DyGMamba: Efficiently Modeling Long-Term Temporal Dependency on Continuous-Time Dynamic Graphs with State Space Models
Authors: Zifeng Ding
Yifeng Li
Yuan He
Antonio Norelli
Jingcheng Wu
Volker Tresp
Michael M. Bronstein
Yunpu Ma
Venue: Accepted by TMLR ()

Abstract: Learning useful representations for continuous-time dynamic graphs (CTDGs) is challenging, due to the concurrent need to span long node interaction histories and grasp nuanced temporal details. In particular, two problems emerge: (1) Encoding longer histories requires more computational resources, making it crucial for CTDG models to maintain low computational complexity to ensure efficiency; (2) Meanwhile, more powerful models are needed to identify and select the most critical temporal information within the extended context provided by longer histories. To address these problems, we propose a CTDG representation learning model named DyGMamba, originating from the popular Mamba state space model (SSM). DyGMamba first leverages a node-level SSM to encode the sequence of historical node interactions. Another time-level SSM is then employed to exploit the temporal patterns hidden in the historical graph, where its output is used to dynamically select the critical information from the interaction history. We validate DyGMamba experimentally on the dynamic link prediction task. The results show that our model achieves state-of-the-art in most cases. DyGMamba also maintains high efficiency in terms of computational resources, making it possible to capture long temporal dependencies with a limited computation budget.

Published in Transactions on Machine Learning Research (05/2025)
DyGMamba: Efficiently Modeling Long-Term Temporal De- pendency on Continuous- Time Dynamic Graphs with State Space Models
Zifeng Ding" University of Cambridge
zd3200cam.cc uk
Yifeng Li Technical University of Munich
seffnn.li@ltm.ccm de
Yuan He University of Oxford
yuan.he:cs.oor uk
Antonio Norelli University of Oxford
ontommio.norelli@ess.oo
Jingcheng Wu University of Stutigart
jingcheng, wwwkii uni-ssuttgrssut..
Volker Tresp LMU Munich
tresp@dbs..jp ift. de
Michael Bronstein University of Oxford
michael, www.sbbeeeeenssoooo.c...ooo
Yunpu Mat
cognitive. Seriiiiie..uupuuugggnn com
LMU Munich, Munich Center for Machine Learning
Reviewed OpenReview: https: //
https://opennevieeu openreview. net/ fot/ffrmm? id= sg5AJuVuua
Abstract
Learning useful representations for contimuous-time dynamic graphs (CTDGs) is challeng- ing, due the concurrent need to span long node interaction histories and grasp nuanced temporal details. In particular, two problems emerge: (1) Encoding longer histories re- quires more computational resources, making it crucial for CTDG models to maintain low computational complexity to ensure efficiency; (2) Meanwhile, more powerful models needed to identify and seleet the most critical temporal information within the extended context provided by longer histories. To address these problems, we propose a CTDG rep- resentation learning model named DyGMamba, originating from the popular Mamba state space model (SSM). DyGMamba first leverages a node-level SSM to encode the sequence of historical node interactions. Another time-level SSM is then employed to exploit the temporal patterns hidden in the historical graph, where its output used to dynamically select the critical information from the interaction history. We validate DyGMamba exper- imentally the dynamic link prediction task. The results show that our model achieves state-of-the-art in most cases. DyGMamba also maintains high efficiency in terms of compu- tational resources, making it possible to capture long temporal dependencies with a limited computation budger
"Equal Contribution. Zifeng's work done during the visit at the University of Oxford. i "Corresponding author. We release our code the following link:
T7fffenggDigg/DGGMania.... DyGMamba.

---

Published in Transactions on Machine Learning Research (05/2025)
1 Introduction
Dynamic graphs store node interactions in the form of links labeled with timestamps (Kazemi et al. 2020 In recent years, learning dynamic graphs has gained increasing interest since can be used to facilitate various real-world applications. Dynamic graphs can be classified into two types, i.c.; discrete-time dynamic graph (DTDG) and contimuuus-time dynamic graph (CTDG). A DTDG is represented as sequence of graph snapshols that are observed at regular time intervals, where all the edges in a snapshot are taken as existingg simultaneously, while a CTDG consists of a stream of events where each of them is observed individually with its own timestamp. Previous works (Kazemi et al. 2020 Shirzadkhani et al. 2024) have indicated that CTDGs have an advantage over DTDGs in preserving lemporal details, and therelore, more attention is paid to developing novel CTDG modeling approaches for dynamic graph representation learning.
Recent effort in CTDG modeling has resulted in wide range of models. However, most of them are unable to model long-teng long-term temporal dependencies of nodes, despite the existence of abundant historical information. To solve this problem, Yu et al.] (2023) proposc CTDG model DyGFormer that can handle long-ten torm-term node interaction histories based on Transformer (Vaswani et al. 2017). Despite its ability in modeling longer histories, employing a Transformer naturally introduces excessive usage of computational resources due to its quadratic complexity. Another recent work CTAN (Gravina et al. 2024) tries to capture long-term temporal dependencies by propagating graph information based model. Despile the model's high efficiency, Gravina el. al. (024) show that CTAN cannol caplure very long histories and is surpassed by DyGFormer on the CTDGs where learning from very far away temporal information is critical. Based on these observations, we summarize the first challenge in CTDG modeling: How to develop a model that is scalable in modeling very long-term historical interactions? Another point worth noting is that as longer histories introduce more temporal information, more powerful models are needed to identify and select the most critical parts. This reveals another challenge: How to effectively select critical temporal information with long node interaction histories?
non-dissipative way over time with a graph convolution-
To address the first challenge, we propose to leverage popular state space model (SSM), i.c., Mamba SSM (Gu & Dao, 2023 to encode the long sequence of historical node interactions. Since Mamba is proven effective and efficient in long sequence modeling (Gu & Dao 2023). it maintains low computational complexity and is scalable in modeling long-term temporal dependencies. For the second challenge, we address it by learning temporal patterns of node interactions and dynamically selecting the critical temporal information based on them. The motivation can be explained by the following example. Consider a CTDG with nodes as people or songs and edges representing person playing a song at a specific time. If person u frequently plays a hit song e initially but decreases the frequency later on, the time intervals between plays increase. Ignoring this pattern can lead models incorrectly predict that will still play at future timestamps due to their high appearances in each other's historical interactions. If CTDG model recognizes this pattern, it can prioritize other temporal information, such as u increasingly listeming to new song v' before t, instead of focusing on u, e interactions. Since each pattern corresponds to a specific edge, c.g., (u, U. t), we name these patterns as edge-specific temporal patterns.
To this end, we propose a new CTDG model named DyGMamba. DyGMamba first leverages a node-level Mamba SSM to encode historical node interactions. Another time-level Mamba SSM is then employed to exploit the edge-specific temporal patterns, where its output is used to dynamically select the critical information from the interaction history. To summarize: (1) We present DyGMamba, the first model using SSMs for CTDG representation learning; (2) DyGMamba demonstrates high efficiency and strong effectiveness in modeling long-term temporal dependencies in CTDGs; (3) Experimental results show that DyGMamba achieves new state-of oftt----tttttt on dynamic link prediction over most common CTDG datasets.
2 Related Work and Preliminaries
2.1
Related Work
Dynamic Graph Representation Learning. Dynamic graph representation learning methods can be categorized into two groups, i.e., DTDG and CTDG methods. DTDG methods (Pareja al. 2020 Goyal
2

---

Published in Transactions on Machine Learning Research (05/2025)
et al. 2020 Sankar et al. 2020 You et al. 2022: Li et al. 2024a] can only model DTDGs where each of them
is represented as sequence of graph snapshots. Modeling a dynamic graph as graph snapshots requires time discretization and will inevitably cause information loss (Kazemi et al. 2020). To overcome this problem, recent works focus more on developing CTDG methods that treat a dynamic graph as a stream of events,
where each event has its own unique timestamp. Some works (Trivedi et al. 2019 Chang et al. 2020 model
CTDGs by using temporal point process. Another line of works (Xu et al. 2020; Ma et al. 2020 Wang
iet al., b Gravina et al. 2024) designs advanced temporal graph neural networks for CT Besides,
some other methods are developed based memory networks (Rossi et al. 2020 Liu et al., 2022). temporal
random walk (Wang et al. 2021c Jin et al.] 2022 and temporal sequence modeling (Cong et al.
2023 Yu
ltt al. 2023 Tian et al.) 2024). Since some real-world CTDGs heavily rely on long-term temporal information
for effective learning, a number of works start to develop CTDG models that can do long range propagation
of information over time Yu et al.
2023 Gravina et al.
2024
State Space Models. Transformer Vaswani et al. 2017 7 is de facto backbone architecture in modern deep learning, However, its self-attention mechanism results in large space and time complexity, making it unsuitable for extremely long sequence modeling (Duman Keles et al.) 2023). To address this, many works focus on building structured state space models that scale linearly or near- lear-linearly with input sequence length
(Gu et al., 2021 2022b a Smith et al.) 2023 Peng ct al.) 2023 Ma et al. 2023 Gu & Dao 2023). Most
structured SSMs exhibit linear lime invariance (LTT), meaning their parameters are not inpul-dependent and fixed for all time-steps. Gu & Dao (2023) demonstrate that LTI prevents SSMs from effectively selecting relevant information from the input context, which problematic for tasks requiring context-aware reasoning. To solve this issue, Gu & Dao (2023) proposes S6, also known as Mamba, which uses selection mechanism to dynamically choose important information from input sequence elements. Selection mechanism involves learning functions that map input data to SSM's parameters, making Mamba both efficient and effective in modeling language, DNA sequences, and audio.
State Space Models for Graphs. Recently, there have been several works employing Mamba SSM for representation learning static graphs (Wang et al. 2024: Behrouz & Hashemi 2024). Wang et al. (0244) integrate a Mamba SSM into graph transformers, employing input-dependent node selection and prioritization mechanisms to efficiently capture long-range multi-hop dependencies in static graphs. [Behrouz] & HHsshemi (2024) adapt Mamba by creating a bidirectional SSM for static graphs, utilizing hierarchical tokenization and an optional positional encoding mechanism to improve scalability and performance on diverse graph-based downstream lasks. Both of them aim to efficiently encode long-range multi-hop graph information. Unlike our work, they do not have dedicated structures to process temporal information and are thus not suitable for CTDG modeling. There is also another work {Li et al. 2024b) that tries to use Mamba to model spatial-temporal graphs (STGs). Li et al. (2024b) employ Oppatil-Tempprt Selective State Space Module for adaptive spatial-temporal feature selection and Kalman Filtering Graph Neural Networks to dynamically integrate and optimize embeddings from various temporal granularities. Although this work achieves high efficiency and superior forccasting accuracy on STGs, it is not applicable to modeling CTDGGs because (1) STGs are inherently DTDGs, and (2) this model primarily aims at predicting node states in STGs (e.g., traffic sensor readings) rather than explicitly capturing the continuous and irregular structural evolution characteristic of CTDGs. In our work, we propose Mamba-based model to capture evolving graph dynamics by encoding long-range historical event streams and introducing a specialized module designed explicitly to encode edge-specific temporal patterns. This approach significantly differs from existing Mamba-based graph reasoning methods, enabling more effective temporal reasoning on CTDGG.
2.2 Preliminaries
CTDG and Task Formulation. We define CTDG and dynamic link prediction follows.
Definition 1 (Contimnous-Time Dynamic Graph). Let N and T denote a set of nodes and timestamps, respectively- A CTDG is sequence of [9] chronological interactions g {(1,, ((u,,,;44)) 41) with 0 t $ ta ... (g.,, where Wi V, E N are the source and destination node of the i-th interaction happening at t4 E T, respectively. Each node u E N con be equipped with a node feature x,, E Rdy, and each interaction (u, v,
3

---

Published in Transactions on Machine Learning Research (05/2025)
be associated with link (edge) feature eyu Rd If g is not attributed, we set node and link features to zero vectors.
Definition 2 (Dynamic Link Prediction). Given CTDG 9, a source node u E N, destinalion mode N,, , timestamp t E T, and all the interactions before t, i.e,, (, Ui,ti)tt < t, (us Vs 4)) E G}, dynemic link prediction aims to predict whether the interaction (u, u. () erists.
$4 and Mamba SSM. S4 and Mamba (Gu et al. b022b Gu Dao 2023 are inspired by a continuous system which can be described as z(r) Az + Bq(T) and r(T) Cz(T). q(T) R and r(7) E R are the 1-dimensional input and output over time respectively. A E Rdi xd,, B E Rddx11 C E Rixd, are three parameters deciding the system. Based on it, both S4 and Mamba include time-scale parameter A R and discretize all the parameters to adapt to discretized system
2 Az,- + Bpr,
Cz+; A = AA)( B (Ax)) - I)AB
(1)
Here, I is also discretized to denote the position of a sequence element. Given Eq. with S4 and Mamba can be written as computing an output sequence with convolution
|
sequence processing
q q p KSSM, where KSSM = (CB, CAB, ..., CÀ PRI-11
(2)
P E Iill and q E RiP are input and output sequences, where |p| is the sequence length of p. * denotes the dementtwise multiplication. When the dimension size of each element Pr in p becomes higher (i.e., Pr E Rdd2 vector and d2 > 1), both $4 and Mamba are a Single-Input Single-Output (STSO) fashion, processing each input dimension in parallel with the same of parameters. We follow Gu & Dao (2023) and denote the computation in Eq. 2 on the input sequences with vector elements function SSM A.MA.B.C( Different from S4 which uses same parameters to process each element, Mamba changes its parameters into input-dependent by employing several trainable lincar layers to map input into B, C and A. The system is evolving as it processes different elements in the input sequence, making Manba lime-variant and suitable for modeling temporal sequences.
3 DyGMamba
(u, v. ?
MLP Predictor for Downstream Tasks, e.g., Dynamic Link Prediction
CTDG
Node Output MLP
Time Output MLP
Node Output MLP
Temporal Pattern-Aware Dynamic Information Selection
u's Temporal Neighbors
v's Temporal Neighbors
Node-Level SSM Block
Time-Level SSM Block
Nove- SSM Block
u's Encoded Neighbor Features
v's Encoded Neighbor Features
Feature Encoder & Patching
Time Difference Features
Figure 1: Model overview of DyGMamba.
Fig. 1 iillstrrates the overview of DyGMamba. Given potential interaction (u, v, t), CTDG models asked to predict whether it exists not. DyGMamba extracts the historical one-hop interactions of node and v before timestamp t from the CTDG G and gets two interaction sequences St. = {(u, u', t')]t'' < t, (u, t') e G)U(("' u, l')lt' < 1, (u', M, u, u,,') E 9) and S = {(v,' d'(')) < L, (u,,o,'') E GGU e, ())( < (v', E G} containing u's and v's one-hop temporal neighbors Net, = {(w', ((w't')) (u) u', ,') or (u', u!t E G, < t] and Ne = {(v', (1)((, v", o',()) or (v',v, v,(") E 9} (link features are omitted for clarity). Then encodes the neighbors in Net, and Net, to get two sequences of encoded neighbor representations for u and To learn
We 7 rather than indicate time in continuous system distinguish from the time in CTDGs. Input and output of SSM (SMA.B. (.B.C()) are matrices where each row vector corresponding element. See App. more details SISO and the function.
4

---

Published in Transactions on Machine Learning Research (05/2025)
the edge-specific temporal pattern of (u, v, t), we find the interactions between and before t, compute the time difference between each pair of neighboring interactions, and build a sequence of time differences Su Finally, DyGMamba dynamically selects critical information by assigning different weights to different encoded neighbors based on the learned temporal pattern, and uses the selected information to achieve link prediction.
3.1 Learning One-Hop Temporal Neighbors
Encode Neighbor Features. Given one-hop temporal neighbors Net, of the source node we sort them in the chronological order and append (u, t) at the end to form a sequence of Ned, +1 temporal nodes. We take their node features from the dataset and stack them into feature matrix X E [ +1)xdy, Similarly, we To incorporate temporal information, we encode the time difference between u and each one-hop temporal neighbor (u', tt) using the time encoding function introduced in TGAT (Xu et al. 2020): (1/dr(cos(u1
build & link feature matrix E, E [B((Ned))+1)xd
cos(waatt (ost)).)...,,(((((ttt t') + Φdr)] dr is the dimension of time representation. W1 War and Φ1 Odr are trainable parameters. The time feature of u's temporal neighbors are denoted as TV E 11)der We follow the same way to get X +1)xd E e [I +1) and T E I((NNE[++))xdd for v's temporal neighbors. Following Tian et al. (0024), we also consider the historical node interaction frequencies in the interaction sequences S and S of source u and destination v. For example, assume the interacted nodes of u and (arranged in chronological order) are {a, a} and {b, b, a), the appearing frequencies of a, b in u/v's historical interactions are 2/1, 0/2, respectively. And the frequeney of the interaction involving u and v is 1. Thus, the node interaction frequency features of t and v are written as Ft = [[2, 1), [1, 1), 12.1),10 [0. T]]T and F: = [[o. 2). [o, 2), 1), (2. 1), TJ], respectively. Note that the last elements ((O, 1] and [0, 1]) in F and Ftt correspond to the appended (u, t) and (v, t) not existing in the observed histories. We initialize them with [0, number of historical interactions between u, v). An encoding multilayer perceptron (MLP) f()) R Rd is employed to encode these features into representations:
F 4 - (( 0J) + (P.1:1)) E IR(MEEI)))ddd
- = 0)) + S(P 1]) E (((Nee)
Patching Neighbors. We employ the patching technique proposed by (Yu et al., 2023) to save computa- tional resources when dealing with a large number of temporal neighbors. We treat p temporally adjacent neighbors as patch and flatten their features. For example, with patching, X E (R results in a new patched feature matrix X E R () (we pad X with zero-valued features when |ee + 1 cannot be divided by p)- Similarly, we get E E [[[Nee])++pplxxpdddd
T2 E R [[NNe[)+1)/))xxx and F2 [ [[1(Ne+++1)/pl x(p-dx) (9 is either u or v). Each row of feature matrix corresponds to an element of the input sequence sent into SSM later. Recall that SSMs process sequences in recurrent way. Patching decreases the length of the sequence by roughly times, making great contribution in saving computational resources.
Node-Leval SSM Block. We first map the padded features of u's and e's onc-hop temporal neighbors to the same dimension d, ie., == Jx(h): E = (E,), T2 == (TT,, F1 := (r())). fx(+) : RR + Rd fE(+) Bg + Rd, fr(-) dr + Rd fr(-): Rr + Rd are four MLPs for different types of neighbor features. We take the concatenation of them as the encoded representations of the temporal neighbors, i.e., H. X E TT to learn the temporal dependencies of temporal neighbors. The node-level SSM block consists of Ix layers, where each layer is defined as follows (Eq. 3 First, we input HI into a Mamba SSM
([((N)()+1)/))x4) We input H. and H. separately into node-level SSM block
B = WB, B, E [1NNe)]+11/pp)ddss C = H&W., E (R[(NNe((+1)p
(3a)
A Softplus(rroadcastt.FF (H, Parama,) E TR TO +1)/p]x4d, A = Ax)(AIA1), B, = (AIAA) (a (exp(A B)A;
(3b)
(3c)
H == H + SSM SSM C, (H)).
(3d)
WB,, We, E daxxdasu and WA, E R4xx1, A,, B, E R [[(Ne(1+1)/) x4dx are discretized parameters.
Parama, [1(NNe(111/// is a parameter defined by Gu & Dao (023). Broadcast is function that
5

