DyG-Mamba: Continuous State Space Modeling on Dynamic Graphs
Authors: Dongyuan Li
Shiyin Tan
Ying Zhang
Ming Jin
Shirui Pan
Manabu Okumura
Renhe Jiang
Venue: NeurIPS 2025 poster ()
Keywords: Temporal Graph
Link Prediction
Long-term Sequence Modeling
Primary area: deep_learning
TL;DR: Inspired by Ebbinghaus' forgetting curve, we propose a continuous state space model for dynamic graph modeling and achieve state-of-the-art performance.

Abstract: Dynamic graph modeling aims to uncover evolutionary patterns in real-world systems, enabling accurate social recommendation and early detection of cancer cells. Inspired by the success of recent state space models in efficiently capturing long-term dependencies, we propose DyG-Mamba by translating dynamic graph modeling into a long-term sequence modeling problem. Specifically, inspired by Ebbinghaus' forgetting curve, we treat the irregular timespans between events as control signals, allowing DyG-Mamba to dynamically adjust the forgetting of historical information. This mechanism ensures effective usage of irregular timespans, thereby improving both model effectiveness and inductive capability. In addition, inspired by Ebbinghaus' review cycle, we redefine core parameters to ensure that DyG-Mamba selectively reviews historical information and filters out noisy inputs, further enhancing the model’s robustness. Through exhaustive experiments on 12 datasets covering dynamic link prediction and node classification tasks, we show that DyG-Mamba achieves state-of-the-art performance on most datasets, while demonstrating significantly improved computational and memory efficiency. Our code is available at https://github.com/Clearloveyuan/DyG-Mamba.

DyG-Mamba: Continuous State Space Modeling on Dynamic Graphs
Dongyuan Li', Shiyin Tan', Ying Zhang', Ming Jin",
Shirui Pan', Manabu Okumura', Renhe Jiang'
'The University of Tokyo, IInstitute of Science Tokyo,
SRIKEN Center for Advanced Intelligence Project, *Griffith University
lidyécsis t-tooyo..... ac. jp. tanshiyin@lr pi. titech. ac.jp, ying. zhang&riken, jp. mingj inedu@gmai......... com, s. panagriffith edu..edu.aa au, okuîpi. oku@pi.titech.. titech. ac. jc.jp. jiangrh@csis u-tokyo.ace ac.jpp
Abstract
Dynamic graph modeling aims to uncover evolutionary patterns in real-world systems, enabling accurate social recommendation and early detection of cancer cells. Inspired by the success of recent state space models in efficiently capturing long-term dependencies, we propose DyG-Mamba by translating dynamic graph modeling into a long-term sequence modeling problem. Specifically, inspired by Ebbinghaus' forgetting curve, we treat the irregular timespans between events as control signals, allowing DyG-Mamba to dynamically adjust the forgetting of his- torical information. This mechanism ensures effective usage of irregular timespans, thereby improving both model effectiveness and inductive capability. In addition, inspired by Ebbinghaus' review cycle, we redefine core parameters to ensure that DyG-Mamba selectively reviews historical information and filters out noisy inputs, further enhancing the model's 's robustness. Through exhaustive experiments on 12 datasets covering dynamic link prediction and node classification tasks, show that DyG-Mamba achieves statc-of-the- demonstrating significantly improved computational and memory efficiency. Code is available at [https: /tt///gttttttttt/ com/ Claarlovyyuannyy--- -Mamba.....
performance on most datasets, while
Introduction
Dynamic graph modeling represents entities nodes and timestamped relationships edges, aiming to explore the underlying evolution patterns of real-world systems [1]. It has attracted great attention in various fields, e.g., social networks [2], traffic systems [3], and recommender systems [4].
Despite the great success of current methods, there are still two limitations. Firstly, existing methods lack the ability to effectively and efficiently track long-term temporal dependencies in dynamic graphs. Specifically, RNN-based methods, e.g., JODIE [2] and TGN [5], model temporal evolution through recurrent updates of node embeddings. Although theoretically capable of capturing long-term dependencies, they suffer from vanishing/exploding gradients in practice, limiting their effective- ness long sequences. On the other hand, Transformer-biss based models, e.g., DyGFormer [6] and SimpleDyG [7], address gradient issues through the self-attention mechanism but require prohibitive quadratic (N22 computational complexity for sequences of length N. Recent efficiency improve- ments through patching [6] or temporal convolutions [8] inevitably sacrifice temporal resolution, forcing dffeetiieenss--effciiene
trade-off. Other recent methods, using multi-layer perceptions (MLP), egg., GraphMixer [9], FreeDyG [10], or graph neural networks (GNN), e.g., TGAT [11], primarily focus short-term dependencies, and their performance often decreases the sequence
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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length increases [12]. Secondly, existing methods lack robustness against noise. Real-world dy- namic graphs frequently contain various types of noisy events [13]. RNN-based and GNN-based methods are naturally susceptible to noise interference, leading to unstable performance [14, 15]. Although Transformers partially mitigate historical noise via self-attention, they remain susceptible to noisy data and cannot fully eliminate its impact [16]. How to filter out noisy history information more effectively and efficiently remains a challenge [17].
To address these issues, we propose DyG-Mamba, a novel timespan-informel continuous state space model (SSM), for dynamic graph modeling. Firstly, compared to Transformer- based methods that rely on large number of trainable parameters, DyG-M DyG-Mamba employs only one trainable step size parameter At to capture forgetting laws of historical information, along with a small set of parameters in the encoder and decoder layers. Under the same GPU memory constraints, DyG-Mami can directly process the entire long-term sequence without pooling, thereby preserving temporal details and effectively modeling long-term dependencies. Furthermore, inspired by Ebbinghaus' forgetting curve [18], which posits that forgetting primarily correlated with timespans rather than content, aim to equip DyG-Mamba with the same forgetting mechanism. Specifically, we design monotonically increasing and learnable timespan function to redefine At, enabling the dynamic system to automatically learn how to compress historical information across different timespans, i.., the model forgets historical information in a fast-then-slow pattern as the timespan increases, thereby enhancing both its effectivvness and inductiveness. Additionally, compared Transformer's 's quadratic time complexity. DyG-Mami adopts the same hardware-awee aware parallel scan optimization Mamba [19], enabling it to efficiently capture long-tern dependencies with linear time complexity. Secondly, inspired by Ebbinghaus' review cycle that periodic review can counteract forgetting [20], to further enhance robustness, we redefine SSM's core parameters B and C' to be input-dependent and add spectral norm constraints to ensure Lipschitz continuity. This strategy enables DyG- selectively review historical information and thus remain robust against noise. Main contributions:
To the best of knowledge, graph modeling. By redefining the core SSM parameters, DyG-Mamba achieves high efficiency and effectiveness in capturing long-term temporal dependencies.
the first to introduce SSMs for continuous-ting dynamic
Inspired by both the forgetting curve and the review eycle that counters it, propose timespan- informed continuous SSM that adopts timespans to control system forgetting while incorporating dependdepnndent parameterization. This design improves DyG-Mamba's s capability to model long- term sequences with irregular timespans, enhancing its effectiveness, inductiveness and robustness.
Extensive experiments on 12 benchmarks show that DyG-Mamba achieves state-of-the-a perfor- mance with superior effectiveness and robustness.
Table Comparison of continuous-time dynamic graph baselines from six aspects. With batch size of 200 and sequence length of 512, a model is considered time and memory efficient if the running time and memory usage are less than GraphMixer. i.e., running time 250 seconds and memory usage 30,000 MB. Adding 50% noisy temporal edges, the performance drop < 10% indicates robustness.
JODIE DyRep TGN CAWN TGAT EdgeBank GraphMixer TCL DyGFormer DyG-Mamba
Long-Term Capability
x
x
X
X
x
Time Efficient
X
X
X
X
v
/
x
Memory Efficient
X
X
X
X
Vf
V
x
x
Irregular timespan Supportive
X
X
X
X
X
X
X
Inductive Supportive
V
V
V
V
Noise Robust
X
X
X
x
2
Related Work
Dynamic Graph Modeling. Discrete-ting methods segment the dynamic graph into snapshots at predetermined time granularity. then employ GNN (snapshot encoder) with recurrent module (dynamic tracker) to learn node embedding [21-24]. However, fixing the time granularity in advance ignores the fine-grained temporal order within each snapshot. In contrast, continuous-time methods directly use timestamps learn node embedding. Based on their neural architectures, they can be categorized into four types, including RNN-based methods, e.g., JODIE [2], based methods,

