann entropy for topological evolution as well as variational approximations to feature-related information measures. Experimental results demonstrate both the effectiveness and the robustness of the proposed framework.", "questions": "- The TGT used in the paper relies on GAT as its backbone, I am curious about the dependence on backbones.\n- Yet another recommendation: It would be nice to have a visualization of what those learned thumbnails look like and to what extent, at least in a human-understandable sense, that the learned thumbnails efficiently captures the temporal patterns in the temporal graph datasets.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper reasonably leverage ideas from information theory and proposed an elegant framework that systematically combines sophisticated approaches, with a solid foundation.\n- The empirical results and ablation reasonably verifies the design choice of TGT.", "weaknesses": "- While the overall presentation of the paper is good, the proposed framework is nonetheless complicated that would potentially benefit from a more accessible description, for example I would recommend the authors to draw a conceptual markov chain that indicates how the thumbnail compresses multiple graph snapshots, as well as how the thumbnail representation is utilized to guide (or serving as a constrained) in the representation learning phase.\n- See ``questions`` below", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "DQh1Hjsp4Y", "reviewer": "Reviewer_rkWZ", "summary": "This work presents a novel approach for modelling dynamic graph representations, focusing on link prediction, called Temporal Graph Thumbnail (TGT). The authors identify that recent methods often overlook the global evolution of temporal graphs, which limits their ability to capture long-term dependencies effectively. To address this, the paper introduces a method that models thumbnails of the temporal graph using von Neumann graph entropy and node mutual information. This thumbnail captures the global evolutionary trajectory of the graph and is leveraged as an optimization guide to learn robust and generalizable node embeddings. By incorporating global evolution information, TGT significantly enhances robustness and representation quality in dynamic graph learning.\n\nEmpirical experiments demonstrate that TGT surpasses baseline methods both in performance on clean temporal graphs and in robustness under adversarial attacks. These attacks include perturbations on graph structures, node features, and temporal ordering.", "questions": "- The concept of learning a graph evolution trajectory over time was previously explored in GraphPulse[11]. Could the authors clarify the key differences between TGT and GraphPulse in terms of approach and performance?\n\n- How does TGT scale on large-scale datasets, particularly those in the TGB[12]?\n\n- The paper employs random Gaussian noise as a node feature attack. How robust is TGT against other sophisticated feature attack methods, such as Nettack, PRBCD, SGA, and FGA, as mentioned in the identified weaknesses (See **W1**)?\n\n- Robustness evaluations focus on untargeted attacks, which might be misleading. How often do these attacks realistically perturb nodes or edges involved in link prediction? Targeted attacks on specific nodes and links provide more informative robustness assessments.\n\n- The term “snapshot” is frequently used. Is this research focused on discrete-time dynamic graphs (DTDG)? If so, what is the typical size or timespan of these snapshots?\n\n- How robust are the learned node embeddings from TGT under adversarial attacks on node property prediction tasks as defined within TGB[12]?\n\n---\n[11] Shamsi, Kiarash, et al. \"GraphPulse: Topological representations for temporal graph property prediction.\" *The Twelfth International Conference on Learning Representations*. 2024.\n\n[12] Huang, Shenyang, et al. \"Temporal graph benchmark for machine learning on temporal graphs.\" *Advances in Neural Information Processing Systems* 36 (2023): 2056-2073.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- **Novelty.** The paper introduces a novel approach that models a thumbnail of a temporal graph using von Neumann graph entropy and incorporates this thumbnail to guide optimization for learning superior and robust node representations.\n\n- **Superior Node Representation.** By leveraging von Neumann entropy to constrain the information bottleneck associated with global structural evolution, TGT outperforms seven strong baseline methods across three diverse datasets.\n\n- **Robust Node Representation.** Modeling the thumbnail and utilizing von Neumann entropy enhances robustness against various adversarial attack settings, including: Random Gaussian noise disrupting node features, Nettack adversarial perturbations on graph structure, and Permuting snapshot chronological order causing temporal disruptions.\n\n- **Ablation Studies.** clearly highlight the contributions of different model components, including evolution constraint, structure and feature mutual information, thumbnail and von Neumann Entropy.", "weaknesses": "**W1: Robustness evaluation.** The robustness evaluation of TGT is limited to random and out-of-date adversarial attacks on a static graph. To improve the strength and generality of the evaluation, it would be more convincing to evaluate the robustness of TGT under feature attack methods[1,2,3], recent structure attacks on static graphs [4,5], and adversarial attacks on dynamic graphs [6].\n\n**W2: CTDG vs DTDG.** The paper does not clearly state whether the focus is on continuous-time dynamic graphs (CTDG) or discrete-time dynamic graphs (DTDG). Including a discussion on the differences between these two temporal graph settings and explicitly highlighting which type the work addresses (ideally in Section 2) would benefit clarity and contextualize the contributions.\n\n**W3: TGNN baselines.**  The work compares the performance of TGT against out-of-date baselines. A naive but effective baseline, such as EdgeBank[7], and recent TGNN models[8,9,10].\n\n*Minor*\n\n**W4: Evaluation metrics.** The paper mainly reports AUC and AP for link prediction. Including Mean Reciprocal Rank (MRR) under the Temporal Graph Benchmark (TGB) evaluation settings would provide a more comprehensive view of TGT's ranking performance and enhance comparability with other temporal graph models evaluated on TGB[12].\n\n---\n[1] Zügner, Daniel, Amir Akbarnejad, and Stephan Günnemann. \"Adversarial attacks on neural networks for graph data.\" *Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining*. 2018.\n[2] Chen, Jinyin, et al. \"Fast gradient attack on network embedding.\" *arXiv preprint arXiv:1809.02797* (2018).\n\n[3] Li, Jintang, et al. \"Adversarial attack on large scale graph.\" *IEEE Transactions on Knowledge and Data Engineering* 35.1 (2021): 82-95.\n\n[4] Geisler, Simon, et al. \"Robustness of graph neural networks at scale.\" *Advances in Neural Information Processing Systems* 34 (2021): 7637-7649.\n\n[5] Alom, Zulfikar, et al. \"GOttack: Universal Adversarial Attacks on Graph Neural Networks via Graph Orbits Learning.\" *The Thirteenth International Conference on Learning Representations*. 2025.\n\n[6] Dai, Yue, et al. \"MemFreezing: A Novel Adversarial Attack on Temporal Graph Neural Networks under Limited Future Knowledge.\" *Forty-second International Conference on Machine Learning*. 2025.\n\n[7] Poursafaei, Farimah, et al. \"Towards better evaluation for dynamic link prediction.\" *Advances in Neural Information Processing Systems* 35 (2022): 32928-32941.\n\n[8] Yu, Le, et al. \"Towards better dynamic graph learning: New architecture and unified library.\" *Advances in Neural Information Processing Systems* 36 (2023): 67686-67700.\n\n[9] Lu, Xiaodong, et al. \"Improving temporal link prediction via temporal walk matrix projection.\" *Advances in Neural Information Processing Systems* 37 (2024): 141153-141182.\n\n[10] Ding, Zifeng, et al. \"Dygmamba: Efficiently modeling long-term temporal dependency on continuous-time dynamic graphs with state space models.\" *arXiv preprint arXiv:2408.04713* (2024).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)