---

Published in Transactions on Machine Learning Research (05/2025)
copies its vector input for 4d times to form matrix with 4d identical columns (following the definition in Gu & Dao 20233 ). I is an identity matrix. Then we use an MLP Smode()) R2dd - R on SSM's output
H = Ha + fuode (TayerNorm(H:))
After Iy layers, we have H. and H. that contain the encoded information of all one-hop temporal neigh- bors for the entities and as well as the information of themselves. Since we sort temporal neighbors chronologically, our node-level SSM block can directly learn the temporal dynamics for graph forecasting.
3.2 Learning from Sllge-Specffic Temporal Patterns
Live-Level SSM Block. To capture edge-specific temporal patterns, we use another time-level SSM block consisting of IT layers. We first find out temporally nearest historical interactions between u and v before and sort them in the chronological order, i.e., {(u, v, to)..... (u, v, f4-1)) < ... < fx-1 f}. Then we construct timestamp sequence {to, (4..11,.,.111,. based on these intcractions and the prediction timestamp L. We compute the time difference between each neighboring pair of them and further gel a time difference sequence {t1 to. f2 - t1,..., t- -t-11,, representing the change of time interval between two identical interactions. Each element in this sequence is input into the time encoding function stated above to get a edge-specific (specific to the edge (u, U))) time feature. The features are stacked into a feature matrix H u,v E Rxxdr and mapped by an MLP [mmp1()) : Rdd - Rnd (7 E [o, 1] a hyperparameter)) i.e.,
H == fwap1 Swapi (H, A time-level SSM layer takes Hy, as input and computes
B₂ H WB, Rexdess C2 H Wc, R RexdaMM
(5a)
22 Softplus(Broadent (Broadcast (H u,u Wa,) + Parama,) E gexnd
(5b)
A = A2)( B,
- I)A,B2
(5c)
H H + SSSMSM C,, (H.
(5d)
WB,, Wc, E RRddxdss and WA, E Rıdx1_ Az, B2 E RexxdddesM are discretized parameters. Parama, E glxyyd is a parameter defined as same as Parama, In practice we set k to a number much smaller than Nedll e.g., 10. This ensures that time-level SSM will not incur huge computational burden and the model focuses more on the recent histories. Note that we cannot always find recent historical interactions wetween each pair of nodes, leading to varying lengths of time difference sequences for different (u, v,t,) t) a batch of data. To enable batch processing, we set the time difference without found historical interaction to very large number 1000 For example, if k = 2, and for (u, v, we can only find (u, 2, to). The time difference sequence will be {10'0,t t.). 1010 is much larger than t t- to. indicating that u and have not had an interaction for an extremely long time, same as existing no historical interaction. We further explain why we use SSM to learn temporal patterns in App. M
Dynamic Information Selection with Temporal Patterns. After the lime-level SSM block, we com- pute a compressed representation to represent the edge-specific temporal pattern by averaging over k encoded time intervals: h McanPooling tern specific to the edge (u, v, (). To leverage learned temporal pattern, we use to dynamically select the information from the encoded temporal neighbors H₃
As result, we have h Rnd to represent the temporal pat-
hi = /map2(h', E BRdd
(6a)
hi = Wagg THh- E Rdd, where Wagg = fmapa (Hz) E R (1Ne61+1//p),
(6b)
au f(hh)) hi E Rdd, a (b()) hig E gdd;
(6c)
h = Be H where So = Softmax(H/ E R [((Ne((11)/p))
(6d)
fmap2()) : Rdd + 4d and [-)pa(+) Rddd + Rd are two mapping MLPs. f'(+) : R - Rdd is another
MLP introducing training parameters. Note that aw/a, is computed by considering both the edge-specific
temporal pattern and the opposite node v/u. In the node-level SSM block, we separately model the one-hop
6

---

Published in Transactions on Machine Learning Research (05/2025)
temporal neighbors of each node 0, making it hard to connect and v. Computing ag Eq. 6c helps to strengthen the connection between both nodes and meanwhile incorporates the learned temporal pattern. f0 is derived by transforming the queried results based on ag into weights. It is then used to compute a weighted- sum of all temporal neighbors for representing 8 at t, i.c., hg. The neighbors assigned with greater weights from Be are selected as more critical and will contribute more to hý- Finally, we output the representations of u, v and the edgo-specific temporal pattern by employing two output MLPs somal()) : R -> Rddx and (+u Rnd Rdd, i.e., hg == foucs (by) Rdv, h fo E Rdd.
3.3 Leveraging Learned Representations for Link Prediction
We leverage h2 and h for dynamic link prediction. We employ a prediction MLP, i... fLP(t) Radx - R,, as the predictor. The probability of existing a link (u, is computed as v' = Sigmoid(fir (h/, h//h ())))))) For model parameter learning, we use the following loss function
L
1
Σ 2M
(vlu, v, logls/ u, ()) + (1 - y(u, v, 1)) log(1 -y vl(,,,,))))
2M
y(u v,() is the ground truth label denoting the existence of (u, v, 1) (1/0 means the total number of edges existing in the training data (positive edges). We follow previous work (Yu et al. 2023) and randomly sample one negative edge for each positive edge during training. Therefore, in total we have 2M edges considered in our loss L.
M is
4
Experiments
In Sec. 4.2.1 we validate DyGMamba's ability in CTDG representation learning by comparing it with baseline methods on dynamic link prediction We show the effectiveness of model components by conducting ablation studics (Sec. 4.2.2p and analysis on synthetic datasets (Sec. 4.2.3.. In Sec. 4.3.1 we show
DyGMamba's 's elficiency against various baselines. We also show that it achieves much stronger scalability in modeling long-term temporal information compared with the current state-of-the-an DyGFormer (Sec.
4.3.2 and Sec. 4.3.3
4.1
Experimental Setting
CTDG Datascts and Basclines. We consider seven real-world CTDG datascts collected by (Poursafaci et al. )022), i.e., LastFM, Enron, MOOC, Reddit, Wikipedia, UCT and Social Evo.. Dataset statistics are presented App. A.I Among them, we take LastFM, Enron and MOOC as long-range temporal dependent datasets because according to Yu et al. (2023), much longer histories are needed for optimal representation learning on them. We compare MyGMa with ten recent CTDG baseline models, i..., JODIE (Kumar
et al.3 2019). DyRep (Trivedi et al. 2019), TGAT (Xu et al., 2020). TGN (Rossi et al. 2020). CAWN (Wang et al. 2021ch, EdgeBank (Poursafaei et al.) 2022). TCL (Wang et al. 2021a). Graph (Cong et al.
2023), DyGFormer (Yu el al. 2023) and CTAN (Gravina et al. 2024). Among them, only Dy Former
and CTAAN are designed for long- range temporal information propagation. Detailed descriptions of baseline mothods are presented App. B We also implemented FreeDyG (Tian et al. 2024) by using its official
code repository, however, on LastFM, we find that FreeDyG's loss cannot. converge and the reported results are not reproducible. So we do not report its performance in our paper.
Implementation Details and Evaluation Settings. We use the implementations and the best hyper- parameters provided by Yu et al. (2023) for all baseline models except CTAN. For CTAN, we use its official implementation, fixing the number of layers to 5. All models are trained with a batch size of 200 for fair efficiency analysis. For DyGMamba, we report the number of sampled one-hop temporal neighbors p and the patch size p here. On Wikipedia, Social Fvo., and UCT, p & p = 32 1. On Reddit, p & p = 64
"To supplement, we also validate on the dynamic node classification task. Since current mainstream datascts of this task requires long-term temporal reasoning, we put the discussion App. F Additionally, we also benchmark DyGMamba DTDGs in App. This serves supplementary experiment and does not directly connect to our main focus.
7

---

Published in Transactions on Machine Learning Research (05/2025)
2. On MOOC, p & p 128 & 4. On Enron, p & p =256 & 8. On LastFM, p & p 512 & 16. Note that to fairly compare DyGMamba's 's efficiency with DyGFormer, we keep the sequence length p/p input into the SSM as same as the length input into Transformer in Yu et al. (2023), i.e., p/p = 32. All experiments are implemented with PyTorch (Paszke et al. 2019 on a server equipped with an AMD EPYC 7513 32-Core Processor and a single NVIDIA A40 with 45GB memory. We run each experiment for five times with five random sceds and report the mean results together with error bars. Further implementation details including complete hyperparaameter configurations are presented in App. C We employ two evaluation set tings follow- ing previous works: the transductive and inductive settings. As suggested in (Poursafaei et al.) 2022), we do link prediction evaluation using three negative sampling strategies (NSSs): random, historical and inductive. Historical NSS is only considered under the transductive setting. See App. for for detailed explanations. We employ two metrios, i.e,, average precision (AP) and area under the receiver operaling characteristic curve (AUC-ROC)
Table 1: AP of transductive dynamic link prediction. The best and the sccond best results are marked as bold and underlined, respectively. CTAN cannot be trained before 120 hours timeout on Social Evo. so is ranked bottom on this dataset.
Datasets
JODIE
DyRop
TGN
CAWN
EdgeBank
TCL
CraphiMixrr
DyGmerr
CTAN
DyGMumba
LavtFM
70.56
+ 2.91
71.83
2:51
73:30
+ 0.15
75:31
+:11-:100 5.62
85.00 + 0.11
79:10 + 0.30
70.62
1:53
75.36 0.19
97:05
+ 0111
86:11 + 0.30
93.35
0:20
Random
MOGC
Ezzoa
84.55
81.04
8: 0.53
4 3
81.00
75,00
2:00::000 0.77
2:26
85.7
70.70
4: 0.20
4: 1.05
30
86.08
1.05
1.69
80.50 2: 0.10
80.50
0.43
50.53
57.07
: 0.30
: 0.30
83.50
S5.41
2: 0.71
0.90
82.80
62 2 0.30
0.35
12
87.06
2: 0.11
0.58
84:71 2 2.55
02.02 6: 1.20
80.21
02.65
0.08
0.12
Reddit
06.31
8: 0.06
08.18
0.03
05.07
2: 0.01
08.65
0.64
96.11
0.01
94.56
2 0.30
97.78
0.00
67.31 2 0,01
06.22
4: 0.01
97.21
8 0.54
09.32
0.01
Wilrpwllin
16.51
0.22
01.58
0.29
96.85
0.06
98
0.10
08-17
0.01
50:37
0.30
16 (7:75-1
0.01
97.92
0.00
99.03
0.0
96.61 0.70
99.15
0.02
UCE
06.11
2.75
92:33
1.30
50.15
0.41
05,74 1 0.17
06.91
Social
98:45
74.96
0.10
98:36
Tinasnt
94.77
Rank
4.29
6,00
5.97
6,41
3,00
LaotiFM
F1.38
71.83
2.91
71.00-1
0.36
0.90
08:03
0.33
11.00
3:07
72.38
0.37
81.51 0.11
82:00 0.30
83.02
0.10
Eltsorkol
MODC
Bicoos
08.33
78.02
2:43
1.05
73:14
72.04
2.55
1.83
88.83
64.24
4: 1.24
4: 0.73
65.
74:33
06.05-4
2.32
0.59
14.00
63.000
0.53
0.36
(0.71
76.53
0.00
0.201
78-51
72. 10.301
0.66
1.24
17:30
77
0.83
1.222
85. 15 + 0.99
76.00 +: 0.76
07.75
27.24
1.59
2.008
85.89
77.77
0.94
1.32
Reddit
79.56
8: 0.90
79.40
3.90
79.78
4: 0.25
61.00
0.32
90.06
0.25
78.30
= 0.20
77.38
0.20
78.30
0.40
RL.4 2: 1.05
89.77
=: 2.28
$1.50
1.52
Wiktpudiis
61.16
0.73
78.
0.95
87.31
+ 0:36
67:31
0.25
09.77
6.07
75.35
0.00
60.12
1.00
90:24
0.00
70 11.00
98.91
0.10
51:77
1:30
UCE
74.77
a 5.35
52,89
2.83
06.76
+ 0.77
61.32
1.26
64.00
1.78
05.50
0.00
74.02
2.70
83.58
1.06
90.44
+: 1.16
76.02
0.32
51.03
1.00
Social Loo.
M1.00
2:47
M2,80
0.00
90.33
4: 1:30
1.68
85.003
0.11
50.37
0.00
145.90
0.85
90.30
0.33
MT.04 + 0.16
TTmooot
97.35
0.02
Avg. Ronk
6.57
5.14
6,27
4.34
9:29
8.71
7,00
4.71
5.000
4.71
2:14
LawiiFM
5.01
7:05
0..L
67.87
0.35
72.00 0.05
80.08
0.85
75:65
0.54
Indactiv
Eusua
06.78
0.000
0.80
60.56
MODC
1:41
74/14
0.77
90:00
81.11
Roddi
96.50
0.36
0.13
01.80
0.18
56.57
0.30
65.37
0.26
01.00 3.00
50.90
2.19
91.15 0.54
Wikipoliis
74.78
70:55
1.23
77
+: 0.29
55
0.15
06:27
50.88
03.54-1
4.00
58.31
0.20
62.00
14.00
94.18
0.08
79.86
2.16
UCE
06.00
61.61
2.62
67
4:
70:34
0.72
61.00
57.15
78:49
2.21
79.57
00.51 4: 1.83
66:25
2.51
Social Eoo.
01.08:1
3.39
02. 07.84.4
0.88
00.
4: 0:30
94.35 53
1.52
80.50
0.13
83.00
0.00
14:10
0.65
0.30 95.
97 4: 0.12
Timooot
97.68
0.42
Avg. Bank
9.67
6.48
0.67
7.57
6.00
8.00
5.00
6.71
2.45
Table 2: AP of inductive dynamic link prediction. EdgeBBnn cannot do inductive link prediction so is not reported.
NS8
Datasete
JODIE
DyRep
TGAT
TGN
CAWN
TCL
GraphMixer
DyGFormer
CTAN
DyGManba
Lastf'M
1.19
83:47
1.00
78:40
0.50
3:37
89:33
+ 1.55
62.07
94.17 0.10
3.0
94.42
Rambom
Farom
78.07
1.59
73.97
3.00
65.67
78.75
1.60
95.30
0.56
82.61
+ 0.61
75.55
0.81
88.02 0.27
74.61 + 1.64
89.67
0.27
MO0C
80.57
+ 0.52
80.00
+ 0.68
85.28
= 0.30
88.01
+ 1.48
$1.32
0.42
82.25 + 0.99
81.58
0.17
87.05 + 0.51
61.99 + 2:24
88.64
0.08
Reidit
96.43 1: 0.16
96.80
0:26
07.13
1:03::0.0 0.04
07.41
+ 0.12
08
0.01
95.01
+ 0.10
95.21
0.08
96.85 + 0.02
50.07 +: 2:53
08.97
0.01
Wikipodii
92.21
96
96-07
96.61
0.01
0.66
98.17
UCI
79.73
1,68
58.39
1 238
79.00
1 0.69
87.81
1 1.32
92:61
0.35
86.19 1 1.37
91.17 0.20
94.45 1. 0.13
49.78 1 5.02
04.76
0.19
Social Ewo.
91.72
+ 0.66
60.10 + 1.50
91.47
0 0.10
90.74
+ 1.40
79.83
0.14
92.51 + 0.11
91.89 +: 0.05
90.05 +: 0.10
Timmet
93.13
0.05
Avg. Rank
6.29
8.00
7.00
5.14
4.43
5.57
5:86
2.14
9.57
1.00
Lasti'M
71.37 a: 3.45
09.75 à: 2.73
79.26
01.34
08.47
4. 0,07
71.28
0.43
08.79 J. 0.93
76.47 J: 0.31
75.07 J: 1.45
55,60 :: 3.91
76.76 =: 0.43
Iostaesiow
Earon
60.00
+ 1.11
62.64
+ 2.33
10.95
1.00
64.51
+ 1.66
03.61
0.63
68.93 + 1.34
71.71
1.33
67.21
0.77
68.66 + 2:31
69.77 + 0.60
MOGC
6167
1.18
02.00
77.48
2.84
74.36
70.95
1.46
71.N7
0.94
1.34
80.76
1.00
Recdit
62.54
0.52
61,7
0.86
63.96
0.25
05.27
1 0.57
64.10
61.45
0.25
04,82
0.30
65.03
T8.35
1.05
Wikipodia
68:22 + 0.96
61.07
+ 0.82
84.19
0.36
81.96
+ 0.62
02:34
6.79
71.46 + 4.95
67.47
+: 0.25
57.90
11.06
92.61 = 0.90
71.14 + 2.44
UCI
63.57 4: 2.15
52.63
1.87
09.77
01.43
031.44
4: 0.50
61:44
1.52
74,39 4: 1.81
81.40
0.52
70:25 6: 2.022
52.31 #: 2.67
72.17 4: 2:220
Social
67.30
80.30
94.56
Timmet
96.83
Avg. Rank
0.85
8.57
5.29
7.43
4.86
3.14
4.43
4.2 Performance Analysis
4.2.1
Comparative Study on Benchmark Datasets
We report the AP of transductive and inductive link prodiction in Table 1 and 2
(AUC-ROC reported in
Table 12 and 13 in App. Ej. We find that: (1) DyGMamba constantly ranks top under the random NSS,
8