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e.g., DySAT [22], MLP-based methods, e.g., FreeDyG [10], and Transformer- methods, e.g.,
SimpleDyG [7]. Additional techniques, such ordinary differential equations [25, 26]. random walks [27], and temporal point processes [28], have also been introduced to capture continuous temporal information. Table provides a detailed comparison between DyG-Mamba and SOTAs, in- cluding JODIE [2], DyRep [29]. TGN [5], CAWN [11], TGAT [11], EdgeBank [30], GraphMixer [9], TCL [23], and DyGFormer [6] from the following angles: if the method can effectively handle unseen nodes during training (i.e., inductive), capture long-tern -erg-term dependencies with both time and memory efficiency, exhibit robustness against noise, and effectively leverage irregular timespans.
State Space Models. SSMs have attracted great attention for long sequence modeling [31, 321. Mamba [19] designs data-dependent selection mechanism with parallel scan optimization, achieving SOTA performance on many fields [33]. Graph Mamba [34, 35] applies SSMs to static graphs
for embedding learning. DG-Mamba [17] and GraphSSM [36] extend Mamba to discrete-time
dynamic graphs by modeling snapshot sequences with fixed time intervals. And STG-Mamba [37] adopts Mamba layers on
graphs. However, these methods are not applicable to continuous-timg dynamic graphs with irregular timestamps, and thus are not directly comparable to our setting. PIVEM [38] learns dynamic node embeddings by approximating temporal evolution through piecewise linear interpolation, based on a latent distance model with piecewise constant and node-specific velocities. It can be viewed as a special case of a first-order SSM, where the hidden state corresponds node velocity and evolves linearly over time. In contrast, our method generalizes this idea by introducing learnable memory decay and imput-adaptive updates, allowing it to better capture irregular temporal dynamics and model more complex patterns in dynamic graphs.
3 Preliminary
Dynamic Graph Modeling. Dynamic graphs can be modeled sequence of non-decreasing chronological interactions G [[11, 11, f1.........
..... (Ur, denote the source and destination nodes of the i-th link and V denote all nodes. Each node is associated with node feature æ E 1d and each interaction has a link feature e' E R where dy and dæ denote dimensionality. Given the source node u, destination node v, timestamp i, and all their historical interactions before i, dynamic graph modeling aims to learn time-aware node embedding for them. We validate the learned node embedding via two common tasks: (i) dynamic link prediction, which predicts whether two nodes are connected in future; and (ii) dynamic node classification, which infers the class of nodes.
T)} with 0 S t S т. where u4 E V
Continuous SSMs. They define linear mapping from t-th input u1)) E R 1xd to output y(! E Rdd via a hidden state variable h(t) E RRmxd formulated by:
h' h'(t) Ah(t) + Bu(t),
(1)
(t) = Ch(() + Du()),
(2)
where E mm B E mx1
C E IR. 1xm are trainable parameters, and D since Du(t) can be viewed as skip connection. (1,2) could be discretized for controllable optimization via the zero-order hold (ZOH), formulated by:
h, Ah + Bu,,
y: = Chh,
(4)
where A= = Ctp(AtA), B (AtA) -1 (A ()t))) C = C, and At is predefined step size.
4 Methodology
The overview of DyG-Mamba is shown in Figure 1. First, in Section 4.1, we introduce dynamic graph encoding and encoding alignment. Then, in Section 4.2, we introduce two main limitations of current SSMs, and DyG-Mamba can alleviate these issues by redefining four core parameters of SSMs. Finally, Section 4.3, we apply DyG-Mamba on downstream tasks and show its complexity.
4.1 Dynamic Graph Encoding
In Figure 1, we first extract the first-hop interaction sequence S of node u before timestamp from dynamic graph, where S = {uu, k1, k1, (u, kjul. {ju))} with |u] denoting the sequence length.
3

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time (days)
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Analogous Edbinghans Forgetting Carve
Dynomii Grophe Fecoding
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Bedeflin 8 Lincar, Onput [[ail]
Elshinghass'
Continures-tree Dynamic Graph
Graph Ingut
Redefiee c Uneerg(apprt 5 s.. [M
Cycle
Figure I: Overview of our proposed DyG-Mamba with four redefined core parameters A. B and C. Pseudocodes are in Appendix
Node and Edge Encoding. We directly adopt the node and edge features provided by datasets as node encoding X R xdy and edge encoding X IR "|xdg for S respectively. If the graph is non-attributed, we simply set the node or edge encoding to zero vectors.
Absolute Temporal Encoding. We encode the absolute timespans between timestamp ij and the final prediction timestamp by using an encoding function (Ts(w(T-1)) to obtain the absolute temporal encoding X E R julxddr, where w = (i-11)))/ Ji=1 with a and / as trainable parameters. Following [9], keep constant during training facilitate easier model optimization.
Co-occurrence Frequency Encoding. Two nodes that frequently interact with the same neighbors tend to have similar embeddings. Thus, capture this feature by adopting co-occurrence frequency encoding. Formally, let the neighbors of u and v be S, = {a, b} and S, = {b, b, c, a). the co- occurrence features of u could be denoted by C = [[1, 1] [1, 2|]. where [1, 1] denotes the occurrence frequency of in S, and Sb, respectively. Then, we define function f (-) R Rdc to encode the co-occurrence features by:
X (J (C (3,0]) (S (.1)))cc+ bc, where AL. x with de denotes dimensionality, We and be are trainable parameters. We implement f (-): - Iede by two-layer perception with ReLU activation.
(5)
Encoding Alignment. We align the above-mentioned encoding to the same dimension d:
Z X W. + b.. where * E {N, E, C}.
(6)
where W, E Rdd xd and b. E d are trainable parameters. Finally, we concatenate aligned encoding for S as - Z I| Z 2 IZ, and Z E Rlu|x4d.
4.2 DyG-Mamba: Dynamic Graph Mamba
4.2.1
Rethinking SSM Dynamic Graph
To learn node embedding of u, SSM first encodes Zī, using linear layer followed by ID convolution layer and SiLU activation function, which could be formulated by
M = SiLU (ConvID (Linear (Z.)) )) E R gllu/xdd
(7)
Then, SSM initializes four core trainable parameters: A E glu[x8dx8d governs state transition,
B TR. xul and C R|u|x8d governs input/output projections, and At controls system's 's update step size [19]. And the k-th output of the SSM is given by:
h = Asha-1 + Bams,
(8)
m = C,ha
(9)