---

Published in Transactions on Machine Learning Research (05/2025)
showing a superior performance; (2) Under the historical and inductive NSS, DyGMamba achieve the best average rank compared with all baselines. More importantly, it shows more superiority on the datasets where encoding longer-term temporal dependencies is necessary, e.g., on LastFM, Enron and MOOC. (3) Among the models that can do long range propagation of information over time (i.c., DyGFormer, CTAN and DyGMamba); DyGMamba achieves the best average rank under any NSS setting in both transductive and inductive link prediction. On the long-range temporal dependent datasets, DyGMamba outperforms DyGFormer and CTAN in most cases; (4) CTAN achieves much better results in transductive than in inductive link prediction. This is because CTAN requires multi-hop temporal neighbors to learn node representations, which is difficult for unseen nodes. By contrast, DyGMamba. and DyGFormer require only one-hop temporal neighbors, thus performing much better in inductive link prediction.
Table 3: Ablation studies under transductive setting. R/H/I means random//uistoriinn panddom//iittorica//nnduiiii NSS. Metric is AP.
Dataseta
LuxFM
Euron
MOGC
Roddit
Wikipedia
UCI
Social
Models
Varaut
98.04
71.30
01.35
55:46
63:25
17.04
61.60
90..00
80.00
91.55
91.50
50.12
9017
7247
73.51
73:47
81.15
M1.37
71.13
M2.10)
17.34
Varint
92.71
82.55
72.86
02.40
78.64
88.90
85.23
51.02
51.74
91.05
39.06
79:14
73.49
96.55
81.00
91.71
97.25
Varane
99.24
53-57
72.65
00:52
77.07
78-05
88.71
85.26
51:00
99:97
89:
91.07
38.08
81.33
79.79
05:57
82.33
72.98
91.74
97.17
Variant E
65.80
633.98
70.09
80.20
70.85
55.45 42
83.25
82.18
70.06
99.00
82:31
01.02
98.52
81,21
76.62
05.12
88211
71.04
04.38
96.07
57,54
Varion F
67.47
07.11
64.10
68.
09.16
63.98
82. 42
76.18
74.12
90.05
76.08
65.75
98.76
72.41
70.05
94.74
65.91
63.50
91.17
96.15
06.20
DyCMumba
93.33
83.02
73.63
92.65
77.77
80.40 96
89.21
85.89
81.11
09.32
$1.50
01.15
99.15
81.77
79.56
85.01
81,03
71.95
04.77
97.35
97.68
Table 4: Ablation studies under inductive setting. R/I means random/induutee NSS. Metric is AP.
Datasets
LastFM
MOOC
Reddit
Wikipedia
UCI
Evo.
Models
R
B
R
I
R
R
Variant A
94.12
73.08
S5.31
61.
64.25
76.16
98.84
65.19
98.40
70.58
553.23
70.84
92.39
96.54
Varient
94.25
$0.13
67.5
75.08
58.92
92.41
70.76
90.42
91.11
Varisnt
94.18
78.44
80.30
68.55
58.59
80.39
98.90
64.07
98.65
60.82
94.47
72.05
93.07
96.20
Variant D
94.21
70.64
89.44
67.9 07.91
88.29
80.86
98.91
65.10
98.09
71.10
94.51
73.50
93.10
96.75
Variant
89.08
82.24
76.72
65.20
98.40
90.
74.64
92.76
Variant
87.71
06.45
84.112
06.85
81.13
78.85
98.54
61.74
98.37
67.33 50
592 81
64.59
92.42
90.38
DYGMambo
94.42
76.76
89.67
68.77
88.64
80.75
98.99
65.30
98.77
71.14
94.76
72.17
93.13
96.88
4.2.2 Ablation Study
We conduct four ablation studies to study the effectiveness of model components. In study A, we make a model variant (Variant A) by removing the time-level SSM block and restrain our model from learning temporal patterns (information selection is substituted by mean pooling over the output of Eq. 4 In study B, we make a model variant (Variant B) by removing the Mamba SSM layers (Eq. 3 in the node-level SSM block. In study C, we switch the computation of the selection weights 3n in Eq. 6d to 30 = Softmax sel (Hz)) (feal()) 1Rd + p2dd) to create Variant C. In study D, we base on Variant C and develop Variant D that further enables information selection from opposite nodes, i.e., B_ = (oftmax((fall (oftmax(faa(H())) 3. = Softmax(fee) (oftmax((ea(HH)) Both ablation and D do information selection without learning temporal patterns. In study F. and F, we devise Variant E and Variant F' by removing the time features and the node interaction frequency features, respectively, during neighbor encoding. From Table 3 and 4 we find that: (1) Variant A is constantly beaten by DyGMamba, showing the effectiveness of dynamic information selection based on edge-specific temporal patterns; (2) MyGManba always outperforms Variant B, indicating the importance of encoding the one-hop temporal neighbors with SSM layers for capturing graph dynamics; (3) Variant C' and D perform better than Variant A in most cases, implying that selecting temporal information is generally contributive; (4) Variant C generally lags behind Variant D, meaning that information selection from opposite node beneficial; (5) DyGMamba performs better than both Variant C and D in almost all cases, proving that information selection based on temporal patterns is more effective; (6) Variant E and F in general achieve worse results than DyG validating the contribution of both time and node interaction frequency features.
4.2.3 A Closer Look into Temporal Pattern Modeling with Synthetic Datasets
We observe from ablation studies that dynamic information selection based on temporal patterns contributes to better model performance real-world datasets. To better quantify its benefits, we construct three
9

---

Published in Transactions on Machine Learning Research (05/2025)
synthetic datasets, i.e., S1, S2 and S3, that follow different patterns and compare our model with DyGFormer, CTAN well as Variant A, C, D on them. Each synthetic dataset contains 7 nodes, where the interactions of each pair of two nodes follow a certain pattern along time. And for each node, we generate interactions with all the other nodes. Assume we have a pair of node u and v and they have interactions at {t]] in S1, the time intervals between neighboring interactions {11 - lo: ..., IN IN-13 follow an increasing trend with a constant velocity of 0.05, i.c., (1+22- (441) - (f++1 - 1,) 0.05 In S2, we set the time intervals to a decreasing trend with the same velocity, Le., (l++1 - 4,) (4442 (--1) 0.05. And in S3, we modify S1 by repeating several periods of increasing patterns taken from S1 to form a periodic dataset. In this way, we have three datasets demonstrating diverse temporal patterns: increasing/ prrraaiinn/derrasiiingg /ecreasing///cc/c////eeea////eee petrraasing/prroddiiii periodic time intervals between neighboring interactions. Details of dataset construction and statistics are provided in App. A.2 From Table 5| we observe that DyGMamba greatly outperforms DyGFormer and CTAN. More importantly Variant A, and D show similar performance to DyGFormer, meaning that our time-level SSM block is able capture temporal patterns and modcling such patterns for dynamic information selection is important CTDG reasoning. For more implementation details on synthetic datasets, please refer to App. C.2
Table 5: Performance (Random NSS) on synthetic datasets.
(a) AP on synthetic datasets.
Datasets
DyGFormer
CTAN
Varlant
Varlant
Varlant
DyGMamba
S1
35.19 + 0.98
51.25 2.11
53.72
0.04
35.45 +: 0.32
54.52 + 0.71
81.58 1: 1.31
$2
37.80 + 1.61
51.17
0.93
60.16
2,20
61.71 2.33
61.51 + 3,00
85.36 + 2.55
S3
79.20 + 0.60
51.46 0.19
77.61 2.31
77.61 2.31
79.41 # 2.13
86.59
(b) AUC-ROC on synthetic datascts.
Datasets
DyGForrer
CTAN
Variant A
Variant C
Variant D
DyGMamba
S1
56.27 + 0.54
51.25 + 2.38
53.16 + 0.39
57.41 + 0.04
55.83 : 1.03
86.61 +: 1.30
59.06 6.07
51.46 + 0.91
62.50
+ 2.28
61.93 + 2.59
62.06 + 3.40
89.94 2.70
82.89 1.11
52.12 + 0.41
81.78 + 0.40
82.73 1.97
81.20 : 3.57
91.72 0.11
4.3 Efficiency Analysis
We evaluate the models' efficiency based on the following aspects: model size (number of trainable param- eters), per-epoch training time, and GPU memory consumption during training. Since DyGMamba shares similarities with DyGFormer, i.e., both of them model large sequences of one-hop temporal neighbors and use patching to enhance scalability, we further analyze the impact of patch size on their scalability and performance. We also compare the complexity of DyGMamba and DyGFormer to highlight DyGMamba's 's efficiency.
4.3.1 Model Size, Per Epoch Training Time and GPU Memory Comparison Across Various Models
Fig. 2 compares DyGMamba with five baselines in terms of number of parameters (model size), per epoch training time, and GPU memory consumption during training We find that: (1) DyGMamba uses very few parameters while maintaining the best performance, showing strong parameter efficiency. Only CTAN constantly uses fewer parameters than DyGMamba, however, its performance is significantly worse with the only exception of Enron; (2) DyGMamba is always more efficient than DyGFormer with the same length of input sequence (p/p =32) while achieving the same performance; (3) Although DyGMamba generally consumess more GPU memory and takes more time to train per epoch compared with most baselines, the gap of consumption is modest. To model more temporal neighbors for long-range temporal dependent datasets, DyGMamba naturally requires more computational resources, thus enlarging the consumption gap. DyGFormer shows the same trend DyGMamba since it also captures long-term temporal dependencies but at a higher cost; (4) CTAN requires very few computational resources. However, on long-range temporal dependent datasets, it is beaten by DyGFormer and DyGMamba by large margin, e.g., on LastFM and
TTe baselines included here are either extremely inefficient (e.g., CAWN) or inferior performance (e.g. DyRep). Complete statistics of all baseline models presented App. G.T
10

---

Published in Transactions on Machine Learning Research (05/2025)
LastdFM
Time// Raporhtphhhh (min)
LastFM
LastPM
MDOC
Train fosch (min)
MOOC
MODC
40
Epoch
Parram
20
30
GPU Memory
2.8
2.6
20
2.4
0.5
Average Prediion Enron
Average Precision
Averge Preditoon
Average Precision
Auerage Predsion
Average Predsion
Train Timee/ (min)
Enron
UCI
Train tire/fooc (min)
UCI
Pheame (M4B
GRU Mamory (08)
84
2.8
22
1.4
Average Predsion
Average Procision
Average Precision
Avarage Praciion
Averaga Preciion
Average Precisio
TGN
TCL
GraphMiixer DyGFormer
CTAN DyGMamba
Figure 2: Efficiency comparison on four datascts among DyGMamba and five baselines in terms of number (#) of parameters, training time per epoch and GPU memory. The performance metric here is AP of transductive link prediction under random NSS. The greener, the better snrformance/effccienve to other methods, DyGMamba consistently shows strong overall capability across different datasets. More Further comparison Reddit and Social Evo. is presented in App. G.1
In contrast
explanations in Sec. 4.3.1
15900
Precision (Bardom NSS)
Dyditmmer DyCiMabba Varians
DyChermeer
DyGPomer
15410
Oy&Former OyCMombbe
Number of Parometers (MB)
DyGk
-
Dy DyGaamba
92.06
1.4
4.24
GiU Memory (M03)
12000
Thain melepppch
F2
92.35
10000
253
d
8000
3P12
163
4462
20
29
44
100
20
200
250
100
150
200
250
300
100
150
200
250
300
100
150
200
250
300
Sequence Length
Sequence Length
Sequence Length
Sequence Length
(a) Enron performance.
(b) Enron # params.
(c) Enron GPU memory.
(d) Enron train time/epoch.
Figure 3: Impact of patch size on DyGFormer, DyGMamba and Variant A, given a fixed number of sampled temporal neighbors p Enron. Patch size varies from 8, 4, 2, 1. Sequence length p/p increases patch size decreases. Performance is the transductive AP under random NSS.
Enron. Besides, CTAN is also hard to converge. Although it takes little time to train single epoch, it needs more epochs to reach the best performance, leading to a long total training time. See App. G..2 for total training time comparison among DyGFormer, CTAN and DyGMamba. To supplement, we also present in App. G.3 another experiment to compare DyGMamba with basclines in modeling an increasing number of temporal neighbors on Enron with limited total training time.
4.3.2 Impact of Patch Size on Scalability and Performance
Patching treats temporal neighbors as one patch and thus decreases the sequence length by p times. This is very helpful in cutting the consumption of GPU memory and training/evain evaluation time. However, patching introduces excessive parameters because it done through fN. fE. fr and fP whose sizes increase as the patch size grows. Fig. 3h shows the numbers of parameters of DyGFormer and DyGmamba with different patch sizes on long-term temporal dependent dataset Enron. We find that patching greatly affects model sizes. To further study how patching affects DyGMamba, and track DyGMamba's performance (Fig. [3a) as well as efficiency (Fig, 3b to 3d] on Enron. Meanwhile, we also keep track on DyGFormer under the same patch size for comparison. We have several findings: (1) Whatever the patch size is, DyGMamba always consumnes fewer parameters, less GPU memory and per epoch training time, showing its high efficiency; (2) While both models require increasing computational budgets the patch size decreases, the speed of incrcase is much lower for DyGMamba, demonstrating strong scalability in modeling longer sequences; (3) Different trends in performance change are observed between two models. While DyGFormer performs worse, DyGMamba can benefit. from smaller patch size,
decrease the patch size gradually from 8 to 1
11

---

Published in Transactions on Machine Learning Research (05/2025)
indicating its strong ability to capture nuanced temporal details even if the sequence becomes much longer. Note that the models use fewer parameters under smaller patch size. This also shows that DyGMamba can achieve much stronger parameter efficiency by reducing patch sizes. To further study the reason for finding (3), we plot the performance of Variant A under different patch sizes in Fig. 3a We find that Variant A's performance degrades when sequence length is more than 64. This means that dynamic information selection based on odge-specific temporal patterns is essential for DyGMamba to optimally process long soquences. To supplement, we provide additional analysis on MOOC in App. J
Reason for Diverse Performance Trends between DyGMamba and DyGFormer. Patching de- creases sequence length by applying linear transformation on patch of node embeddings, giving model much more parameters to tune (as indicated in Fig. 3b). Larger patch size mixes more sampled neighbors in each patch, introducing more training parameters while losing nuanced temporal details brought by the temporal order of the neighbors within each patch. For DyGFormer, the negative influence brought by mix- ing neighbors within patches is smaller than the positive influence brought by more trainable parameters, the performance constantly increases with a growing patch size in Fig. 3a By contrast, for DyGMamba, the negative influence brought by mixing neighbors is much greater than the positive influence brought by more trainable parameters, so it shows better performance when the patch size is smaller. For Variant A, given an increasing patch size, it follows the trend of DyGFormer when the patch size is below a threshold and shows degrading performance after that. This means that there is a tradc-off between the lost temporal details and the additional parameters when we modify patch size. Also, by comparing Variant A and DyG- Mamba, we can tell that the difference in performance trend roots from the dynamic information selection module. Smaller patch size leads to longer sequences with more temporal details. The strong capability of the dynamic information selection module in long-range temporal reasoning let DyGMamba benefit from more nuanced temporal details, which is more influential than increasing Training parameters.
4.3.3 Complexity Analysis: DyGMamba vs. DyGFormer
DyGMamba follows the current statc-of-the-art DyGFormer by learning from one-hop temporal neighbors for temporal reasoning. We analyze the complexity of both models to show DyGMamba's efficiency. Sequence length is the key factor affecting the consumption of computational resources in DyGFormer and DyGMamba. Following the computation of previous work (Zhu et al., 2024), the complexity of Transformer and Mamba in DyGroreer and MyGMamba can be written as
O(Tranforrmer) = 4(ppp) + ((d) = 8d(p/p)?, O(Mamba) =
(8a)
+ (p/D)(Ad)dssM 16dssMd(p/p).
(8b)
This means that DyGMamba holds a computational complexity linear to p/p, while DyGFormer's complexity is quadratic to p/p. As result, as the sequence length grows (cither p increases or p decreases), DyGForrer is less scalable compared with DyGMambe Some may argue that increasing the patch size p to a large enough value can offset the negative influence of the higher complexity of Transformer. However, as discussed in Sec. 4.3.2 increasing patch size will substantially increase model paramcters, causing burden in parameter optimization, and meanwhile lose temporal details. This indicates that patching is not always beneficial and DyGMamba's s low complexity provides an alternative way to maintain great efficiency while considering more temporal information. With the same number of sampled temporal neighbors and equivalent computational resources, DyGMamba can leverage a smaller patch size, mitigating the negative effects of lost temporal details and simplifying parameter optimization, as implied in Sec. 4.3.2
Modeling Increasing Number of Temporal Neighbors with Fixed Patch Size. Sequence length is decided by the sampled temporal neighbors p and the patch size p. If we want to model a huge number of temporal neighbors, e.g. P = 4096, keeping the sequence length unchanged as 32 means we need to set p to 128. Considering the potential negative impact brought by excessively large patch size (i.e., too many lost temporal details, harder optimization and much lower parameter efficiency); it is also important to see
.As we set & (number of edgo-specific historical interactions discussed in Sec. 3.2 number much smaller than the number of sampled hop temporal neighbors, Le., sequence length p/p, we omit here the contribution the time-level SSM in complexity analysis.
12

---

Published in Transactions on Machine Learning Research (05/2025)
if model is scalable to p. with fixed p. In Fig. show the consumed GPU memory of DyGMamba and DyGFormer with varying from 64, 128, 256, 512, 1024, 2048, 4096 and 8192 on Enron, under fixed patch size 8. We find that DyGMamba shows superior scalability over DyGFormer. While DyGFormer can only process 2048 nodes, DyGMamba can deal with more than 8192 (at least times) on a single 45GB NVIDIA A40 GPU. This implies our model's polential to process huge number of temporal neighbors with limited GPU memory budget, while keeping a moderate patch size. The experiment provides conerete example to demonstrate DyGMamba's advantage in efficiency brought by its low complexity.
GPU Memory (GB)
40
42.3
Out of Memory
39.35
30
20
13.95
20.41
DyGFormer DyGMamba
10
5.86
10.85
3.79
6.2
0
512
1024
2048
4096
8192
Number of Sampled Neighbors
Figure 4: GPU memory consumption on Enron with increasing neighbors. For clarity, memory is shown only when neighbors are more than 512.
5
Discussion
In this section, we discuss where DyGMamba excels and where it falls short, and explain choice to consider only onc-hop temporal neighbors. We additionally provide a full justification of our motivation in App. recommended for reading after the main paper.
Where DyGMamba Excels and Where It Falls Short. We summarize here the conditions under which DyGMamba performs well and scenarios where may fall short. DyGMamba is expected out- perform previous state-of-the-art models in the following scenarios: (1) when modeling long-range temporal information is critical; (2) when there is a clear temporal pattern, iee, the density of repeated edges follows specific trend. For the firsi scenario, we have demonstrated in Table α and 2 that DyGMamba surpasses previous models on long-range temporrl-dependent datasets (LastF'M, MOOC, and Enron) in both trans- ductive and inductive link prediction under the random NSS setting. Under the historical and inductive NSS settings, DyGMamba consistently ranks among the top 3 models and even achieves the top position in out of 9 cases. Regarding the second scenario, Table 5 confirms that DyGMamba's dynamic information selection module significantly enhances its capability model temporal patterns, enabling superior per- formance datasets characterized by clear temporal trends. Converscly, DyGMamba might underperform previous state-of-the-art models on certain datasets (e.g., Wikipedia and UCT) under historical and inductive NSS settings, particularly when the number of available one-hop temporal neighbors for nodes in the link prediction query is limited (the availability of historical neighbors across datascts for both link prediction tasks under all NSS settings presented in Table 22 DyGMamba rely on abundant historical information effectively capture graph dynamics and temporal patterns. Insufficient historical context may hinder its performance, leading potential misclassification of negative edges and higher false positive rates.
Why We Only Consider One-Hop Neighbors. We only consider encoding one-hop temporal neighbors in DyGMamba due to two reasons. First, our primary motivation is to design efficient model that effec- lively captures long-range lemporal dependencies in CTDGs. While incorporating higher-order information is an intuitive approach, introduces several challenges. Maintaining the same computational budget would require reducing the number of first-order (one-hop) temporal neighbors, leading a less representative one-hop neighborhood. The intenattve-allocatinn more computational resources such as increased GPU memory goes against objective of efficiency in this work. Moreover, modeling higher-order temporal
13

---

Published in Transactions on Machine Learning Research (05/2025)
neighbors requires additional specialized modules to differentiate the inffuence of neighbors from different hops, which can further increase computational cost. and difficulties in modeling. Besides, recent CTDG models that only consider one-hop temporal neighbors, such as DyGFormer and FreeDyG, have demon- strated strong performance, and DyGMamba further validates this observation. Some of our baselines, such as CAWN and CTAN, do incorporate higher-order information. However, as shown Table and 2 they are outperformed by DyGMamba in most cases. A potential reason for this is that these models may still struggle to optimally differentiate information from different hops. Efficiently integrating higher-order in- formation into CTDG modeling is promising research direction, and would be happy to explore it in future work.
6 Conclusion
We propose DyGMamba, an efficient CTDG representation learning model that can capture long-term temporal dependencies. DyGMamba first leverages node-level SSM to encode long sequences of historical node interactions. Tt. then employs a time-level SSM to learn edge-specific temporal patterns. The learned patterns are used to select the critical part of the encoded temporal information. DyGMamba achieves superior performance on dynamic link prediction, and moreover, it shows high efficiency and strong scalability compared with previous CTDG methods, implying a great potential in modeling huge amounts of temporal information with limited computational budget.
7 Limitation
One limitation of our work is that DyGMamba is designed to only model CTDGs. DTDGs are represented as a sequence of graph snapshots, where all the edges a snapshot are taken as existing simultaneously. This poses challenge DyGMamba because can only encode edges soquentially, which are not suitable for modeling concurrent edges. A possible solution to first employ methods such graph neural networks (GNN) to encode each graph snapshot and then use SSM to model the temporal graph dynamics (similar to approaches used in recent works on temporal knowledge graph modeling, e.g., HHan et al. (2021 How- ever, the introduction of GNNs inevitably requires more computational resources, which would lower model efficiency.
Acknowledgements
Zifeng Ding receives funding from the European Research Council (ERC) under the European Union's Hori- zon 2020 Research and Innovation programme grant AVeriTeC (Grant agreement No. 865958). He also acknowledges travel support from the European Union's Horizon 2020 research and innovation programme under ELISE Grant Agreement No 951847. Yuan He is supported by Samsung Research UK (SRUK), EP- SRC projects UK FIRES (EP/S019111/1) and ConCur (EP/ (BP/V500069/)) Michael Bronstein and Antonio Norelli are supported by EPSRC Turing AI World-Leading Research Fellowship No. EP/X040062. Al Hub on Mathematical Foundations of Intelligence: An "Erlangen Programme" for A1 No. EP EP/Y028872/1. Antonio Norelli is also supported by Project CETT. Jingeheng Wu is funded by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) SFB 1574 Project number 471687386. This work is partially supported by the Munich Center for Machine Learning and is performed on the HoreKa supercom- puter funded by the Ministry of Science, Research and the Arts Baden- Württemberg and by the Federal Ministry of Education and Rescarch. The authors also gratefully acknowlodge the Gauss Centre for Super- computing V. for funding this project by providing computing time on the GCS Supercomputer HLRS High-Performance Computing Center Stuttgart.
EPSRC
References
Ali Behrouz and Farnoosh Hashemi. Graph mamba: Towards learning on graphs with state space models. CoRR, abs/2402. 22222220200777 08678,, 2024. doi: 10.48550/ARXIT 2402.08678. URL
arXiv 2402. 08678
14

---

Published in Transactions on Machine Learning Research (05/2025)
Xiaofu Chang, Xuqin Liu, Jianfeng Wen, Shuang Li, Yanming Fang, Le Song, and Yuan Qi. Contimuous-time dynamic graph learning via neural interaction processes. In Mathieu d' Aquin, Stefan Dietze, Claudia Hauff, Edward Curry, and Philippe Tudré-Maurons (eds.), CIKM '20: The 29th ACM International Conference on Information and Knowledge Management, Virtual Event, Ireland, October 19-23, 2020, pp. 145-154. ACM, 2020. doi: 10.1145 10.1145/3340531 /3/33333435333 111.1411446
URL https://doi. wtt/ 100.001111/3330530111 34119466
Weilin Cong, Si Zhang, Jian Kang, Baichuan Yuan, Hao Wu, Xin Zhou, Hanghang Tong, and Mehrdad Mahdavi. Do we really need complicated model architectures for temporal networks? In The Fleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. Open- Review.net, 2023. URL https: https::/openreeiieet http://ppparrreeeeeeeeeedd/aeaea
Feyza Duman Keles, Pruthuvi Mahesakya Wijewardena, and Chinmay Hegde. On the computational com- plexity of self-attention. In Shipra Agrawal and Francesco Orabona (eds.), Proceedings of The 34th In- ternational Conference on Algorithmic Learning Theory, volume 201 of Proceedings of Machine Learn- ing Research, pp. 597 619. PMLR, 20 Feb 23 Feb 2023. URL https: attpss:/pproceedingg.ou mlr. wwwp:emwr..rpeeeeeeeeeeee press/v201/ duman-keles23a. html
Palash Goyal, Sujit Rokka Chhetri, and Arquimedes Canedo. dyngraph2vec: Capturing network dynamics using dynamic graph representation learning. Knowl. Based Syst., 187, 2020. doi: 10.1016/J.KNOSY. 2019.06.024. URL httpps://doi..org////// 10. wtw./////.....////...1.. k.woo...ssss.s 2019. .0....666.0 024
knosys.
Alessio Gravina, Giulio Lovisotto, Claudio Gallicchio, Davide Bacciu, and Claas Grohnfeldt. Long range propagation on contimuous-time dynamic graphs. In Forty-first International Conference on Machine Learning, 2024. URL http:////pppppeeeeeeeeeee/e////r
Albert Gu and Tri Dao. Mamba: Lincar-time sequence modeling with selective state spaces. CoRR. abs/ 22120232, 2023.
doi: 10.48550/ARXI1 2312.00752. URL https: /ttp::////o..r//1..... wrgg//...855//r/11.... 48550/arXiv.
2312.00752
Albert Gu, Isys Johnson, Karan Goel, Khaled Saab, Tri Dao, Atri Rudra, and Christopher Ré. Combining recurrent, convolutional, and continuous-Lime models with linear state space layers. In Marc' Aurelio Ran- zato, Alina Beygelzimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan (cds.), Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Sys- tems 2021, NewrIPS 2021, December 6- 6-14, 2021, virtual, pp. 572 585, 2021. URL http:::/ neurips. ctc.ps.cc/ppp cttccc/paaa hwtc//pppr//00000 2021 htw.//aahh/eaae http/////....aaa11ccceeeeeeeeeer Abstrract..bbt....oc.am..cc..... .htrla
Albert Gu, Karan Goel, Ankit Gupta, and Christopher Ré. On the parameterization and initializa- tion of diagonal state space models. In Sanmi Koyejo, S. Mohamed, A. Agarwal, Danielle Belgrave, K. Cho, and A. Oh (eds.), Advances in Newral Information Processing Systems 35: Annual Confer- ence on Neural Information Processing Systems 2022, NewrIPS 2022, New Orleans, LA, USA, Novem- ber 28- December 9, 2022, 2022a. URL http://papers www.:iiips.....e.ee wwtppcppppatcaaeeeeeeee pales//paper2222//nee 90532faaee4790044494449999999999
Censtract-Confereen
html
Albert Gu, Karan Goel, and Christopher Ré. Efficiently modeling long sequences with structured state spaces. In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net, 2022b. URL
Zhen Han, Zileng Ding, Yumpu Ma, Yujia Gu, and Volker Tresp. Learning neural ordinary equa- tions for forecasting future links on temporal knowledge graphs. In Marie-Francin Moens, Xuanjing Huang, Lucia Specia, and Scott Wen-tan Yih (ods.), Proceedings of the 2021 Conference on Empiri- cal Methods in Natural Language Processing, EMNLP 2021, Virtual Event / Punta Cana, Dominican Republic, 7-11 November, 2021, pp. 8352 8364. Association for Computational Linguistics, 2021. doi: 658. URL scttp::/ddo..rrg// 10. 18653/v1/2021 emnlp-main 658
10.18653/ V //222221122PPP.
Ming Jin, Yuan-Fang Li, and Shirui Pan. Neural temporal walks: Motif-aware representation learn- ing on contimious-time dynamic graphs. In Sanmi Koyejo, S. Mohamod, A. Agarwal, Danielle Bel- grave, K. Cho, and A. Oh (eds.). Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NewrlPS 2022, New Orleans, LA, USA,
15

---

Published in Transactions on Machine Learning Research (05/2025)
November 28- December 9, 2022, 2022. URL bttp:///ppppaer wwtpssiippp..c...niaieeeeee http:////ppporrlllee//////////// (haa555er77800666660000001111220
html
Seyed Mehran Kazemi, Rishab Goel, Kshitij Jain, Ivan Kobyzev, Akshay Sethi, Peter Forsyth, and Pascal Poupart. Representation learning for dynamic graphs: A survey, J. Mach. Learn. Res., 21:70:1-700 70:73,
2020. URL http:///mmm...oggppppsrss//////s
Srijan Kumar, Xikun Zhang, and Jure Leskovec. Predicting dynamic embedding trajectory in temporal interaction networks. In Ankur Teredesai, Vipin Kumar, Ying Li, Rómer Rosales, Evimaria Terzi, and George Karypis (eds.), Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD 2019, Anchorage, AK, USA, August 4-8, 2019, pp. 1269-1278. ACM, 2019. doi: 10.1145/3222 03////22200033008855 URL https://doi. org/10. 1145/3292500 3330895
Jintang Li, Ruofan Wu, Xinzhou Jin, Boqun Ma, Liang Chen, and Zibin Zheng. State space models on temporal graphs: A frrst-rinciples study. arXiv preprint arXiv:2406. 00943, 2024a.
Lincan Li, Hanchen Wang, Wenjie Zhang, and Adelle Coster. Stg-mamba: Spatial-temporal graph learning URL via selective state space model. CoRR, /2403. 240..1422, 2024b. doi: 10.48550/ARXIV.205 battps//doi..oom org/ 10 48550/ 2403. 12418
Yunyu Liu, Jianzhu Ma, and Pan Li. Neural predicting higher-order patterns in temporal nelworks. In Frédérique Laforest, Raphaël Troncy, Elena Simperl, Deepak Agarwal, Aristides Gionis, Ivan Herman, and Lionel Médini (eds.), WWW '22: The ACM Web Conference 2022, Virtual Event, Lyon, France, April 25- 29, 2022, pp. 1340-1351. ACM, 2022. doi: 10.1145/3485447.22 37.84447.351188. 10. 1145/4484//4471// 351121111 3512181
URL https://doi.... org/
Xuezhe Ma, Chunting Zhou, Xiang Kong, Junxian He, Liangke Gui, Graham Neubig, Jonathan May, and Luke Zettlemoyer. Mega: Moving average equipped gated attention. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. URL httts::/ wttp://wwwee//ddddddddllleeeeeee
Yao Ma, Ziyi Guo, Zhaochun Ren, Jiliang Tang, and Dawei Yin. Streaming graph neural networks. Tn Jimmy X. Huang, Yi Chang, Xueqi Cheng, Jaap Kamps, Vanessa Murdock, Ji-Rong Wen, and Yiqun Liu (eds.), Proceedings of the 43rd International ACM SIGIR conference on research and development in Information Retrieval, SIGIR 2020, Virtual Event, China, July 25-30, 2020, pp. doi: 10. 00.115//3397271.440000 URL http::///...orr/11..111... 03414///33372211.1009222
ACM, 2020.
Aldo Pareja, Giacomo Domeniconi, Jie Chen, Tengfei Ma, Toyotaro Suzumura, Hiroki Kanezashi, Tim Kaler, Tao B. Schardl, and Charles E. Leiserson. Evolvegen: Evolving graph convolutional networks for dynamic graphs. In The Fhirty-Fourth. AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAT Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, URL https://doi. http:::://oi..rg/1....o ott./...10/9..9999
pp. 5363 5370. AAAI Press, 2020. doi: 10. aaai v34104.5995 5984
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshcin, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Z. Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, Biggh-eeformance deep learning li- brary. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d' AAlc6--Buc, Emily B. Fox, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 32: Annual Confer- ence on Newral Information Processing Systems 2019, NearIPS 2019, December 8-14, 2019, Vancou- ver, BC, Canada, pp. 8024 8035, 2019. URL https://proceeddngn hwwp:ssneniiiiiccoo hwtprippssc/pppppr2/1/////aa falcc228fee0119222558857722222 htt.://aaa..0001222727-0rrr.eee. Abstract. html
Bo Peng, Eric Alcaide, Quentin Anthony, Alon Albalak, Samuel Arcadinho, Stella Biderman, Huanqi Cao, Xin Cheng, Michael Chung, Leon Derezynski, Xingjian Du, Matteo Grella, Kranthi Kiran GV, Xuzheng
16