---

where my represents the k-th input, A₁ ex B (Atx (AR ()t,
C = C22, and h2 denotes node's k-th hidden state representation.
Finally, SSM adopts skip connection to avoid gradient vanishing and generates the sequential output:
Zuoout (Mg @ SiLU(Linear(ZZ)) W out bout;
(10)
where 3 denotes element-wise product, W E Rddx4d and bout IR are trainable parameters.
As shown in (8.99. SSM contains three core parameters, Au. B₁ and Ck. which determine the effectiveness for long-term sequence modeling. Specifically, (i) A controls the forgetting of historical information, determined by Alę and A, Existing SSMs, such as Hippo [39] and Mamba, typically initialize A randomly and either fix the step size At as constant or adopt a data-dependent strategy to set At = (inne(L Z.)). However, these SSMs do not account for the crucial role of irregular timespans in real-world sequential input, leading to suboptimal performance. Moreover, directly tying the input Z to At further weakens SSM's effectiveness and inductiveness, it will encounter a large variety of unseen input during testing. (ii) Existing SSMs struggle to effectively filter out noisy historical information. Although Mamba initializes B and C' as data-dependent parameters, allowing B& and C₁ to selectively copy important past information, input noise can still affect their initialization, thereby weakening their robustness [17]. To address these issues, we propose DyG-Mamba, a timespan-informed continuous SSM designed for dynamic graph modeling.
4.2.2 Timespan-Infon Informedd Continuous SSM
To better utilize irregular timespans and enhance the effectiveness and inductiveness of SSMs, we first redefine At and A. Ebbinghaus's s forgetting curve describes how memory retention decreases exponentially over time and can be formulated as R = cxp(-t/S) [40, 41]. where R denotes memory retention, t is the time interval, and S is a decay constant. Inspired by this formulation, we decay coefficient, enabling the model to apply temporal decay
reinterpret R as a sinespan-dependeen to historical states proportionally to the elapsed timespan. Accordingly, we design DyG-Mamba with a similar exponential forgetting mechanism, where the core parameter Aç. which governs the degree of forgetting, decays exponentially as the k-th timespan (fx+1 fx)) increases. Since A₁ is jointly determined by both At and A, this mechanism is realized by redefining these two variables.
Redefining Parameter At. To establish the forgetting curve relationship between timespans and Ag. we first define the connection between timespans and the step size At. Since At could directly influence Ag. Specifically, we define monotonically increasing, learnable timespan function to redefine the step size parameter At as
to+1 t2 exp (ww2 e-exp
At w1 1 1-ex
(11)
T T- t1
where Afe denotes the k-th step size, (1++1- 1 (x) denotes the k-th timespan, w E Rd and w2 E &d are trainable vectors designed capture fine-grained timespan features, with each element constrained to be positive. 1 is an all-ones vector, @ denotes element-wiss multiplication, and r and t1 are the last and first appearing timestamps, respectively.
This redefinition of At establishes a direct relationship between timespan length and step size scaling. Its monotonically increasing property ensures that longer timespans induce stronger decay in A exp( At, As). Consequently, careful initialization of A is required to maintain a balance between effective forgetting and numerical stability.
Redefining Parameter A. We consider two factors when redefining the initialization of A. (i) A should maintain the forgetting curve relationship, i.e., exp(Atx AR)) should diminish exponentially (ii) A determines the stability of recurrent updating in long-term sequence modeling [42], Le., it should prevent gradient vanishing or explosion over time. To satisfy these two requirements, we initialize A as diagonal matrix whose eigenvalues all have negative real parts. Theorem 4.1 confirms that this initialization strategy satisfies both conditions.
as A/a increases.
Theorem 4.1. Let A ds=diag(A1,, ... An), where the real parts of the eigenvalues satisfy Re(X) O. For any timespan Atg. we have A =iiggeee Atu. Atu......, e...,,,,an (h, and B Bx=diag(X; A (exi ...., 1) B) X= (cAn - BR,n), where Atx; and BR. are the i-th elements of At, and B The i-th coordinate of h2 is denoted as h A.bie he h - (c): 1)
5

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(i) Theorem 4.1 guarantees the forgetting curve relationship. When Ate,; is sufficiently small, then ex, Ats, = 1, i.e., hg,; R 11i- This indicates that for small timespans, the model retains historical states and disregards the current input. On the other hand, as the timespan increases, e Ale. gradually approaches 0 decreasing rate over time, causing the system to forget previous states and place greater emphasis on current input m,g.s (ii) Since Re(A,) < 0, the term le Ark.i is bounded by I and decreases as Atg, increases. This prevents the hidden state from diverging over extended sequences and mitigates gradient explosion, ensuring the stability of recurrent updates.
Traditional SSMs struggle to effectively filter noise from the input sequence. To solve this issue, we redefine B and C' as input-dependent parameters and introduce spectral norm constraints enhance robustness. Specifically, Ebbinghaus" review cycle indicates that periodic review of previously learned information helps counteract memory decay [20]. Inspired by this, we design DyG-Mamba continuously review important node while forgetting irrelevant or noisy inputs.
Redefining Parameter B and C. To align the SSM with the review cycle, we first define B and C' as input-dependent parameters, e.g., B Linearg (M Then
filter out noise by Theorem 4.2. Theorem 4.2. Let B and C be input-dependent parameters and my denote the k-th input in M Update process for SSMs, as shown in (8), could be further decomposed as
k-2
m
II
A Bim +... CBBBmis
(12)
1=0
cQ2ii AL Cy B.B.m +... +C.Bmmp
According Theorem 4.2, C2 can be interpreted as the query corresponding to the k-th input, while By and my serve as the key and value, respectively. Thus, the product C, represents the importance of the j-th historical input my to the k-th input. While Theorem 4.2 enables automatic filtering of irrelevant and noisy historical inputs, the construction of B and C using only linear layers makes them susceptible to input noise. To address this issue, we introduce spectral norm constraints on the initialization of B and to achieve dual objectives, formulated by
B + bB, C - + be. s.t. Wall2 < 1, ||Wll 1.
(13)
where WB and We are weight matrices, and || is the spectral norm, i.e., the largest singular value of Wic. The spectral norm constraints guarantee Lipschitz continuity, ensuring that B and C remain stable under input perturbations. Overall, DyG-Mamba is robust to input noise, preventing irrelevant samples from being erroneously reinforced. This robustness guaranteed by Theorem 4.3.
Theorem 4.3. Given [|W S 1, |[ell2 S 1, 7 = max; Re(X: < O, and T as the total sequence duration, the oulpui perturbation satisfies:
Am < M (1- |1
(14)
4.3 DyG-Mamba for Downstream Tasks
For dynamic link prediction, we first process the first-hop interaction sequences of source node u and destination node through two independent DyG-Mamba models. Based on Eq.(7-10), two models generate sequential output embeddings 2 oou and To respectively. Then, we adopt readout function, i.e., MEAN pooling, to obtain their node embedding, defined as 2 = MEAN (r,.. with E {u, v.. Finally, we concatenate two node embedding and adopt an MLP for link prediction:
(ReLU 2 (2)
(15)
We adopt binary cross-entropy loss for optimization
[8
LIP
ly log D: + (1 - y.) log(1 0;)],
(16)
B

---

where [B] denotes the batch size containing both positive and negative samples, and y: and ds represent the i-th ground-trume truth and predicted label, respectively.
For dynamic node classification, we use one DyG-Mamha model and discard co-occurrence encoding, while keeping other components identical to the link prediction setup.
Computational Complexity. Given batch size b, feature dimension d, and sequence length L. DyG- Mamba achieves linear memory and time complexity of (bddL), while DyGFormer has quadratic complexity of O(bdL²). This highlights the efficiency of DyG-Mamba. Details in Appendix B.
Table 2: Transductive: AP for dynamic link prediction with random (rnd), historical (hist), and inductive (ind) negative edge sampling. bold and underlined emphasize best and 2nd-best results.
Dunasees
JODIE
DyRep
IGN
CAWN
TGAT
EdgeBank
GraphMiser
TCL
DyGiomee
MyGbb
Wikipedlis
96. 26.0010114 14
94 06000006
98.4510006
98.7610.03
56
0000
10.00
07:251:000
56
16
000010000
99.86.0.0 0.01
Reddi
96.31:0.14
20.00.44
06.63:0006
99.11
0000
04.6620.00
07:11=0011
02
00:32-00:1
09.25-0000
MOOC
80.2342.44
919770442
19.15+1.0
1.60
80.15=0.22
15
57.97=00.00
82.78-0.15
91.17=0.19
LanFM
20.85-2111
21:22-2..21
7.0743997
86.99990.06
03.42=021
79.29:20000
75.61=0:24
27.2742116
9.00000112
94.22-0.04
Earon
84.77200.30
32.38
86.33411.11
89.56:10.09
71.12:0577
83.53-20.00
N2.25:0.16
79.70
02.47:00.122
03.22-0.03
Social Evo.
89.89110.55
85.3710030
90.57.1.117
84,96.1..99
93.1610.117
74,95
03.271.0007
93.13.10.16
94.75_0.00
UCT
$9.43111009
64:12.00.30
90.34:1:004
95.18.10.06
79.6310..70
76.20110.00
03.2510..7
39.57.1..63
05.7910.17
96.79-0.06
Can. Parl.
60.26::0.31
24:22.76
20.88:2:.34
69
70.73:00.72
64. 05:00.00000
77:04:0:46
68:67723667
07.36:20.55
0.07
US Legis
75.05 52
75 75.54=00.90
05.99=0.55
70.58
68
66.52=23.1
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70.74:1100
69.
69.50::048
71:11=0.59 50
74.11
74
Trade
64,9610031
63:21 10,91
6.303:1.337
65.3910.12
61.47:0.18
60,4110.00
62:61:0227
62.211003
65
66.46.11 1.29
68.5
63.9122081
62.81
65.72::2.17
12.84:0.10
52.21-098
58 48:00.0000
52.11:00116
30:00:::00
55.552043
65:60:1110
Cantact
95.91
95.98=0.15
96.9+0.56
90.26:0.255
96.28=0.09
02.58=00.00
91.02=003
02,44+0122
20-00.0
V1.37:0.01
Avg. Kank
6.088
6,00
4.42
8:42
6:33
6.92
4:92
6.58
2.92
2.42
Wikipodia
83.01:0.66
79
06.66-0333
1..21+1667
87.3--011
29.35-00.00
9.190-0.10
22:23-2554
N2.12=1.22
Reddit
90.0340.36
79:55-220
03.59
81.57-0.77
MOOC
75.60 78.9611.22
57.0
74,0510.99
10.00
85.85 1.566
87.33 1,46
LasFM
74.35113.81
76.92.12.46
76881446
69 19.86.1043
71 59:044224
13.03:10.00
39.30.1..31 0.49
01.57.1048
04.090444
Faron
69.85::2.70
71. 21:16:2.76
73 176
64.73-0.3 36
64 04007-100
76 0300000
77.98 0.92
7O 70.66-0000 39
75
75
Evo.
87.4811.766
91.651036
85,5
10.5710..00
94.9110,11
94,7410.31
97.38 0.14
UCI
75.26115.80 80
10:13.14
20,43:12122
65,30:1.43
68 27
65.0000000
14
30,25:1:.74
82, 02:17:0.80 80
Can. Parl.
$1.79:1063
1.221
68
3.07
66.51-2.77
67.1 17:10:0084
63.0020.00
74 047777
65 65.03-11000
97 00:00:111 51
07.22:20299
US Legis.
51.71#5576 76
2.222.5
74 7.57
68 62-223
12114:000
63 0.00
81.65=1022
30.53+3005
85 35.50
Trade
61.3941MM
19.19:1.35
56.443351
55.71
05.74-0911
81.52-0.00
17.05=1.22
55.90+1..17
64.41=140
19:
Vote
69.30:11.12
09.3743393
51,26:50,34
52-96-2114
84.89.0.00
51:20:1:00
52,30-:2.35
60:4::155
Concact
95.311213
96.39 16.39.100
93.05
84.16.0.19
96.05 96.05....52
88.81
93.16.10:11
93.86.10221
97.57.10.00
97.80 10.14
Asg. Rank
6.08
6.00
4.42
8,42
6:33
6.92
4.92
6.58
3.00
2.33
Wikipedia
75.65100779
1..51
43.62:10.44
80.63-10.00
767777772
78
59
B4 0.7777
Reddit
36.98:10.16
86.30:10.26
08.101024
89259
85.48
85.48
KS 0.110
87.45.10299
91.11:1.400
0.44
MOOC
65.23112.19
61 61.66.10.95
29.501191
73.51 10.44
75.95
1004
19.43 10.00
74.27
74.51
81.24 0.69
11 1
LasFMM
62.67:1-4.49
22:01:2:00
65
16.951550 98
0.
71 1822017
0..09100000
68 03:
58 58.110089
73.97-0.30 50
74
68.96:20.98 98
67.70=1155
20.89-2222
75.1520.55
62-02-18
0.00000000
11.29-2033
Social Evvo,
$9.87:2-411
93.28-00.48
93.13-036
88 32.22-0.27
94
83 0.00
04
04.00-0006 36
96.1--022
UCI
65.99:11.40
14.79:11.76
70944071
64.61
08
57 57.05
00.10-
76.01+1.111
72.25:11.71
75.71
Can. Pari.
48.42:00.66
58.61
65 66.341-287
67.75
88
1111
62.16100000
69 01648
05
15.85
95 05.44
96. 0.79
US Legis.
50.27.1.5113
63.4111166
67.57
65.81 18.52
61
5.0
64.74 64.00
1..631...44
18.15
3.66
65.03 0.69
60,42111.48
61.041601
62.5410.87
60.811 1.44
73.97.1000
55.79.1100
88
Vote
67.79::1.4
6..53-11.98
67
12.000034
16
66
30:00.00
51 0.:00:0773
51.91 01:01:0000
1.161
Conract
93.4841778
94.18-0.10
90.18-3328
89.31
94 05
85. 20.00000000
90.8721.35
91. 91.35+0021
94.75=0.25
94.57 57..222
Asg. Kank
7.35
7.25
5.17
6.17
5:25
6:75
5:42
5.58
3.75
2.33
5 Experiments
5.1 Experimental Setup
Datasets and Baselines. We evaluate performance on 12 datasets, each split into 70%/15%//15% for training, validation and testing. Details in Appendix D.I. We select nine SOTA baselines for comparison, e.g., four RNN-based methods: JODIE [2], DyRep [29], TGN [5] and CAWN [11]. a
GNN-based method: TGAT [1]], masory-based method: EdgeBBnnk [30], MLP-based method: GraphMixer [9], and two Transformer-base methods: TCL [23) and DyGFormer [6].
Implementation Details. For fair comparison, we use DyGLib [6] reproduce all baselines via the same training and inference pipeline. We set the same input length for DyGFormer and DyG-Mamba to fairly compare the long-term dynamic graph modeling ability. We train each model for 100 epochs and select the best-performing checkpoint for testing. We repeat each experiment 10 times with different random seeds and report the mean and standard derivation. Details in Appendix D.2.
Evaluation Details. We evaluate baselines in transductive and inductive settings, where the former predicts future links among nodes seen during training, and the latter focus on unseen nodes [6]. For negative sampling, we follow [30] and adopt random (rnd), historical (hist), and inductive (ind) strategies (see Appendix D.3). Metrics are Average Precision (AP) and AUC-ROC.
7