---

Published in Transactions on Machine Learning Research (05/2025)
He, Haowen Hou, Przemyslaw Kazienko, Jan Kocon, Jiaming Kong, Bartlomiej Koptyra, Hayden Lau, Ji- aju Lin, Krishna Sri Ipsit Mantri, Ferdinand Mom, Atsushi Saito, Guangyu Song, Xiangru Tang, Johan S. Wind, Stanislaw Wozniak, Zhenyuan Zhang, Qinghua Zhou, Jian Zhu, and Rui-Jie Zhu. RWKVV reinvent- ing rins for the transformer cra. In Houda Bouamor, Juan Pino, and Kalika Bali (eds.), Findings of the Association for Compulational Linguisties: EMNLP 2023, Singapore, December 6-10, 2023, pp. 14048- 14077. Association for Computational
Lingutaaatiiiiiaiininiiiiiiiit Linguisties, 2023. doi: URL https://doi.org// www.oo//............o. 18653/81/ http://022...iiddnne-emmmmmmpp 936
/666/V1/2223/ WANDINGS-EMNN.PEEE
Farimah Poursafaci, Shenyang Huang. Kellin Pelrine, and Reihanch Rabbany. Towards better evalu- ation for dynamic link prediction. In Sanmi Koyejo, S. Mohamed, A. Agarwal, Danielle Belgrave, K. Cho, and A. Oh (eds.), Advances in Neural Information Processing Systems 35: Annual Confer- ence on Neural Information Processing Systems 2022, NewrIPS 2022, New Orleans, LA, USA, Novem- ber 28- December 9, 2022, 2022. URL http://papers: aapp..c/paper.. wccppaperffitte Setton---sttttt-ttttaaaa
ptlps//paper//22///// and Benchmarks. html
Emanuele Rossi, Ben Chamberlain, Fabrizio Frasca, Davide Eynard, Federico Monti, and Michael M. Bron- stein. Temporal graph networks for deep learning on dynamic graphs. CoRR, abs/2006. 10637, 2020. URL
https://arxil: http::///arriv.orrr//////00// wtgg//a//////////oee
10637
Aravind Sankar, Yanhong Wu, Liang Gou, Wei Zhang, and Hao Yang. Dysat: Deep neural representation learning on dynamic graphs via self-attention networks. In James Caverlee, Xia (Ben) Hu, Mounia Talmas, and Wei Wang (eds.), WSDM '20: The Thirteenth ACM International Conference on Web Search and Data Mining, Houston, TX, USA, February 3-7, 2020, pp. 519-527. ACM, 2020. doi: URL https:///oi. 10. 1111/1333//11111 3371845
Razich Shirzadkhami, Shenyang Huang, Elahe Kooshafar, Reihanch Rabbany, and Farimah Poursafaci. Tem- poral graph analysis with tgx. In Proceedings of the 17th ACM International Conference on Web Search and Data Mining, WSDM 224 pp. 1086-1089, New York, NY, USA, 2024. Association for Computing Machinery. ISBN 9798400703713. doi: 0..1145/3618855363338
URL httts://ddi.org/10 1145/
Jimmny T. H. Smith, Andrew Warrington, and Scott W. Linderman. Simplified state space layers for sequence modeling. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. URL https::/ http:///w...eeeepe//////aaaaaaaa
Yuxing Tian, Yiyan Qi, and Fan Guo. Freedyg: Frequency enhanced contimuous-time dynamic graph model for link prediction. In The Twelfth International Conference on Learning Representations, 2024. URL
met/fforunnddd22M551111
llya O. Tolstikhin, Neil Houlsby, Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Thomas Unterthiner, Jessica Yung, Andreas Steiner, Daniel Keysers, Jakob Uszkoreit, Mario Lucic, and Alexey Dosovitskiy. Mlp-mixer: An all-mlp archilecture for vision. In Marc' Aurelio Ranzato, Alina Beygelvimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan (eds.), Advances in Neural Information Processing Systems 34: Anreual Conference on Neural Information Processing Systems 2021, NewrIPS 2021, December 6-14, 2021, virtual, pp. 24261 24272, 2021. URL http:://procedinggs. wttp:/gs...nni...neeeeeppeaeaaee httpe//ppppppp////aeeeepee 201///2221//hhh///// Sa b89fe Abstract html
Rakshit Trivedi, Mehrdad Farajtabar, Prasenjeet Biswal, and Hongyuan Zha. Dyrep: Learning represen- tations over dynamic graphs. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.nct, 2019. URL http:///oppnreeee.eeeeeeeeee id=HyePrhR5KX
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkorcit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett (eds.), Advonces in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pp. 11998-6008 2017. URL http:://prooeeang...nuu.... occ.aa/ee/.T/haasaSaaeeeeaeaaaaa Abstaaattt.a.aaaaaatttttaa.aa...
17

---

Published in Transactions on Machine Learning Research (05/2025)
Roger Waleffe, Wonmin Byeon, Duncan Riach, Brandon Norick, Vijay Korthikanti, Tri Dao, Albert Gu, Ali Hatamizadeh, Sudhakar Singh, Deepak Narayanan, Garvit, Kulshreshtha, Vartika Singh, Jared Casper, Jan Kautz, Mohammad Shoeybi, and Bryan Catanzaro. An empirical study of mamba-based language URL attps://doi.org//
models. CoRR, abs/ //////24600778777 2024. doi: 10.48550/ 10.48550/ARXIV.22 48550/arXiv. 2406. 24060078877777777
Chloe Wang, Oleksii Tsepa, Jun Ma, and Bo Wang. Graph-mamba: Towards long-range graph sequence modeling with selective state spaces. CoRR, 00s//2402000755 2024. doi: (18550/ARXII 2402.00789. 2402 00789
URL https:///oi. htt.://or//....5////aaaa...
Lu Wang, Xiaofu Chang, Shuang Li, Yunfei Chu, Hui Li, Wei Zhang, Xiaoleng He, Le Song, Jingren Zhou, and Hongxia Yang. TCL: transformer-base based dynamic graph modelling via contrastive learning. CoRR, abs/2105.07944 2021a. URL htttps://arxatt
wwwgg/bb/211000000 07944
Xuhong Wang, Ding Lyu, Mengjian Li, Yang Xia, Qi Yang, Xinwen Wang, Xinguang Wang, Ping Cui, Yupu Yang, Bowen Sun, and Zhenyu Guo. APAN: asynchronous propagation attention network for real-time temporal graph embedding. In Guoliang Li, Zhanhuai Li, Stratos Idreos, and Divesh Srivastava (eds.), SIGMOD 21: International Conference Management Data, Virtual Event, China, June 20-25, 2021, pp. 2628 2638. ACM, 2021b. doi: /311//44.811.1.55564 URL https://doi org/ 10tg/10..00 1145//444444816 3457564
Yanbang Wang, Yen-Yu Chang, Yunyu Liu, Jure Leskovec, and Pan Li. Inductive representation learn- ing in temporal networks via causal anonymous walks. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.nel, 2021c. URL
https://openrevieww met/foormmmm//xxX4433333)
Da. Xu, Chuanwei Ruan, Evren Körpeoglu, Sushant Kumar, and Kannan Achan. Inductive representation learning on temporal graphs. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, net, 2020.
id=rJewlyHywH
Jiaxuan You, Tianyu Du, and Jure Leskovec. ROLAND: graph learning framework for dynamic graphs.
Aidong Zhang and Huzefa Rangwala (eds.), KDD '22: The 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, August 14 18, 2022, pp. 2358 2366. ACM, 2022. URL https://doi. org/10. 1145001411333347.... 0553/333336733333300 3539300
doi: 0144455534778835
Le Yu, Leilei Sun, Bowen Du, and Weifeng Lv. Towards better dynamic graph learning: New ar- chitecture and unified library. In Alice Oh, Tristan Naunann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), Advances in Newral Information Processing Systems 36: An- nual Conference on Neural Information Processing Systems 2023, NewrIPS 2023, New Orleans, LA, USA, December 10 16, 2023, 2023. URL ltttp://papers... nips www.iii....nppppeee cc/paper.ppppp.... pilles/apper2222//eaa/ Abstract-Contee Conttt-ooofoeeeeee html
a61 0511009afaar00047777956551111111
Lianghui Zhu, Bencheng Liao, Qian Zhang, Xinlong Wang, Wenyu Liu, and Xinggang Wang. Vision mamba: Efficient visual representation learning with bidirectional state space model. In Forty-first International Conference Machine Learning, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.ned 2024.
URL https::/oppereviiee het.
A CTDG Dataset Details
A.1 World Benchmark Datasets
We present the dataset statistics of all considered CTDG datasots in Table D. All the datasets our experi- ments are taken from Yu al. (2023). We chronologically split each dataset with the ratio of 70% 19611%/155 /15% for training/ ceaiing/valinn van.aaaaaaoonneesttteaa /esttig. Please refer to it for detailed dataset descriptions.
18

---

Published in Transactions on Machine Learning Research (05/2025)
Table Dataset statistics.
# N&E Feat means the numbers of node and edge features.
Datasets
# Nodes
# Edges
# N&FE Feat
Bipartite
Duration
# Timestamps
Time Granularity
LM
1,980
1,293103
0 & 0
True
month
1,283,614
Unix timestamps
Enron
184
125,235
0 & 0
False
3 years
22,632
Unix timestamps
MOOC
7,144
411,749
0 & 4
True
months
345,600
Unix timestamps
Reddit
10,984
672,447
0 & 172
True
1 month
669,065
Unix timestamps
Wikipedia
9,227
157,474
& 172
True
1 month
152,757
Unix timestamps
UCI 1,899
59,835
0 & 0 0 &
False
196 days
58,911
Unix timestamps
Social Evo.
74
2,099,519
False
8 months
565,932
Unix timestamps
A.2 Synthetic Datasets
For all of our three synthetic datasets, node and link features not involved during datasct construction. The construction details are as follows:
S1: For each interaction pair and u, their first interaction is at timestamp 0 and the second interaction is generated randomly. Thus, the first time interval is also determined. Starting from the second interval, they follow an increasing trend with a constant velocity of 0.05, i.e., (f+42 4+11 - (4++1 4))= 0.05. The number of interactions for each node pair is randomly determined and the interaction numbers
node pairs sum up to 100000.
S2: For each interaction pair u and v, their first interaction at timestamp 0 and the second interaction is generated randomly. However, it should be large enough that the interval will not drop to zero negalive afterwards. Starting from the second interval, they follow decreasing trend with constant velocity of 0.05, ie., (tr+1 - +4) (t++2 4441 0.05. The number of interactions for cach node pair is randomly determined and the interaction numbers of all node pairs sum up to 100000.
S3: S3 contains periods. In each period, the interactions of cach node pair u and v are generated following the same pattern in S1. The number interactions for each node pair is randomly determined and the interaction numbers of all node pairs sum up to 12000 in the period.
We present the statistics of all synthetic datasets in Table 7 chronologically split each dataset with the ratio of 70%/15% 15% for training/validationn validation/testing To better visualize the temporal patterns in each dataset, we pick one pair of interacting nodes and plot the time intervals between neighboring interactions in cach dataset in Figure 5. Note that for the periodic dataset S3 (Figure 5c). cach of the train, validation and lest sets contains at. least one start of new period. This ensures that models have to caplure periodic temporal patterns in order to achieve good performance during evaluation, rather than only learning the increasing time intervals as specified in S1.
Furthermore, we provide the information about the numbbers of interactions regarding interacting node pairs in Table M We show that each node pair equipped with a substantial number of interactions, meaning that temporal patterns in our synthetic datasets span across long time periods. This encourages models to consider long-term temporal dependencies for better graph reasoning.
Table 7: Synthetic dataset statistics.
Datasets
# Nodes
Edges
# Timestamps
Time Range
S1
7
100,000
96,869
0 163241.65
S2
100,000
98,004
1573561.52
S3
7
95,657
95,370
0 1771300.40
19

---

Published in Transactions on Machine Learning Research (05/2025)
Table 8: Interaction information of node pairs in synthetic datasets. Complete Dataset includes the numbers of interactions across the whole datasets, including training, validation and testing.
Datasets
Avg. # Interactions
Min # Interactions
Max # Interactions
S1
1,428.57
1,205
1,663
Training Set
S2
1,428,57
1,424
1,433
S3
1,367.91
1,364
1,372
S1
2,010.20
1,879
2,150
2,010.20
1,921
2,097
Complete Dataset
S3
1,952.18
1,721
2,193
$3 (each period)
244.02
215
274
TIme Interval (Timestampp Difference)
220
Tram
Time Interval Limestamp Uifference)
Train
Time Interval (Timestamp Difference)
834
Ieani
1092
ICST
750
700
40-
240
40
802
250
500
150
1000
7350
1500
3750 2003
250
900
750
1000
1250 1505
1750 2000
300
1000
7500
7000
Interaction Index
Interaction Index
Interaction Index
(a) Synthetic dataset S1.
(b) Synthetic dataset $2.
(c) Synthetic dataset
Figure 5: Time intervals of node pair u, in synthetic datasets S1, $2 and $3.
B
Baseline Details
We provide the detailed descriptions of all baselines here. The baselines can be split into two groups: the methods designed/ noo designed for long-range temporal information propagation.
B.1 Baselines Not Designed for Bang-Range Temporal Information Propagation
JODIE [Kmaar et al. 2019:
JODIE employs recurrent neural network (RNN) for each node and uses a projection operation to learn the future representation trajectory of each node.
DyRop [rivedi al. 2019):
DyRep updates node representations as events appear. It designs a bwo-lime scale deep temporal point process approach for source and destination nodes and couples the structural and temporal components with temporal-attentinn aggregation module.
TGAT Xu et al. 2020 TGAT computes the node representations by aggregating each node's temporal neighbors based on self-attention module. A time encoding function proposed learn functional representations of time.
TGN Rossii et al. 2020): TGN leverages an evolving memory for each node and updates the memorry when a node-relevant interaction occurs by using a message function, a message aggregator, and memory updater. An embedding module is used generate the temporal representations of nodes.
CAWN WWng et al. 2021c:: CAWN is a random walk- based method. It does multiple causal anonymous walks for each node and extracts relative node identitics from the walk results. RNNs are then introduced to encode the anonymous walks. The aggregated walk information forms the final node representation.
20

---

Published in Transactions on Machine Learning Research (05/2025)
EdgeBank (Poursafaci et al. 2022): EdgeBank is non-parametric method purely based mem- ory. It stores the observed interactions in its memory and updates the memory through various strategies, An interaction, ie., link, will be predicted as existing if it is stored in the memory, and non-existing otherwisc. EdgeBank uses four memory update strategies: (1) EdgeBank where all the observed edges are stored in the memory; (2) EdgeBankkw-te where only the edges within the duration of the test set from the immediate past are kept in the memory; (3) EdgeBankkw-e: where only the edges within the average time intervals of repeated edges from the immediate past are kept in the memory; (4) EdgeBanku, where the edges with appearing counts higher than threshold are stored in the memory. The results reported in our paper correspond to the best results achieved among the four memory update strategies.
TCL (Wang et al. 2021a;: TCL first extracts temporal dependency interaction sub-graphs for source and interaction nodes and then presents a graph transformer to aggregale node information from the sub-graphs. A cross-attention operation is implemented to enable information communication between two source and destination nodes.
GraphMixer designs link-encoder based on MLP-Mixer Tol- stikhin et al. 2021 to learn from the temporal interactions. A mean pooling- based node-encoder is used to aggregate the node features. Link prediction is done with a link classifier that leverages the representations output by link-encoder and node-encoder.
GraphMixer (Cong et al. 2023
Note that TGN uses a memory network to store the whole graph history, making it able to preserve long-range temporal information. However, as discussed in Yu al. (2023), faces problem of vanishing vanishing/explodiing /exploding gradients, proventing it from optimally capturing long-term temporal dependencies. EdgeBank can also
preserve very long graph history, but we can observe from the experimental results (Table | 12 that without learnable parameters, it is not strong enough on long-range temporal dependent datasets.
B.2 Baselines Designed for Long-Range Temporal Information Propagation
DyGFormer Yu et al. 2023 : DyGFormer is Transformer-based CTDG model. It takes the long-term one-hop temporal interactions of source and destination nodes and uses a Transformer to encode them. A patching technique is developed to cut the computational consumption and a node co-occurrence encoding scheme is used to exploit the correlations of nodes each interaction. DyGFormer achieves long-range temporal information propagation by increasing the number of sampled one-hop historical interactions. The patching technique ensures that even with huge number of sampled interactions, the length of the sequence inpul into Transformer will not be too long, making it possible to implement DyGFormer with limited computational budget.
CTAN (Gravina et al. 2024]: CTAN is deep graph network for learning CTDGG based on non- dissipative ordinary differential equations. CTAN's formulation allows for scalable long-range temporal information propagation in CTDGs because its non-dissipative layer can retain the infor- mation from specific event indefinitely, ensuring that the historical context of a node is preserved despite the occurrence of additional events involving this node.
C Implementation Details
We train every CTDG model except for CTAN for a maximum number of 200 epochs. Maximum epochs for CTAN training is 1000. We evaluate each model on the validation set at the end of every training epoch and adopt an carly stopping strategy with a patience of 20. We take the model that achieves the best validation result. for testing. We use the implementations? provided by Yu et al. (2023) for all baseline models except
CTAN, For CTAN, we use its official implementation [ All models are trained with batch size of 200 for
fair efficiency analysis. All experiments are implemented with PyTorch (Paszke et al. 2019 on server
21

---

Published in Transactions on Machine Learning Research (05/2025)
equipped with AMD EPYC 7513 32-Core Processor and single NVIDIA A40 with 45GB memory. We run each experiment for five times with five random seeds and report the mean results together with error bars.
C.1 Hyperparameter Configurations Real-Worid Datasets
For all the basclines except CTAN, please refer to Yu et al. (2023) for the hyperparameter configurations on real-world datasets. For CTAN, we present its hyperparameter configurations in Table g We keep its hyperparameters unchanged for all real-world datasets. Note that we set the number of graph convolution layers (GCLs) CTAN to its maximum, i.e., 5, order to maximize its performance in capturing long- long-term temporal dependencics.
Table Hyperparameter configurations of CTAN on all real- datasets. y here denotes the discretization step size introduced in Gravina et al. (2024), different from the one in MyGMamba.
Model
# GCL
Embedding Dim
CTAN
5
0.5
0.5
128
We report the hyperparameter searching strategy of DyGMamba real-world datasets and the best hyper- parameters in Table 10 To achieve fair efficiency comparison with DyGFormer, we fix the length of the input sequence into the node-level SSM to 32, i.e., & p= 32. The results reported in Table I 2 12 13 achieved by DyGMamba with & p= 32. In practice, we can decrease to have a better performance given more computational resources (as discussed in Sec. 4.3 DyGMamba keeps the embedding size as same as DyGFormer on all real-world datascts, i.c., dN dg 172, dr = 100, de = 50. We also set y = 0.5 for all experiments of DyGMamba. The dimension of SSMs dssM 16 remains the default value of mamba SSM's official repository | For all experiments,
are all
set the numbers of layers in both node-level and time-level SSMs 2. We also set to 0.5 for all datasets. A smaller lowers the computational resource consumption in time-level SSM, potentially at the cost of performance. Raising y's value does not necessarily lead to better performance but will lower efficiency. We search y's value in {0.1, 0.5, 0.7, 1} and find that 0.5 brings a good balance between performance and efficiency.
Table 10: DyGMamba hyperparameter searching strategy on real-world datasets. The best settings are marked as bold.
Datasets
Dropout
p p
LasiFM
(0.0, 0.1, 0.2)
(1024 32, 512 16, 256 & 8)
{30, 10, 5)
Euron
(..0, 0.1, 0.2)
[512 & 16, 256 & 8, 128 & 4)
[30, 10, 5]
MOOC
(0.0, 0.1, 0.2)
(512 16, 256 8, 128 4)
{30, 10, 5)
Reddit
{0.0, 0.1, 0.2)
(128 & 4, 64 & 2, 32 & 1}
[30, 10, 5
Wikipedia
(0.0, 0.1, 0.2)
(64 & 32 1)
[30, 10,
UCI
(0.0, 0.1, 0.2]
[64 & 2, 32
& 1)
[30, 10, 5)
Social Evo.
(0.0, 0.1, 0.2]
[64 & 2, 32
1)
[30, 10, 5)
C.2 Hyperparameter Configurations on Synthetic Datasets
We use the same settings of CTAN on real-world datasets when we experiment on synthetic datasets. For DyGFormer and DyGMamba, we fix the length of the input sequence into the Transformer and the node-level SSM to 32, i.c., p/p 32. For DyGFormer, same default values
set the hyperparameters except p and to the real-world datasets, and search for the best p & p within (512 & 16, 256 & 8, 128 & 4, 64 & 2, 32 & 1}. For DyGMamba, we scarch for the best & within the same scarch range and further search for the best. k. All the other hyperparameters are set as same as the setting on LasiFM. We
hhtpe///gghhbb.mm//tttttpppppee
22