---

Table 3: Inductive: AP for dynamic link prediction with random negative edge sampling strategies. The notations are the same as Table 2.
Datasets
JODIE
DyRep
TGN
CAWN
TGAT
TCL.
GraphMixer
DyGFormer
DyG-Mamha
Wikipedia
94,82+00200
92,43+0.37
97,83+0..44
98. 08.24=0003
96 22:0.07
96.65=0.02
96.22=0.17
08.59+0.03
98.66=20.02
Reddit
96.50:10.13
96.0910.11
97.50:10.07
98.62:0.01
97.09-0.04
95.2610002
94,09=0.07
9..41-002
98.9110001
MOOC
79.63+1.92
81.07=0.44
89.04#1.17
81442:0.24
85.50=0.19
81,41=0221
80.60=0.22
86.9620.45
89.98:0.04
LastPM
81,61+3382
1.4022+148
41.4544..9 2.
89.42:0007
78.63-0:11
82.11+0.22
73.53=1.66 66
94.23--0.09
95.16+0..05
Euron
10.72+1.39
74.55
7..9441.02
86.35=0..51
67.05=1.51
75.88=0:48
76.14=0.79
$9.76220.3
90.97:10.01
Social Evo.
91,96=20048
90,04+00.47
90.77++0.86
79.94=0.18
91.41=0.16
91.86=0006
91.55=00099
93.14=00.04
93.17=00055
UCI
19.86++1.48
57.48-+1.87
88.12:42.00
92.73-+0.06
79.54-0.48
91.19:0.42
87.36-2003
94.54:00.12
93. 93.38-+0.21
Can. Purt.
53.92.10.94
54.02.10.76
54.10.10.93
55.80.10.69
55. 08.79
55.91
54.30.00.66
87.74.10.71
96.64.1.004
US Legis
54,93442.29
57.28-+0.71
58.63-+-0.77
13.17-+1120
51.00-.111
50.71+0.76
52.59-0977
54.28+22877
55.25-+-.54
UN Trude
59.65.10.77
57.02.100669
58.31.13.15
25.24.10:21
6.103
62.17.0.31
62.21-10.12
64.551.0.62
67.04.10.2 0.20
UN Vote
56.64110996
54.62:12.22
58.85:12.51
49.9420.45
52.24=1.46
50.68:0.44
51.60:0.97
55.93:10.39
58 18..8810..55
Contact
94.34::1.45
92.18:10.41
93.82.10.99
89.55
95.87
90.59:0.05
91.11=0.112
98.03-1-.02
98.10:10.02
Avg. Rank
5.83
6.83
4.67
4.92
6.21
6.00
6.71
2.50
1.33
5.2 Effectiveness Evaluation
Table 2 and Table 3 show models' performance in dynamic link prediction under transductive and inductive settings. AUC-ROC score is reported in the Appendix D.4. From these tables, we observe that DyG-Mamba achieves the best performance on most datasets and achieves the best average rank in both AP and AUC-ROC across three negative edge sampling strategies, demonstrating its higher effectiveness and better generalization compared to SOTA baselines. The primary reasons for DyG- Mamba's 's superior performance can be summarized in three key aspects. (i). DyG-Mamba employs an SSM architecture that effectively handles long-tern sequences. In contrast, DyGFormer requires patching under the same input length, which compresses the sequence data and leads to information loss. (ii). DyG-Mamba leverages irregular temporal information to control the compression of historical states, thereby making more efficient use of time information and enhancing model's generalization capability. (iii). DyG-Mamba selectively filters out past noise or irrelevant information, resulting in more robust node embedding and improved prediction accuracy.
To further verify the scalability and effi- ciency of DyG-Mamba on million-edge tem- poral graphs, we conducted additional exper- iments on tgbl-coin-ve [43], which con- tains 638K nodes and 22.8M temporal edges. Following the standardized training pipeline and hyperparameter setup in Yu et al. [44], we ensured a fair comparison with the ex- isting baselines. As summarized in Table 4, DyG-Mamba not only achieves superior predictive performance but also demonstrates remarkable training efficiency, further validating its scalability on real-world large-scale dynamic graphs
Table 4: Scalability on million-edge temporal graphs.
Method
Performance
Running Time
GPU Usagee
DyRep TGN
45.20:1:.60
49:38:39
48116M
58.60:13.770
38:26:48
48116M
GraphMixer DyGFormer
75.31:10.21
11:59:20
12204M
75.17:00.38
45:19:11
41348M
DyG-Mamba
75.17:10.38
12:32:04
18094M
Scalability of Effectiveness, As shown in Fig- ure 2, to highlight DyG-Mamba's 's ability to cap- ture long-term temporal dependencies, we com- pare it to three best-performing baselines. Ob- viously, DyG- DyG-Man'''s performance improves substantially with longer sequences and out- performs baselines even at shorter sequence lengths, showing its effectiveness in modeling long-term dependencies on dynamic graphs.
GraphMiser
DyGiermmer
D-Gmbie
0.95
0.90
Averge Presuicon
0.95
0.55
0.75
0.65
10
log: length of ieput LastFM
log: longth inpui on Can
Figure AP score w.r.t. varying sequence lengths.
Ablation Study. In Table 5, we conduct an ablation study on three datasets to evalu- ate the effectiveness of each component in DyG-Mamba. Specifically, we examine five variants: (i). [w/o timespan] replaces times- pan with input sample as control signals, i.e., the same setting with vanilla Mamba with Al- = (Linear((((()) (i). [w/o Time- encoding] removes the absolute temporal en-
Table 5: Results (AP) of time information ablations.
Scttings
Can. Parl.
Enron
USLegis.
w/o timespan
96.90-0.15
92, 92.1440.12 12
72.26+:0.76
w/ Time-encoding
97.80.10.43
92.83110006
73.33.1..15
w/o Time
96.87++0.18
92.08+0.112
72.19+0.72
w/o Selective
96.32+1.16
91.33400.10
71.62+0.77
w/o Data-dependent
79.24110.58
12.2510.16
70.57.10.4
DyG-ma
98.37:10.07
93.22-10.03
74.1112..3
8