---

Published in Transactions on Machine Learning Research (05/2025)
Table 11: DyGFormer and DyGMamba hyperparameter searching strategy on synthetic datasets. The best settings are marked as bold.
Models
DyGFormer
DyGMamba
Datasets
p &
&
k
S1
(512 & 16, 250 & 128 & 64 & 2. 32 &
(512 & 16, 256 & 8, 128 & 4.
61
&
2,
32
& 1]
[30, 10, 5]
S2
[512 & 16, 256 & 8, 128 & 4. 64
& 2,
32 &
[512 & 16, 256 & 8, 128 & 4
64
&
2.
32
& 1]
[30. 10, 5)
S3
{512 & 16, 256 & 8, 128 & 4, 64 & 2, 32 &
[112 & 16, 256 & 8, 128 & 4,
64
&
2.
32
&
1]
(30, 10, 5]
Table 12: AUC-ROC of transductive dynamic link prediction.
N88
JODIE
DyRop
TGAT
TGN
CAWN
ExgesEank
TCL
GraphMisur
DyGPuemnr
CTAN
DyGMammn
LanFM
71.41
212
76.64 1.06
55,12
0.31
T1.00 1:44
73.51
0.34
10.00 0.1
00.31
Roadon
MOOC
Euaoa
84.00
63.00
2:20
3.87
1.46
68.72 0.96
50.34
50.45
0.25
0.50
0.00
56004
54.16
85.08
0.90
90.34
Redeit
59.2
WiMipedie
10:37
98.54
10.27 0.00
0.00
10.002
UCT
08:46
2.05
78.06
1.10
00.08
08:81
0.35
85.40
0.90
0.52
91.45
0.23 08.23
94.77
22.18
00.87
0.02
94.03
0.06
06.31
57.84
0.30
06.45 0.21
96.21
0.07
Tiaoout
00.38
0.02
Avg.
8.86
4.50
7.20
7.14
2.14
7:20
Losti'M
70,05 + 4.03
70:631
0:
2.56
64.23 +: 0:45
T8-00) +: 2.17
6TM2
0.322
71 8: 0.001
2.54
64:16
8: 0.33
T5.M0 +: 0.012
70.50
0.52
71.82
0:227
Iistoriol
75.21
8 1:27
00:30:11
1:32
02:35-1
+: 1:07
70.75 +: 1.40
05:02-
0.30
70.30
6 01.00
71.77
1:24
74.89
3.00
17:33 6: 0.61
81.95
1.68
71.73
01.11
MO0C
82.38
4: 1.75
80:71
2.05
81.53 4: 0.79
90.30
4: 2,10
71.74
0.58
61.00
=: 0.30
70.22
1.21
77.09
0.55
87.25 4: 0.81
73.87
2.77
87.91
0.00
Rediin
80.30
8: 0..00
79.06
0.23
79:00 + 0.09
81.04
0.23
60:12
0.30
78.58
0.00
70.53
0.11
87.85
0.35
86.61 1: 0.58
90.63
2.28
51.71
0.40
Wiltpedle
77.49
K1.28
0.005
05.74
3.46
0.47
0.20
72.78 6.65
95.43
0.07
79:09
1.20
UCT
06.63
3.58
07:12
07
1.11
1.102
27.46
70.71
75.43
Social
97.16
Thnoout
Rank
7:14
0.20
2,86
LastiFM
2.20
03.05
1: 4.13
67.00 0.44
64:76
1.51
0.20
07.57
78.70
0.87
70.75
07:57
2:21
39.75
1.12
73:392
75.99
0.06
75.00
00:74
1.10
70.73
1:08
74067
25:40
78.47
MOOC
07,50 4: 1.76
022
1.27
74.44:2
11:11
76,89
4: 2:33
70:05
0.33
48.18
=: 0.20
71.50 4: 1.00
72:23
0.57
90.78 +: 0.99
68.17 3.77
10.08
0.82
Reddin
63.40 4: 0.53
82.
0.96
67.68
0.10
64.57
+: 0.11
68.19 +: 0.20
$5.98
0.00
64.41
0.18
82:24
0.24
88.25 4: 0.04
91.42 = 2.18
50.35
0.52
Wikipodins
70.41
1: 0.50
07.57
0.94
61.64
+: 0.31
R1.21
2: 0.50
68:45
8.64
$1.78
= 0.20
70.31
1.58
84.20
0.36
61.00 +: 9:75
98.87 = 0.11
76.64
2:42
UCT
61.18
1.25
54.10
2:71
39
0.61
03:76
+ 0.00
87.85
0.30
58.05
0.00
05 06 9:07
74.25
0.71
61.00
+ 0.83
06.51 + 0.75
06:85
Social Evo.
91.1
1.00
02.77
2.64
93.64 4: 0:45
94.56 4: 1,50
50.10 6: 0.11
$7.68
0.30
56.23
0.50
94.30
0,26
95,01 :: 0.18
Thnoout
07.37
0.20
Avg. Rank
7.86
8:57
6.14
5.43
0.79
6.43
6.71
6.00
5.14
3.98
2.57
report the hyperparameter searching strategy as well as the best settings of DyGFormer and DyGMamba on synthetic datasets in Table [1] For all experiments with DyGMamba, set the numbers of layers in both nodo-level and time-level SSMs as 2.
Negative Edge Sampling Strategies during Evaluation
We justify why we do not do historical NSS for inductive link prediction. As described in Poursafaei et al. (2022), historical NSS focuses on sampling negative edges from the set of edges that have been observed during previous timestamps but are absont in the current step. In the setting of inductive link prediction, models are asked to predict the links between the nodes unseen in the training dataset. This means when doing historical NSS, models only need to care about the previously observed edges in the test set (or validation set during validation) for choosing negative edges. This makes historical NSS the same as inductive NSS in the inductive link prediction, where inductive NSS samples negative edges that have been observed only in the test set,, but not training set. Empirical results shown in Appendix C.2 Table 13 and 14 of Yu et al. (2023) also prove that there is no difference between historical and inductive NSS in inductive link prediction. So we omit the results of historical NSS in our paper.
E AUC-ROC Results Real-World Datasets
Table 12 and 13 presents the AUC-ROC results of all baselines and DyGMamba on real-world datasets. We have similar obscrvations as the AP results shown in Table and P DyGMamba still demonstrates superior performance and can achieve the best average rank under any NSS setting in both transductive and inductive link prediction.
23

---

Published in Transactions on Machine Learning Research (05/2025)
Table 13: AUC-ROC of inductive dynamic link prediction. EdgeBank cannot do inductive link prediction so is not reported.
Datasets
JODIE
DyRtep
CAWN
GraphMixer
DyGFormer
CTAN
DyGMamba
LM
82.49 +: 0.94
82.82
1.17
70.75 + 0.22
52.61 +: 2.60
67.92 + 0.15
76.05
1.34
80.51 + 0.144
94:10
0.00
61.49 +: 2.75
94.37 = 0..3
Raudom
Foron
1.50
1.20
1.77
2.24
31000
83.22
0.30
83.42
0.77
86.77
0.24
91.88
0.74
81.76
0.46
52:42
81.76
0.33
87.76
66.38 1.59
80.34
0.12
Reddit
96.42
4: 0.13
96.87
4: 11:21
97.02
2: 1..04
97:30
= 0.12
98.42 2: 0.01
94.63
2: 0.08
94,55
2: 1.08
98.71
0.02
82.85 4: 4.08
98.38
0.01
Wikipedia
96.43 + 0.26
01.31
0.40
06.03
0.10
07.71
0.18
98.05
+ 0.05
97.03
0.08
96.25
+ 0.04
95.19
+ 0.02
92.00 0.70
98.72
0.03
UCI
68.84
0.65
1.49
92:40
6.02
92.70
0.335
911.20
2.05
08.52 =: 0,315
0.90
84.73
0.20
94.63
0.06
0,303
95.30
Timcout
95.30
Avg. Rank
6.00
7:43
7.00
4.57
4.71
0.14
6.14
2.14
9:71
1:14
LaseFM
1,20
5.77
63.15
0.17
69 80
3:07
70.50
Induedivv
Earon
65.05
1.27
02.20
2.15
58.52
0.84
0.94
62.07
0.72
67.90
1.34
67.39 1.33
65.07
68.70
1.82
68.98
MO0C
60.37
+ 11.36
62.37
2.05
74.34 + 11.50
70.36
= 2:91
71.18 + 0.54
71.30
1.21
72.15 + 11.65
80.42 0.72
58.06
#: 0.89
81.32
0.63
Reddit
61.84 a: 0.44
011.33
11.33
64.92 4: 11.08
00:24 +: 0.06
05.37 4: 0.12
61.80
0..1
64.36 4: 0..25
61.00 0.53
81.70
4.71
61.91 :: 0.80
Wikipadin
66.31
0.50
58.21
2.23
UCI
1,82
61.50
2.08
61,27
+ 0.78
02.07
0.67
55.60
1.22
65.87
1.90
70.72 + 0.70
61.37 0.98
51.88
2.60
2:22
Social Evo.
88.98 1: 0.83
86.43
1.48
92.37 4: 0.50
91:00 4: 2:14
80.84 4: 0.21
96.50 4: 0.91
90.88 1. 0.22
9197
0.36
Timcout
96.65
0.29
Avg. Rank
7.00
8.71
5.14
5.14
7.14
5.14
3.57
4.71
6.14
2:29
F Dynamic Node Classification
We first give the definition of the dynamic node classification task.
Desinition 3 (Dynamic Node Classification). Given CTDG 9, source node u E NN, a destination node e N,, timestamp t E T, and all the interactions before t, i.e., {(us, Uz ti)lt < t, (ui, 2,,t4) 9), dynamic node classification adms to predict the state (e.g., dynamic node label) of or at t in the condition that the interaction (u, v, t) exists.
We follow Rossi et al. (2020] Xu al. 2020 Yu et al. 2023 to conduct dynamic node classification by estimating the state of node in given interaction at specific timestamp. A classification MLP is employed to map the node representations as well as the learned temporal patterns to the labels. AUC-ROC is used
as the evaluation metric and we follow the dataset splits introduced in Yu et al. (2023 170%15%/185 for training/ valiiaaiootteeettttetee classification results on Wikipedia and Reddit (the only two CTDG datasets for dynamic node classfication). we observe that DyGMamba can achieve the best average rank, showing its strong performance. Note that both Wikipedia and Reddit are not long-range temporal dependent datasets, therefore do not include this part into the main body of the paper. Nonetheless, DyGMamba's 's great results on these datasets further prove its strength in CTDG modeling, regardless of the type of the dataset (whether long-range temporal dependent or not).
in chronological order) for node classification. Table 14 shows the node
Table 14: AUC-ROC of dynamic node classification.
Datasete JODIE
DeRep
TGAT
TGN
CAWN
TCL
GraphMixer
DyGFormer
CTAN
DyGMamba
Willpedia 88.10 +: 1.07
87. # 1.94
83.42
2.52
85,51
3.28
64.30 # 1.16
7500 #: 1.18
85.00 4: 1.73
86335 # 2.19
87.38 # 0.11
87.44 à: 0.82 67.70 1.32
Reddit
3.18
63.12
65.22
67.29
0.15
Avg. Rank
5,50
6.00
5.00
7.30
7.00
6.00
6.50
4.50
4.0
2.50
G Efficieny Analysis Complete Results
We first provide the efficiency analysis results of all baselines in this section. We then provide a comparison of total training time among DyGFormer, CTAN and DyGMamba.
G.1 Efficiency Statistics for all baselines
We provide the efficiency statistics for all basclines Table 15 To supplement, in Figure D we further plot the comparison of model size, per epoch training time and GPU Memory across models on Reddit and Social Evo., analogous to the analysis in Sec. 4.3.1 CTAN cannot. be trained Social Evo. before timeout and
24

---

Published in Transactions on Machine Learning Research (05/2025)
therefore does not appear in the plot of Social Evo.. The complete statistics including the numbers in the plots can be found in Table 15
Table 15: Efficiency statistics for all baselines. EdgeBank is non-parameterizee and not a machine learning model so we omit it here. # Params means number of parameters (MB). Time and Mem denote per epoch training time (min) and GPU memory (GB), respectively. The numbers in this table are the average results of five runs with different random seeds.
LasIFM
Enron
MODC
UCI
Reddit
Sncial Rvn.
Models
4.4
0.75
4.711
DvRep
2.64
8.6
2:29
2.64
0.10
1.34
2.64
0.55
2.35
2.64
0.03
1.51
2.64
5.73
1.21
2.64
7.55
1.76
TGAT
4.02
22.75
4.15
4.02
1.28
3.46
4.02
4.08
3.64
4.02
0.90
3.42
4.02
15:33
2.58
4,02
25,50
3,80
TGN
3.68
12:14
2.21
3.68
0.15
3:88
1.03
2.54
3.68
0.06
1.51
3.67
2,05
1.67
3.83
1.78
1,96
6.23
0.90
1,80
2:35
5.06
GospiMMioor
2.45
16:55
2.78
2.45
1.20
2:23
2.45
4.02
2:40
2.45
0.73
2.19
2.44
4.52
1,57
2.45
15.00
2.71
DyGromeer
5.56
47.00
7.57
4
2.73
3.23
4.801
8.32
3,35
4.15
0,52
2.30
4.24
7.00
2.42
4.14
20.00
2.77
CTAN
0:45
3.33
1.44
0.47
0.50
1.33
0.68
3.22
2.50
0.50
0.38
1.30
0.55
0.56
1.54
0.45
2.41
0.63
DYGaaal
Reddit
Reddit
Reddit
Params (MB)
Training Time (min)
GPU Memary (GB)
2.4
2.2
2
1.8
2
1.6
1.4
98
99
97
99
98
99
Average Precision
Average Precision
Average Precision
Social Evo.
Social Evo.
Social Evo.
4.2
Training Time/Epoch (min)
GPU Memory (GB)
Params (MB)
3.7
18
2.5
3.2
2.3
2.7
1.9
2.2
3
1.7
Average Precision
Average Precision
Average Precision
TGN TCL
GraphMixer
DyGFormer
CTAN
DyGMamba
Figure 6: Efficiency comparison on Reddit and Social Evo. among DyGMamba and five baselines in terms of number (#) of parameters, training time per epoch and GPU memory.
G.2 Total Training Time Comparison among DyGFormer, CTAN and DyGMamba
We present the per epoch training time, number of epochs until the best performance and the total training time in Table I6 Total training time computes the total amount of time a model requires to reach its maximum performance, without considering the patience during training. We observe that CTAN requires much more epochs to converge, e.g., LastFM it uses almost 54 times of epochs than DyGMamba reach its best performance.
G.3 Modeling Increasing Number of Temporal Neighbors with Limited Total Training Time
To further show DyGMamba's superior efficiency against bascline methods, do the following experiments. We train five best performing models (as shown in Table 1 on Enron with gradually increasing number of temporal neighbors" and report their performance. The number of sampled neighbors spans from 8, 16,
For CTAN, number temporal neighbors we mean the sampler size each graph convolutional layer, Lee, the size of the sampled temporal neighborhood for each node a timestamp.
25