---

coding X T. (ii). [w/o Time] removes both timespan and time-encodingg (iv). [w/ Selective] follows parameter settings of S4 [45], i.e., meaning all parameters are independent of imput or times- pan. (v). [w/o Data-dependent] changes the parameters B and C' from being data-dependent to timespan dependent without spectral norm constraints. We observe that removing any component from DyG-Mamba adversely affects its dynamic graph learning capability. Specifically, [w/o times- pan] significantly decreases the performance, as it is crucial for capturing irregular temporal patterns. And [w/ Time-encoding] leads to slight decline in performance since absolute temporal information also important. Furthermore, [w/o Selective] also degrades performance since the fixed parameters fail to filter out irrelevant noise. Finally, relying entirely timespan [w/o Data- nata-dependent] also reduces performance, showing the importance of input-dependent setting for parameters B and C.
Effect of Learnable At Function. The design of the learnable function for At in Eq. (11) is inspired by the Ebbinghaus forget- ting curve, which models memory retention as R = exp(-I/S), where / is the elapsed time and a decay constant. We reinterpret this formulation cay coefficient, ensuring that longer intervals induce stronger decay consistent with human memory dynamics, To validate this choice, compare several monotonic alternatives, including Linear, Logarithmic, Sigmoid, and Exponential variants, as well versions without normalization or timespan inputs. As summarized in Table 6, our formulation consistently achieves the best performance across datasets, demonstrating a balanced and smooth decay behavior. In contrast, Linear and Log variants lack boundedness, Sigmoid saturates early, and the Exp variant over-amplifies long timespans, leading to unstable training.
Table 6: Ablation on learnable At function.
Variants
Can.Parl.
Enron
USLegis.
Linear
94.6410.18
91.13
70.28
Logarithmic Sigmoid Exponential w/o T t
97.18=00112
92.35=0005
72.46-2.34
95.84:10.13
92.26:1:.04
72.15::2.336
timespan-dependent de-
94.68-10.11
90.8410.06
11.23
97.32++0.20
92.46-0.116
72.45=00.4
w/o timespan
96.90=
92.14
72.26:10.76
DyG-Mambo
98.37:10.07
93.2211003
74.11
5.3 Efficiency Evaluation
100
Given an input length of 256, Figure and Appendix D.4 show the training time per epoch and the size of trainable parameters
CAWN
Average Percision
90-
JODIE PERFIAAO
TyRRe
TGN J.5WAM
on Enron data. Obviously, CAWN requires the longest training time and a substantial number of parameters, since it conducts random walks on dynamic graphs to col-
80-
93
ClaphMMune
92
70-
TGAT XXr0000
DyG-Mamha
DyGPormer
91
200
400
lect time-aware sequences. In contrast, sim-
pler methods, e.g., GraphMixer and JODIE, have fewer parameters, but exhibit signif- icant performance gap compared to DyG- Former and DyG-Mamba. Overall, DyG- Mamba achieves the best performance with a small number of trainable parameters and a moderate training time required per epoch.
200
400
600
800
1000
1200
Epoch Times (s) on ENRON Dataset
Figure 3: Comparison of efficiency and effectiveness.
Scalability of Efficiency. In Figure 4, highlight DyG-Mamba's ability to ef- fectively capture long-term temporal de- pendency dynamic graphs, we pro- vide more detailed efficiency compar- ison between Transformer-bases DyG- Former and DyG-Mamba. For fair comparison, we make the same exper- imental setting for both frameworks. We observe that, with increasing input sequence length, DyG-Mamba demon- strates a linear growth trend in both runtime and memory consumption, highlighting its efficiency. Specifically, DyG-Mamba is 8.9 times faster than DyGFormer and reduces GPU memory consumption by 77.2% a sequence length of 2,048. This is because MyG-Mamba only needs few parameters to compress hidden state and adopts hardware-ad parallel scanning for training.
2000
DyG-maaaha Dyooee
Dyti Manba
Epuch Timejo))
1300
EPP MomeyyGii)
DyGFenner
Baaller
1000
11.11
500
70
125
0-
3
10
10
logs length
logs length
(A) Speed Comparison
(B) GPU Memory Comparison
Figure 4: Speed and memory comparison of two layers DyGFormer and DyG-Mamba with varying lengths.

---

Table Training convergence comparison on Can Parl, dataset.
Model
Epoch=20
Epoch=40
Epoch=60
Epoch=80
Epoch=100
Vanilla Mamba
0.2229
0.2026
0.1964
0.1922
0.1914
DyG-Mambs
0.1691
0.1478
0.1482
0.1480
5.4
Training Efficiency
To ensure stable and efficient training, DyG-Mamba incorporates several lightweight yet effective designs. The learnable A/ function is implemented as an element-wise exponential decay through scalar-wise MIP,, enabling adaptive modeling of irregular intervals with negligible overhead. A spectral norm constraint regularizes the and matrices without adding parameters, preventing instability and ensuring bounded outputs under input noise supported by Theorem 4.3, Moreover, redefining and C' as input-dependent linear mappings introduces minimal cost while improving the model's adaptability. As shown in Table 7, DyG-Mamba converges smoothly within 100 epochs the Can. Can.Parl. dataset, confirming its fast and stable optimization behavior across datasets.
5.5 Robustness Evaluation
We conduct robustness test by randomly inserting 10% to 60% noisy edges with chronological timestamps during the eval- uation. In Figure 5, when the proportion of noisy edges increases, DyG-Matt DyG-Mamba ex- hibits only a minor performance decline, indicating stronger noise robustness com- pared to baselines. We attribute robustness the review-based selective memory en- hancement, which enables the model to identify most relevant information and filter out noise.
0.999
1.00
Averago Procision
Avarage Precision
0.95
0
1.0%%
0.90
0.97
0.96
0,80
TGN
GraphMiner
32.30%
DyG Mamba
4.40%
0.7
DyC- Mamba
0.0
0.3
moise rato on Wikipedia
noöse rate on Wikipodia
Figure 5: Inserting noisy edges from 10% to 60%.
5.6 Case Study
In Figure 6, randomly extract a middle part sample from one long- term input sequence of the Wikipedia dataset, e.g., {1713, 160, 1667, 1667, 1667, 1713, 1667). visualize DyG- Mamba's 's efficiency capabillity for long- term sequence modeling. Figures 6(A)- (B) show the normalized cosine simi- larity of node embedding between the last and second-last hidden states. We observe that DyGFormer shows high diagonal similarity and nearly uniform similarity across all neighbors, indicat- ing that it struggles to distinguish important historical information. In contrast, DyG-Mamba assigns greater weights to the historical reappearing destination nodes, better enhancing the node embedding and filtering out irrelevant and noisy historical information.
1712
17D
037
6.00
8:30
0.09
Tilddn states
0.12
0115
3.13
0.44
8.13
0.15
8:30
0,09
0.00
1913
Seocon hidden
Budden
(A) DyGFomner Attention Map
MyG-Mamba Attention Map
Figure 6: Investigate sequential modeling via attention map between source and destination nodes on link prediction.
6 Conclusion
In this work, we propose a novel SSM framework called DyG-Mamba to effectively and efficiently capture long-term temporal dependencies on dynamic graphs. To achieve this goal, we incorporate irregular time spans as controllable signals, thus establishing strong correlation between dynamic evolution patterns and time information. We also implement a review-based selective memory enhancement to further improve the model's robustness. Experimental evaluations on various downstream tasks show DyG-Mamba's 's higher performance and better robustness. the future, we plan to deploy DyGmaamba in more real-world applications.
10

---

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[44] Le Yu. An empirical evaluation of temporal graph benchmark. arXiv preprint arXiv:3307.12510. 2023.
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[46] Weihao Yu and Xinchao Wang. Mambaout: Do we really need mamba for vision? In Proceedings of the Computer Vision and Pattern Recognition Conference (CVPR), pages 4484- 4496, 2025.
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A Limitations
Although our DyG-Mamba delivers strong performance on dynamic graph modeling, it still has several notable limitations. (i) More Interactions. We focus primarily on edge addition, which is widely studied interaction type previous research. Extending our framework to other interaction types, such as node addition/deletion, edge deletion, and node/edge feature transformations, remain an avenue for future research. (ii) Larger-scale Datasets. Existing benchmarks are relatively small- scale datasets. And it is unclear whether these findings will generalize to larger more complex real-world scenarios. (iii) More Domains. Although DyG-Mamba has achieved good results on dynamic network modeling, is still unknown whether this conclusion can be extended other domains. Recent insights from Mambaout [46] suggest that Mamba especially suited for auto- regressive and -engeenuence tasks. Even though dynamic link prediction and node classification are not strictly auto-regressive, we still obtain state-of-the-art performance, indicating that Mamba's broader effectiveness across diverse tasks warrants further exploration.
B Computational Complexity
In our implementation, we employ two DyG-Mamba layers with batch size b. feature dimension d, expanded state dimension 2d, and SSM dimension dysm. Note that while Section 4.1 sets the feature dimension of node embeddings as 4d, we use here for simplicity. On GPUs, high-bandwidth memory (HBM) provides larger capacity, whereas static candom-access memory (SRAM) offers higher bandwidth. Building on Mamba, DyG-Mamba first reads ((bL(2d) + (2d)dzam) bytes of (A, A, B, C) from slow HBM fast SRAM. then derives the discrete A, B of size (b. L. 2d, doam) in SRAM, executes the SSM operation in SRAM, and writes the output of size (b, L. 2d) back HBM. This approach reduces I/O overhead from (2d)
(bd + (2d)d yielding memory complexity of (2d) + (2d)d.sm). Since daam is relatively smaller compared to bL, we simplify as (bbdd) The time complexity calculate B, C, A is (2d) and the SSM process takes ((d)(2d) Compared to transformer-basee methods with quadratic time and memory complexity (bb,2d), DyG-Mamba scales effectively to large sequence lengths.
C Algorithm Details
Here, we list the detailed workflow of DyG-Mamba in Algorithm.1. We also list the procedures of DyG-Mamba for dynamic link prediction in Algorithm.2 and dynamic node classification in Algorithm.3.
Algorithm Continuous SSM
Input: Hidden states Z (#1.1, -i.2 L L. B
(B, L,, d), Control signals At (Ar1, Af2..... Atz.: (B. I, d) SSM
dimeusion dram. Expanded dimension: 2d. // Normalize the input sequence.
Initialize Parameter/:: (2d.daum). P (damm) Initialize a:: (B. 2d) <- Linear Z). (B, L. 2d) Linear" (Z), At: (B. L, 2d) Linear (Att), for o (forwand, backward]
a (B, L, SiLU (x))
B.: d2am)) Linear 2 (x,.)
deam) Linear C (mx))
// Control signal za control the solection of historical information.
Aa: (B. L, + Using Eq((!))
A. L. d.am) < 4, @ Parameter
B.: (B, 2d, deam) +- A, B.
y.: (B, 2d) SSMs A., B., B.. (o)(s))
end
Gated y-
Vorwes (B, 2d) Miseard @ SILU(z)
Residual connection.
Wowtoont (B, 2d) +- Muatoann (z)
2: (B, d) Linear (Veraa )uanan)
Output: Updated hidden states Z
20

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Algorithm 2 Dynamic Link Prediction
Input:
The dynamic interaction set D ((w%, 64. 4)) m Continuous SSM PyGGMMamhu-)) readout function Read() () ouppu projection
layer (K()).
Epocks
for (a11, en, f) E do
node pair (M1-#1). #1). Sampled neighbor sequence: $1, S2.
for in (M1: vi do
Node Features: X u.N & Blulxdy
Edge Features: X E BluixdR,
Time Features: X Rlulxdr,
Co-occurunce Features: X Rlulxde Time Spon: At (Atz, Feature Aligunent.
Z. X W.+ where EN,
Z N I Re v Zi,o
1/ Continuous SSM encodor.
M DyG-Mambal " At..)
end
z (Read) 1). 27, deRead (M; )). Output.
9
D. an 2 cull).
end
Compute Ciink Proliciin-
Output: Dynamic link prodiction lahels.
Algorithm 3 Dynamic Node Classification
Input: The dynamic interaction {(v,, v.) continucus SSM DyG-Mambaf ).. readout function Read(-) ourput projection layer ((-).
for Epochs
for
Sampled neighbor sequence:
S [(ks, (1n,t)) (Mz, f.)..... (Ruu) f1u))).
Node Features: X N. glulxdv
Edge Features: X E. E giulxdE,
Time Features: X T
Time Span: At (Atr. Ata..... Atr) z X W. + b, where N, T.
Z 7: 22, Za.r
M
MoygMMamhu(P Zi. Ata).
d(Read( M.)).
f Softmax (Linear Sortm (RE..U(Lioor 2 )))
end
Compute Cxaae Cherieation
end
Output: Dynamic node classification labels.
Table 8: Statisties of the datasets. N/A denotes that there is node/edge features. # Node denotes the number of nodes.
Datasets
Domains
#Nodes
#Links
#N&L Feature Bipartite
Duration
Unique Steps Time Granularity
Wikipedia
Social
9,227
157,474
N/A & 172
True month
152,757
Unix timestamps
Reddit
Social
10,984
672,447
N/A & 172
True month
669,065
Unix timestamps
MOOC
Interaction
7,144
411,749
N/A &
True 17 months
345,600
Unix timestamps
LastFM
Interaction
1,980
1,293,103
N/A N/A
True
month
1,283,614
Unix timestamps
Enron
Social
184
125,235
N/A N/A
False
years
22,632
Unix timestamps
Sucial Evo.
Proximity
74
2.099.519
N/A &
False
8 months
$65,932
Unix timestamps
UCI
Social
1,899
59,835
N/A N/A
False
196 days
58,911
Unix timestamps
Can. Parl.
Polities
734
74,478
N/A &
False
14 years
14
years congresses
US Legis.
Polities
225
60.396
N/A &
False
12 congresses
12
UN Trade
Economies
255
507.497
N/A &
False
32 years
32
years years
UN Vote
Polities
201
1,035,742
N/A &
False
72 years
72
Contact
Proximity
692
2,426,279
N/A &
False
month
8,064
minutes
21