---

Published in Transactions on Machine Learning Research (05/2025)
Table 16: Comparison among DyGFormer, CTAN and DyGMamba on per epoch training time (Tep (min)), number of epochs until the best performance (# Epoch) and the total training time (Thot (min)). Too Tep x # Epoch. The numbers in this table are the average results of five runs with different random seeds. CTAN cannot be trained on Social Evo. before timcout and thus without # Epoch and Tot
Datasets
LastFM
Enron
MOOC # Fpreh
UCI
Reddit
Social Evo.
Modale
T.p.
o Epenc
Tux
Tip
Fpods
T.p
Fjewch
T.p
I Ejuch
DyCiPurnct
47.00
49.60
2081,20
2.73
32,80
64.20
534.14
0.02
21.00
30.22
001.40
CTAN
3.33
635.00
2114.35
0.50
173.00
88,50
0.22
138.0 00.000
440.36
0.38
206.00
00
50.08
0.53
927.18
173.41
0.45
Timeout 24.40
Tumeont
DyGMamba
28.65
11.80
335.71
2,05
35.00
67.65
4.88
38.00
165.64
0.00
28.00
16.80
3332
25.80
88.96
17.80
451.32
TON
CAMN
Uyceromer ormer
CIAN
UyCiMamba
Average Precision
84
2
24
2
27
Number of Sampled Neighbors (log scale)
Figure 7: Performance comparison among TGN, CAWN, DyGFormer, CTAN and DyGMamba on Enron, with an increasing number of encoded temporal neighbors. The metric is AP under random NSS on trans- ductive link prediction. The time limit for training is 120 min. If a model fails to complete training within this limit, cross is used to mark the data point. Dashed lines indicate that models start to exceed time limit as neighbor number increases.
32, 64, 128 to 256 (Note that these numbers are different from the best hyperparameters reported in Yu et al. (2023)) ). We fix the patch size p of DyGFormer and DyGMamba to 1 in order to maximize their input sequence lengths. We set a time limit of 120 minutes for the total training time. We let all the experiments finish the complete training process and note down the ones that exceed the time limit. In this way, we not only care about the per epoch training time, but also pay attention to how long it takes for models to converge. The experimental results are reported in Fig. 7 The points marked with crosses (x) mean that the training process cannot finish within the time limit (although we still plot their corresponding performance). We find that only TGN and DyGMamba can successfully converge within the time limit when the number of considered neighbors increases to 256. DyGMamba can constantly achieve performance gain from modeling more temporal neighbors while TGN cannot. CAWN is extremely time consuming so it cannot finish training within the time limit even when it is asked to model 16 temporal neighbors. As for the methods designed for long-range temporal information propagation, DyGFormer and CTAN consume much longer total training time than DyGMamba. They fail to converge within 120 minutes when the number of considered neighbors reaches 128 and 256, respectively. We also observe that CTAN's performance fluctuates greatly with the increasing temporal neighbors, indicating that it is not stable to model a large number of temporal neighbors. This also implies that increasing the amount of historical information will gradually make CTAN harder to converge, which might cause trouble in modeling long range temporal dependent datascts.
H Details of S4 and Mamba Operations
Single-Input Single-Output. Given a sequence of vector elements as input, SISO means that the SSM processes each input dimension in parallel with the same set of parameters. For example, sequence of
26

---

Published in Transactions on Machine Learning Research (05/2025)
dg-dimensional vectors will be split into d2 1-dimensional sequences with the same sequence length. Each of them will be computed in parallel as in Eq. 2 with shared set of SSM parameters. After computation, all these d2 sequences will be rearranged back into a sequence of dg-dimensional vectors. SISO fails to mix the information across dimensions cach vector. To address this, S4 and Mamba employs a mixing linear layer mmix()) Rdd + RRda on each dz-dimensional vector to mix the information across d2 dimensions. For more details, please refer (Gu ct al. 2022b), (Smith et al. 2023 and (Gu & Dao 2023
SSM Function. C)(),B,c()) takes a matrix as imput. The input matrix can be considered as a sequence of vector clements, where each row of the matrix corresponds to an element. The output of ASMA.B.C. c..(.) is also a matrix, where each row of the output matrix is the output of its corresponding input vector element. SSM A.B.C( can be viewed as using S4 or Mamba to process a sequence of vectors in the STSO fashion, based on their paramcters A, C.C.
Motivation of Using SSM for Temporal Pattern Modeling
The biggest motivation using SSM for temporal pattern modeling is that helps to maintain good efficiency. If in the future we want to deploy DyGMamba on larger datasets that require much longer historical histories for modeling, the value of k will also increase accordingly and the time difference sequence will not be short anymore. Besides, as have chosen SSM to model historical one-hop temporal neighbors in the node-level SSM, it is natural to employ another SSM for temporal pattern modeling.
J Impact of Patch Size on MOOC
500
DyGFomer DyGMambba
byGFomie
20170
DyGFormer
400
Average Precision (Rardom
80
60-
24
Number Perameters (MB)
4.43
4.4
GPU Menorry (MB)
20000
DyCMambe
Tra TimeeEpoch (3)
450
Mamba
800
cn
$8.45
25000
35.
300
67
10000
16
DyGFummer
17
5000
2528
2038
3054
300
785
20
60
80
100 120
140
20
60
60
00
100
120
140
20
40
60
80
100
120
140
20
40
60
60
100
120
140
Sequence Length
Sequence Length
Sequence Length
Sequence Length
(a) MOOC performance.
(b) MOOC # paramts.
(c) MOOC GPU memory.
(d) MOOC train time//poch.
Figure 8: Impact of patch size DyGFormer, DyGMamba and Variant A, given fixed mumber of sampled temporal neighbors p on MOOC. Patch size varies from 4, 2, 1. Sequence length p/p increases patch size decreases. Performance is the transductive AP under random NSS.
Apart from the analysis on Enron, we further analyze the impact of patch size another long-tern temporal dependent datasct MOOC. Fig. shows the numbers of parameters of DyGFormer and DyGmamba with different patch sizes on MOOC. We confirm that patching greatly affects model sizes. We decrease the patch size gradually from 4 to (the optimal value of DyGMans's hyperparameter & p is 128 & 4) and track DyGMamba's 's performance (Fig. Ba)) as well as efficiency (Fig. 8b to 8d) on MOOC. Meanwhile, we also keep track on DyGFormer under the same patch size for comparison. Same as our analysis on Enron, we plot the performance of Variant A under different patch sizes in Fig. well. We can draw the same conclusions as in analysis Enron.
K Performance of DyGMamba on Discrete-Ting Dynamic Graphs
To better benchmark DyGMamba, we test the performance of DyGMamba on DTDG datasets collected
in YYu al.] 2023) (i.e., Flights, Can. Parl, US Legis., UN Trade, UN Vote and Contact) and compare it
27

---

Published in Transactions on Machine Learning Research (05/2025)
with the 10 recent baselines discussed in Sec. 4.1 (i.e., JODIE, DyRep, TGAT, TGN, CAWN, EdgeBank, TCI, GraphMixer, DyGFormer, C The value of the hyperparameter p & p of DyGMamba is set as same as the one for DyGFormer. The optimal p & for Flights, Can. Parl, US Legis., UN Trade, UN Vote and Contact is {256 & 8, 2048 & 64, 256 & 8, 256 & 8, 256 & 8, 128 & 32 & 17. We report the best hyperparameter k of DyGMamba for each dataset in Table 17 We use the implementations and the best hyperparameters provided by Yu et al. (2023) for all bascline models except CTAN. For CTAN, we use its official implementation, fixing the number of layers to 5. All models are trained with batch size of 200 for fair efficiency analysis. We report the AP of both models on DTDG datasets in the transductive and inductive settings in Table 18 and Table 19 respectively. In addition, Table 20 and Table 21 present the AUC-ROC of both models in the transductive and inductive settings, respectively.
We find that DyGMamba achieves strong overall performance (Avg. Rank) on DTDGs in both transduc- tive and inductive link prediction tasks. However, compared to its performance on CTDGs, DyGMamba's 's performance on DTDGs is not always highly competitive. This expected, as DyGMamba is not originally designed for DTDGs and lacks the ability to optimally handle concurrent edges.
One interesting finding is that the long-range temporal dependent dataset Can. Parl which requires sam- pling 2048 neighbors for modeling, DyGMamba can benefit from such long neighbor sequence and achieve the best performance among all models, indicating the importance of modeling long-term temporal information as well as the strong capability of our model in capturing it. More importantly, we find that DyGMamba can benefit from greater value k as the number of the sampled neighbors p increases. For example, on Can. Parl, as shown in Table 17 the optimal value of k is 100. This makes our selection of using Mamba for temporal pattern modeling more reasonable since Mamba can better demonstrate its advantage in efficiency when there is growing time difference sequence corresponding to the temporal pattern.
Table 17: DyGMamba hyperparameter searching strategy on DTDG datasets. The best settings are marked as bold.
Datasets
Flights
(30, 10, 5)
Can. Parl.
(200, 100, 30)
US Logis.
{30, 10, 5)
UN Trade
{30, 10, 5)
UN Vote
{30, 10, 5)
Contact
(30, 10, 5}
L Detailed Discussion of Motivation
The goal of this work is to introduce a model capable of effective and efficient reasoning over CTDGs using extensive historical information. Our motivation is supported by the following considerations:
Abundant Historical Information in Real-Worke Datasets. We quantify the availability of histor- ical neighbors across various datasets for transductive and inductive link prediction tasks under random, historical, and inductive NSS settings, as shown in Table 22 Observations indicate that real-world CTDG datasets, such as LastFM and Enron, typically possess extensive one-hop historical information. Although it is feasible perform predictions based solely recent interactions, we posit that effectively leveraging longer historical contexts can further enhance model performance.
Limitations of Existing Models in Handling Extensive Historical Information. Two recent state- of-the-art methods, DyGrormer and CTAN, explicitly designed to capture long-range historical dependen- cies, have been considered in our study. Our experimental results reveal distinct limitations: DyGFormer
Although have discussed Sec. 7 that DyGMamba not suitable reason over DTDGs, we still benchmark our model them to show its effectiveness.
28

---

Published in Transactions on Machine Learning Research (05/2025)
Table 18: AP of transductive dynamic link prediction on DTDGs, The best and the second best results are marked as bold and underlined, respectively.
Dutasuta
JODIE
DyRop
TCAT
TCN
CAWN
EdgaBunk
TCL
GraphMisor
DyGPurmer
CTAN
Dy DyCMambb
Caa. Parl
69.58
=: 0.25
68.94
+: 5.65
21,77
1 0.65
09:57
+: 1.86
10.07 = 3.01
64.55 0.00
71:00
1: 0:45
77.40 = 0.62
55.63 a 0.21
87.01 d: 1.10
99.57
:: 0.08
Rondom
Contact
94.75
0.82
96.97
0.16
96.53
0.11
97.21
+ 0.01
10.05
00.6
01.38
01000
91.07
0:111
91 0.07
18. 16:30-2 0:00
98.53 4 0009
10:13 + 0.12
Flihtt
9003
= 1.7
90.49
2 0.49
98.92
2: 0.04
95.01
0.23
58.50 = 0.00
80.35
0.30
91,30
2 0.02
91.2 = 0.04
58. 08.52
0.01
93,64
4: 0.54
98.95
0.06
UN Trade
64:43
0.25
02.8
4: 0.3
61.53
2 0.15
GLU
+: 0.07
65.41
= 0..04
0::11
0.30
02.19
+ 0.07
02 = 0.12
00.00
2: 0.50
09.01
4: 0.38
07.30
0.24
UN Vite
65.35
0.90
68.00
4: 0.5
82.80
0.25
05.16
0.0.
52.77
0.007
68.49 0.00
61.80
2
52.18 0.15
66.50 0.21
70.19
1: 1.38
56.30
0.16
71
7581
2:19
07.83
76,00
0:33
T0.00
56.30
70:0
71.06
1.00
70.44
2:288
Avg. Rank
5.17
0.17
7.50
4.35
7.50
9.67
8:90
7:67
4.00
2.90
3.00
Parl
62
2.75
0.47
66.58
3.54
0.00
1.81
99.77
Contact
96335
16-17
0.17
96.73
0..00
90.10
15 0.118
0.35
17.61 0..00
Flishts
67.63
1.79
22.00
07:77
1.08
64.00
0.64
70.53
0..00
71.11 0.15
71.40
66.83 0.21
76.75
8.04
07,50
2.17
38.35 06.89:1
1.04
06.04
2.43
57:00
1.500
$1.08
0.00
57.91
(22
1.00
07,39
0.32
UN
70:33
1:21
2:91
16
1.83
51.29
0.41
$4.76
0.00
63.78
54.17
1.08
00:30
0.94
74.75
61,07 0.33
Legie.
47.20-2
1.72
H: 7.14
70/12 1: L.15
07:43
+: 6.79
T3.85
14.99
0.00 151.31
83.07
1: 0.888
86.05
1:77
63.19 4: 11 11.58
84.20 1: 0.005
80.15 1.00
Aug. Rank
7.53
0.30
0.50
7:00
9:83
6.00
6.17
5.93
5,83
3.67
Cics. Parl.
48.13
0.42
57.99 +: 2.61
69.54 +: (1.55
63.08 +: 1.39
06.94 =: 11.44
02.20
00.00
65.21 +: 1.42
70:00 =: 0..57
15,01 +: 00.50
97.42 d: 0.71
96.38
0.34
Inductiv
Contact
92.34
0: 1.51
99:19-
6: 01.24
15:14:4
8: 1.19
01:30
+: 0.885
99.61 :: 0..36
65.31
01.00
12:18
:: 0.:32
50.54 01.36
144.50 3: 00.54
07.31 8: 0.119
10.45 6 0..17
Flight
60.39
0: 2.19
71:82
6: 2.44
75.08
2: 0.37
72.27
4: 1.83
00.01
=: 0.355
81.07
0.00
75.10
2: 0.05
74.88 = 0.35
70.31 11:00
76.30 +: 7.96
77.79 5.50
UN Toale
60.55
1.20
60:15
0.52
38.96-1
0:38
59.52
5.79
63.377
9.34
73.00
0.00
01.88
1.43
60.05 0.65
53
00.27
08.00 1 0.733
58.89 00.96
UN Vice
67.81
0.75
68.09
0.21
52.05
1.18
07.58
1.5:
51.90
1.40
06.31 0.00
19.17
1.01
51.26 1.25
53.00
0..23
72.85 1: 1.33
52.34 00.55
46.5
70:16 8:
01.85
T.02
14:35
83,71
0.00
79.11
2.36
Avg. Rank
8:00
0.00
5.50
7.00
7.83
5.83
0.67
5.50
6.83
2.17
4.67
Table 19: AP of inductive dynamic link prodiction on DTDGs. EdgeBank cannot do inductive link prediction SO is not reported.
N8S
Datamels
JODIE
DyRup
TGAT
TGN
CAWN
TCL
GraphiMissr
Dy GlPemeer
CTAN
DyGMMaraba
Can. Parl.
$3.00 1: 1.27
54.28 :: 1.84
55 $5.60.1
1: 0.57
53:47 :: 1.04
55.53 : 0.98
55.06 :: 0.67
56.00 1: 0.01
57.14 +: 1.40
62.99 = 0.34
93.86 :: 2.02
Randow
Contact
05.20
0:51
02:32
0.20
96.18
0.21
02.74
+ 2.86
80.98 0.30
03:95
+ 0.14
00:51
0:03
98.01 + 0.02
70.10-
(1-1--5771 5.71
98.16
0.03
Flighti
91.45
92.80
95.03
1.04
83.54
K3.05
97.79
0.04
76.12
UN Trade
08.04
0.22
50.46
0.00
61.23
66.97
65,30 0.14
02:24
0.27
02.55
64.02 : 11.88
58.83
0.20
70.66
Vote
55.05
55.63
1.43
$0.38
57.86
1.72
10.05 0.77
53:38
0.00
51.96
1.32
56.80 0.16
55.02
50.61
0.13
US Eegis.
51.36
1,00
07.64
1.45
50.21 + 1.77
80.39
2.50
52.79
0:41
07.12 + 0.22
54.68 + 2.07
$4.39 + 1.43
02.55
3:42
56.95 + 1.16
Avg. Rank
6.00
6.17
6,50
5.00
6,33
5.59
6.50
3,00
8:00
1.07
Can. Pael.
53.91
1.35
02.62
1.85
57.42
0.57
03.50 1.96
57.56 0.81
66.59
1.22
56.91 0.55
84.72 + 2.58
63.27
92.68
0.977
Inductive
Contact
00.22
4: 0.35
80.18
d: 0.48
94.93
:: 0.97
N8.74 :: 2,60
74,88 :: 0.81
91.41
=: 0.40
N9.67 :: 0.06
50.75 :: 11.88
70.30
5.008
94.05 :: 0.52
Flights
00.79
1.25
02.54
1.73
55.23
0.42
110.58
:: 1.56
56.76 2: 0.11
65.12
0.09
05.26 4. 0.44
56,45 1: 0.11
66.53
2.41
57.76
1: 2.06
UN Trade
55.30
0.88
54.52
0.52
$4.21
51.84
1.57
56.29
1.86
55.82
15.73
0.20
51,92
0.10
51.91
0.57
32.81
0.18
59:24
52,79
0.54
53,10
2.40
US Eegin
55.50
2.38
50.60
1.64
50.85
2.50
60.19
2.60
55.15
1.35
58.00 00.50
0.90
55.15 0:27
$4.76 + 2.55
52.65
1.00
57.85
0.23
Avg: Rank
5.50
5.00
5.50
5.83
0.50
1.00
5:39
0:33
6.17
4.67
Table 20: AUC-ROC of transductive dynamic link prediction DTDGGs
N88
Datasets
JODIE
Dyltep
TEAT
TGN
CAWN
EdgeBBank
TCL
GraphiMissr
DyGroreee
CTAN
DyGMMamaa
Pari
75.84 0.24
70.50
76.36
0.95
75.05
76.24
66.14
0.00
76.64
0.33
83.22
0.5G
18.00
56.77 1.17
99.09
Random
0.02
Flightx
0.57
0.45
06.92
63
0.15
07.00
56.7
0.00
64.22
0.01
03:97
0.30
70.50
72.00
0.24
71.41
0.21
08.77
53.08
05.02
53.15
9297
0.00
51.73
0:25
50.00
0.00
57.65
73.91
0.12
Lega.
0.50
50.98
1.82
2.19 75.00
35.03 8 0.27
77.10
02.57
0.00
77.11
0.30
70.19
0.28
77.66
86.01
70003
0.26
Avg. Hank
5,30
1..53
9:00
7.53
2.07
12.00
Can. Pari.
62.08
0.00
70.62 = 3.57
71.11
0.95
71.03 4: 0.00
70.44 8: 3.60
62.94
0.00
7407
1.60
18. 56.56-2
0.72
10.16 =: 1.08
60:06
1.14
99.52
0.10
Iistorical
Contuct
96.21
1: 0.67
05.50
0.14
56.01
0.61
96.94
4: 0.24
$3.65 : 0.40
92.15
0.00
$4.70
:: 0.18
52:36
0.30
50.20
= 0.05
$8.57
0.10
97.27
0.06
Flighos
05.61-1
0-0.7)
00:35
1.48
19:35
0:25
03.13-
+ 0.81
05.95-1
0.15
71 81
0.00
70.81
0.7
70.7%
0.32
67.50
0.46
78.67
7.99
08:05
UN Treee
05.60
1.50
63.51
= 1.60
59:35
0.55
01.61
+ 3,58
64.21
+ 2:21
66.14
0.00
62.11
1.05
61.50
0.17
19.17
1.66
68.75
0.30
71.11
0.21
UN Vote
77.54
1.00
76.32
1.04
55.61
2:41
73.31
+: 1.65
50:90
=: 0.72
89.50
0.00
$4.70
1.78
16.20
1.46
64.70
1.35
78.62
1.16
65.17
1.24
DS Loga.
64.20
3.67
88.47
4.92
68.40
4.72
79.20
4: 4.62
0.96 50.85
67.50
0.00
56.20
0.84
90.35
0.73
9.32 70.60
50 07:58
0.54
$8.36
1.7
Avg. Bank
7.55
6.00
0.55
6:85
9.17
3,67
6.55
5.67
5.50
2.17
4.00
Porl
5200 4
63,40
1.74
72:48
11.52
71.06:5
61
1.00
72:29 1.43
T1.25
0..66
1G.77 01.55
$3.20 0.95
06
0.21
Incuctive
0.24
01.11
94.76
94.70
68.26
55
0.00
92.25
06.71
$6.85
0.20
Flighes
71.38 1.84
79.56
72.77
0.04
72.00
0.35
79.52 7.50
71.15
8.1
1.51
UN
71,67
7287
0.00
50:25
7.46
1.06
Legs,
53.14 8.57
96.70 5,01
10 3.05
73.00
79,50 10.01
SN/72 0.00
S153
3.50
55.82
0.39
73.31
56,00
1.31
56006 2.27
Avg. Bank
7.83
5.50
6.17
29

---

Published in Transactions on Machine Learning Research (05/2025)
Table 21: AUC-ROC of inductive dynamic link prediction on DTDGs, EdgeBank cannot do inductive link prediction so is not reported.
Datasets
JODIE
DyRep
TGN
CAWN
GraphMixer
DyGPormer
CTAN
MaGMa
Can. Part.
53.02 = 1.26
55.89 d: 1.77
36.57
5: 0.00
53.36 a 2.11
51.01 :: 0..4
54.92 1: 0.91
19.29 2: 0.00
59.06 i 0.30
59.15 :: 0.36
94.02 I: 3.02
Rondom
Contaus
06.96
+ 0.20
02.00
+ 00:10
90.78
0.11
01.04
2.00
59.00 + -0:32
04.82 + 0.000
02.81 0.00
08.30 + 0.50
90.41 + 4.30
08.44
00.05
Fights
94.82
0.55
50.64
0.44
88.61
0.12
05.81
0.02
52.56
0.13
82.28
97,80
0.05
79.41 5.79
97.98
00.25
UN Trade
0.36
0.21
58.71
0.15
07,25
68:26
0.26
Vone
56.21
1.50
56.72
1.90
52.64
1.09
50.42
2.12
47.02
1:10
51.01
0.60
51.05 1:10
57:47 0.27
53.80
3.51
56.91
0012
US Legis.
07.25
a 1.92
00.86
4. 1.39
36,96
1: 3,44
62:36
4. 2,04
30.91 4: 1.02
58.02 4: 1.37
54.98
2:39
52.96
1.71
53.89 2: 3.55
57.17 4. 1120
Avg. Rank
5.83
0.17
0.67
4.85
0.50
0:93
0.33
3.00
7.85
2.00
Parl.
50.56 2.02
52.07 1.53
58.01
0.45
54.61
0.116
1:15
58.59
2.37
50.05 1.05
92.37 0.18
Indortive
Caniass
01,20 = 0.17
88,90
0.20
94.44
89.78
70.35 0.50
0.28
89.82
01483
74.41
3.94
91.35 0.29
Fighis
00.58 = 0.50
61.64 #: 1.601
60.61
2: 0.82
50.28
0.55
56.51 =: 0.03
00.66 + 0.11
66.07
0.35
55.67 =: 1148
71.35 4: 1.86
16.08 3: 2.12
UN Trade
58.51 + 1.05
56.96 + 0.41
58.71
+ 0.45
54.45 + 1.33
62.005 + 2.01
B1.14 + 1.09
01.00 +0000000000 0.35
55.80 + 0.11
52.05 + 0.55
57.55 + 0.70
UN Veer
02.88
1.62
65,80 + 2.47
51.70 3.99
71.29 2.01
46.01 + 1.35
57.55 + 2.26
45.75 1.04
51.02 0.50
14.31 + 1.14
51.85 + 2.17
US Logis
30.83
2.06
65.64
1.42
47.35
4.25
03:90
2.15
1.06
16.22
63.07
03.64 2.12
Rank
5.37
5.67
3.93
0.83
suffers from high computational complexity, while CTAN exhibits comparatively weaker predictive perfor-
mance. Consequently, neither model achieves an optimal balance between effectiveness and computational efficiency. Our proposed model, DyGMamba, overcomes these limitations by demonstrating superior predic- live capability and efficiency. Exiensive experiments position DyGMamba as the top ranked method across multiple benchmark datasets.
Proven Performance Enhancement with Increased Historical Information. In Appendix G.3 we have included an experiment showing different models' performance with a varying number of temporal neighbors. From Figure 7 we observe that, for datasots with long-range temporal dependencies (such as Enron), an increase in sampled neighbors positively impacts model performance, partieularly for DyGMamba and DyGFormer. We also find that for other baseline models, increasing the number of sampled neighbors does not consistently lead to performance improvements. This suggests that the key issue is not whether increasing sampled neighbors is always beneficial for dynamic graph modeling, but rather whether a model is robust enough to evfectively process larger amounts of historical information. To further support this, we conducted additional experiments on Can. Par! 12 using DyGMamba with an increasing number of sampled temporal neighbors (64, 128, 256, 512, 1024 and 2048). The results, presented in Table 23 confirm that DyGMamba consistently benefits from an increasing neighbor count until reaching the large number of 2048, further demonstrating its ability to leverage long-range temporal dependencies.
Temporal Pattern Modeling is Game Changer. As mentioned in introduction, we wish to capture edge-specific temporal patterns for better CTDG modeling. The motivation can be well supported with the following experiment. We provide a comparison between DyGMamba, DyGFormer, and a modified version of DyGFormer where Transformer is replaced with Mamba (referred to as Variant G). Our results in Table 24 show a performance drop for Variant G across all datasets compared to DyGFormer. Additionally, Variant G performs significantly worse than DyGMamba, with particularly notable declines on the synthetic datasets S1, S2, and S3. We altribute this to two reasons:
Intrinsic Limitations of Mamba: Previous study (Waleffe et al.] 2024) has shown that Mamba is generally less effective than Transformer in tasks requiring st.oong copying or in-context learning abilities. In CTDG modeling, strong copying ability is crucial, as many predictions are based re- calling repeated edges. Consequently, replacing Transformer with Mamba in DyGFormer inevitably leads to a performance drop, despite the gains in efficiency.
The Importance of Temporal Pattern Modeling: Our dynamic information selection module plays crucial role in enabling Mambe-based model to outperform a Transformer-basell model in CTDG modeling, especially when clear temporal patterns exist. This suggests that effectively capturing temporal patterns can significantly enhance the competitiveness of Mamba-based models in this context, further validating the novelty and the contribution of our work.
12 Although Can. Parl a DTDG, requires huge number of sampled historical neighbors for optimal modeling. Therefore, it considered here to prove the performance enhancement with increased historical information.
30

---

Published in Transactions on Machine Learning Research (05/2025)
To summarize,, the design our DyGMamba achieves good balance between efficiency and effectiveness, thanks to temporal pattern modeling.
Table 22: Number of available historical neighbors for all datasets both transductive and inductive link prediction under inductive link prediction, respectively.
/inductive NSS settings. Trans LP and Ind LP denote transductive and
NSS
Random NSS
Historical NSS
Induetive NSS
Trans
Ind LP
Trans LP
Trans LP
Ind LP
Datasets
Avg.
Max
Avg.
Max
Avg.
Max
Avg.
Max
Avg.
Max
LastFM
2,253.03
51,767
2,333.75
51,767
2,393.71
51,767
2,237.20
51,767
2,309.49
51,767
Enron
1,681.80
21,512
1,693.40
21,511
1,734.19
21,512
1,675.21
21,512
1,670.65
21,511
MOOC
2,304.04
19,473
2,312.62
19,473
3,265.62
19,473
2,293.64
19,473
2,275.23
19,473
Reddit
4,129.77
58,726
4,196.20
58,726
4,604.99
58,726
4,128.88
58,726
1,186.38
58,726
Wikipedia
144.39
1,937
149.67
1,937
164.49
1,937
144.17
1,937
149.22
1,937
UCI
172.82
1,546
193.87
1,546
261.31
1,546
173.08
1,546
192.72
1,546
Social Evo.
63,962.35
124,565
78,70133
124,565
60,403.62
124,565
60,395.83
124,565
65,191.24
124,565
Table 23: DyGMamba's performance with increasing sampled neighbors on transductive link prediction
under random NSS.
Can. Parl
64
128
256
512
1024
2048
DyGMamba
72.58
74.46
76.85
82.67
94.88
99.57
Table 24: Comparison among Variant G, DyGFore and DyGMamba on transductive link prediction under random NSS.
Models
LastFM
Enron
MOOC
Reddit
Wikipedia
UCI
Social Evo.
S1
$2
$3
Variant G
92.69
92.24
87.13
98.89
98.70
95.32
94.29
54.40
55.13
72.53
FoGrerr
92.95
92.42
87.66
99.22
99.03
95.74
94.63
55.19
57.80
79.20
MMabaa
93.35
92.65
89.21
99.32
99.15
95.91
94.77
81.85
85.36
86.59
31


Reviews:
[{"review_id": "TjfuVzrkvj", "reviewer": "Reviewer_JfoB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The **main strengths** of this paper can be summarized as follows:\n- **(S1)** The authors propose a novel method for CDG built upon Mamba that has recently gathered significant attention, applying it to the challenging task of temporal dynamic link prediction.\n- **(S2)** The evaluation for the task of temporal dynamic link prediction is thorough, incorporating comparisons on several datasets and baselines.\n- **(S3)** In most cases, DyGMamba offers slightly superior performance compared to CTAN and DyGFormer, while it benefits from improved computation times compared to DyGFormer in the transductive setting. The performance gains of DyGMamba are particularly notable in the transductive/inductive setting and NSS random.\n- **(S4)** On the Enron dataset, DyGMamba performs significantly better than DyGFormer when increasing sequence length (i.e., reducing patch count) with fewer parameters and lower training time.\n\nThe **weaknesses** of the study are the following:\n- **(W1)** In historical and inductive NSS settings, the model is in several cases outperformed or matched in performance by CTAN (and in fewer cases matched by DyGFormer). The efficiency comparison of CTAN in Fig. 2 gives mixed results, but in some cases, CTAN is very competitive.\n- **(W2)** The discussion on graph-based Mamba is limited. Discussion in related work could be extended by comparing the model's main blocks to related models, even if they focus on static rather than dynamic tasks, given their shared characteristics.\n- **(W3)** The model has complex input processing blocks, e.g., incorporating neighbor features with temporal encoding (from TGAT) and historical interaction frequencies (similar to DyGFormer) while employing neighbor patching to reduce resource usage. However, the necessity of these design choices is not well-justified, and their effect has not been studied in ablations.\n- **(W4)** DyGMamba memory- and computation efficiency are improved primarily compared to DyGFormer, which is notably resource-intensive despite being its closest performance competitor. However, several models seem significantly lighter and faster (e.g., the performance-competitive CTAN) than DyGMamba. \n- **(W5)** The efficiency comparison in Fig. 2 (time, memory) would be more informative if extended to larger datasets such as Social Evolution and Reddit, similar to how sequence length impact is analyzed in Fig. 3.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose DyGMamba for Continuous Time Dynamic Graphs (CTDG), leveraging the recent Mamba model and adjusting it for representation learning on temporal dynamic graphs, which is applied to the task of temporal link prediction. Specifically, DyGMamba is based on a node-level Mamba State-Space Model SSM that encodes the sequence of historical node interactions and a time-level Mamba SSM that extracts edge-specific temporal patterns for the dynamic selection of temporal history interactions. DyGMamba is very competitive with the state-of-the-art DyGFormer method and, for several datasets, outcompetes several CTDG baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Based on weakness, the following changes could better support the main claims and findings of the paper:\n- **[Based on W2]** Authors can expand the discussion on graph-based Mamba by including details and differences with methods designed for static graphs. These could better support the methodological novelty of the proposed approach.\n- **[Based on W3]** A justified breakdown of DyGMamba’s preprocessing modules (e.g., features, temporal encoding, patching) and their performance effect in ablations is an important aspect to be covered.\n- **[Based on W4]** The efficiency comparison (time, memory) in Fig. 2 can be extended to include larger datasets such as Social Evo. and Reddit, and possibly settings (except for random NSS/transductive) or methods (e.g., CTAN).\n- **[Others]** CTDG models events as individual occurrences with precise timestamps, whereas the evaluated datasets are largely discrete. It remains unclear whether DyGMamba can adapt to irregular timestamps. If so, can the authors clarify which model components ensure temporal continuity?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "KKfxN8sYYR", "reviewer": "Reviewer_NbKH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- The paper provides a comprehensive list of comparative experiments, ablation studies and efficiency analysis.\n- By incorporating longer histories, the study aims to better learn patterns on the graph that are not specific to pairs of nodes, but to lifespan of interactions, e.g. repeated edges. The example on user-song graph provided in the introduction makes sense intuitively. Having said that, I could not locate any experiments or discussion of results that connect back to that intuition. I think you can relate the performance improvements to density of repeated edges. Following paper makes a good analysis of repeating edges on temporal graphs:\n    \n    Razieh Shirzadkhani, Shenyang Huang, Elahe Kooshafar, Reihaneh Rabbany, and Farimah Poursafaei. 2024. Temporal Graph Analysis with TGX. In Proceedings of the 17th ACM International Conference on Web Search and Data Mining (WSDM '24). Association for Computing Machinery, New York, NY, USA, 1086–1089. https://doi.org/10.1145/3616855.3635694\n    \n- The usage of SSM on the feature matrices built by one-hop neighbourhoods on a temporal setting is an innovative alternative to multi-layer message passing. The order of interactions  provide a natural sequence, thus the potential benefit of SSM on temporal settings is somehow convincing to me. However, there may be a loss of information caused by not considering higher order neighbourhoods in some settings (on datasets that are less vulnerable to over-smoothing). I would love to see a discussion and/or an experiment to address this limitation.\n- As also shown by ablation studies, information selection on temporal patterns rather than node level patterns is an important contribution.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a model, DyGMamba, for temporal graph learning. DyGMamba uses state-space-models (SSMs), to encode long temporal dependencies on node and edge embeddings. The main set of experiments are conducted on 7 benchmark link prediction datasets and compared against 9 state-of-the-art methods. The experiments show the proposed model provides superior performance to existing work  and the individual modules of the model are shown to be contributive by ablation studies.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- A discussion on which settings DyGMamba is expected to perform better than SOTA.\n- A discussion on which settings DyGMamba would fail against SOTA.\n\nPlease see Strengths and Weaknesses.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ZFzgOs6tPt", "reviewer": "Reviewer_TVZU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "***S1*** The overall writing is clear and easy to follow, particularly in the introduction.\n\n***S2*** The experiments are extensive, including complexity analysis and efficiency studies.\n\n--- \n\n***W1*** Limited novelty and effectiveness. The methodology of DyGMamba appears to be a trivial combination of DyGFormer’s design (e.g., the feature encoder, patching) with the existing Mamba architecture. I wonder whether replacing the backbone Transformer in DyGFormer with Mamba would yield comparable performance to DyGMamba. If so, adopting Mamba as the new backbone for continuous-time dynamic graphs (CTDG) would be more of a technical adjustment than a significant research contribution. Despite improvements in efficiency, Tables 1 and 2 suggest that the performance gains of DyGMamba compared to DyGFormer are marginal on most datasets. \n\n***W2*** Unclear motivation. The motivation of the paper centers on addressing efficiency issues, specifically that existing transformer-based methods struggle to model long histories effectively. The authors propose using the existing Mamba model to tackle this problem. However, the paper lacks statistical information, such as the average or maximum number of historical neighbors, which would provide a clearer outline. Additionally, there are no examples demonstrating DyGFormer’s limitations under these conditions. I also wonder whether it is necessary to model all historical information when dealing with extremely large numbers of historical neighbors (e.g., 8192). In such cases, it may be more effective to consider only the most recent neighbors. Although the authors include an experiment with varying numbers of sampled neighbors, I am curious about the impact of different sampled neighbor counts on performance—specifically, whether increasing the number of neighbors consistently leads to performance improvements and to what extent.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies representation learning in continuous-time dynamic graphs and introduces a new method, DyGMamba. Leveraging Mamba as the backbone, this method effectively encodes long historical neighbor information while maintaining efficiency. Specifically, it stacks node-level and time-level blocks to capture both types of information, which are then combined for downstream predictions. Extensive experiments demonstrate promising results.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I recommend that the authors further refine their discussion of the motivation.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper addresses an interesting and important task, continuous-time dynamic graph representation learning, and proposes and investigates a promising novel approach. As such, the findings are likely to be of interest to individuals in TMLR's audience.

The main claims of the paper are that: 
(1)	The introduced DyGMamba is the first model to use state-space models for continuous-time dynamic graph representation learning;
(2)	DyGMamba is efficient and effectively models long-term temporal dependencies; 
(3)	For the conducted experiments on commonly-used datasets, DyGMamba outperforms the baselines for the dynamic link prediction task. 

The paper provides experimental evidence to support these claims.
Decision: Accept (poster)