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D Experimental Details
D.1 Dataset Details
We evaluate our methods on diverse set of dynamic graph datasets, including twelve publicly available datasets collected by Edgebank [30], which are publicly available'. We present the statisties of the datasets in Table 8, where #N&L Feature stands for the dimensions of the node and link features. Note that our calculation of the Contact dataset's statistics (694 nodes and 2,426,280 links) slightly differs from the values reported in [30], although both are derived from the same dataset.
Table 9: Configurations
Configuration
Setting
Learning rate
0.0001
Train Epochs
100
Optimizer
Adam
Dimension of time encoding dr
100
Dimension of co-nccurrence de
50
Dimension of aligned encoding d
50
Dimension of At,'s encoder 4d Dimension of oulput dont Number Mamba blocks Dimension of SSM dyam Expanded factor of Mamba Number of Corss-Attention layer
200
172
2
16
2
1
Table 10: Sequence Length Settings
Dataset
Sequence Length
Wikipedia
64
Reddit
64
MOOC
256
LastFM
512
Enron
512
Social Evo.
64
UCI
32
Can. Parl.
2048
US Legis.
272
UN Trade
256
UN Vote
128
Contact
32
D.2 Implementation Details
Experiment Environment. We conduct experiments Ubuntu 22.04 LTS server equipped with one Intel(R) CTre(TM) 19-10X00X CPU @ 3.70GHz with 10 physical cores and NVIDIA RTX A6000 GPUs (48GB). The code is written in Python 3.10 and the model.
Py PyTorrch 2.1.0 CUDA 11.8 train
Configuration Details. For all baselines, we follow the configurations as DyGFormer reported [6]. For DyG-Mamba, we list all configurations in Table 9. Then, we perform grid search to find the optimal sequence length, with a search range spanning from 32 to 2, 048 in powers of It is worth noticing that DyG-Mamba can handle nodes with sequence lengths shorter than the defined length. When the sequence length exceeds the specified length, quence and preserve the most recent interactions up to the defined length. Finally, we present the sequence length settings in Table 10. All implementation details could be accessed at the link: https//aaaoyymouuu.Appp/DDCCCCaa
truncate the
22

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The inconsistency problem between the description of the transductive setting in DyGFormer and the coding in DyGLib. Thanks to other researchers in this community, we observed that CAW-N explicitly avoids including unseen nodes in the validate/test sets. In contrast, DyGLib, TGAT, and TGN adopt more relaxed version of the transductive setting, where both previously observed and new nodes can appear during evaluation. For a fair comparison, we follow the coding in DyGLib, which uses more relaxed version of the transductive setting. Therefore all baselines in our paper use the same data split with both previously observed and new nodes in transductive setting, ensure the fairness of experimental comparisons. To compare with CAW-N, we also re-implement CAW-N with the relaxed version of transductive setting.
D.3 Detailed Evaluation Settings
Detailed Settings for Effectiveness Evaluation. To provide a more comprehensive evaluation of baseline performance, and following [30], we adopt three distinct negative edge sampling strategies for the temporal link prediction task. We define the training and test edge sets as Euaan and Eiest respectively. The edges of given dynamic graph can then be grouped into three categories: (a) edges observed only during training (Euain Eusa). (b) edges that appear both in training and test (Enrin n Eues), referred to transductive edges, and (c) edges observed exclusively in the test phase (Erest \ Erriin), regarded as inductive edges. Note that inductive negative sampling here is distinct from the usual notion of inductive settings. We elaborate on the three sampling strategies as follows:
Random Negative Sampling (rnd). Negative edges are sampled at random from all possible node pairs. At each timestep, we retain the timestamps, features, and source nodes of the positive edges, but randomly select their destination nodes from the entire node set.
Historical Negative Sampling (hist). In historical negative sampling, we focus on edges that were observed previous time steps but are absent in the current step. This approach assesses whether a method can accurately predict the specific timestamps at which an edge may reappear, rather than simply predicting that it always reoccurs once observed. Formally, for a given time step f, we sample from edges in (Eroiin n E.). If the number of available historical edges is insufficient to match the number of positive edges, we revert to random sampling for the remainder.
Inductive Negative Sampling (ind). Whereas historical sampling centers on edges observed during training, inductive negative sampling evaluates whether a model can capture the reoccurrence of edges that first appear only at test time. Once newly appearing edges have been observed the test, the model is asked to predict if these edges will reoccur subsequent time steps. Formally, at time t, we sample from (Eret O Etrain E.). If there are not enough such inductive edges to match the number of positive edges, the remaining negative edges are sampled randomly.
Detailed Settings for Efficiency Evaluation. In Figures 3,4, in the evaluation of time and memory consumption, we do not choose the configuration as reported DyGLib', as there a significant difference in sequence lengths between different baselines. In the LastFM dataset, the reported number of sampled neighbors for DyRep, TGN, and GraphMixer is set to 10, while for DyGFormer and DyG-Mamba, it is 512. This is unfair because the complexity of time and memory is highly dependent on the number of neighbors sampled. Therefore, for fair comparison in terms of time and memory consumption,
the same number of sampled neighbors across all models: 32 for UCI, 256 for Enron, 512 for LastFM and 64 for Reddit.
Detailed Settings for Robustness Evaluation. As shown in Figure 5, the purpose of the robustness test is to evaluate the ability to against noise in edges and timestamps. the training step, we typically train models on a transductive setting with random negative sampling. In the evaluation step, after neighbor sampling, we randomly select o * L positions to insert noise. Specifically, the noise position's 's node and timestamps are randomly generated. And the noise rate is chosen from 0.1 to 0.6.
"github.com/yule-B BUAW///
23

---

UCI
ENRON
92
MasaMaske
DyGiformer
26.5
80
Grgthtiger A
TOAT ADYKER
90
88
L.guaue
20
100
120
0
200
400
600
800
1000
1200
Epoch Time(s)
Epoch (S)
LASTFM
REDDIT
100.0
DyG Manila
99.5
DyGEommr
70AT
99.0
80
98.5
Dyday
TCL:
98.0
JoBm
00.25
J0DIE
97.5
TCL
60
97.0
2000
1000
2000
3000
4000
5000
Epoch Timels)
Epoch Timets)
Figure 7: The AP score with different sequence lengths. We use the same sequence length for each model (uci=32, enron=256, lastfm=512, reddit=64), The not appearing model means OOM.
D.4 Additional Experimental Results
Training Time and Parameter Size. Figure shows the additional time and parameter size comparison the UCI, Enron, LastFM, and Reddit datasets. The models that do not appear in the figure experienced out-of-memory (OOM) errors. We can see that our model achieves the best performance while incurring low time and memory costs.
Table 1I: Performance on dynamic node classification.
Methods
Wikipedia
Reddit
Avg. Rank
JODIE
88.99
60.37+2.58
5.00
DyRep
86.39-10.98
13.72+1..2
6.00
TGAT
84.0911.27
70.0411.00
5.00
TGN
86.38:12.34
63.27:10.90
7.00
CAWN
84.88 1.33
66.34
6.00
EdgeBank
N/A
N/A
N/A
TCL
77.83+22.13
68.87+2.15
6.00
GraphMixer
86.80.10.79
64.22.13.32
5.00
DyGFormer
87.44.11.08
68.00
3.50
DyG-Mamba
88.58.10.92
70.79
1.50
Performance Dynamic Node Classification. For dynamic node classification, estimate the state of a node in a given interaction at specific time and use AUC-ROC as the evaluation metric. Table 11 shows the AUC-ROC results dynamic node classification. DyG-Mamba achieves SOTA performance on the Reddit dataset and second-hest performance on the Wikipedia dataset. In addition, DyG-Mamba achieves the best average rank of 1.5 compared the second-best DyGFormer with 3.5 AUC-ROC results for all basclines in Table 1.
Robustness Evaluation. We provide additional robustness test results on the UCI and Can. Parl. datasets, as shown in Figure 8. DyG-Mamba consistently exhibits greater robustness compared to DyGFormer and GraphMixer. However, TGN uses memory bank to store previous node embeddings, making it less sensitive to noise in current neighbors.
Transductive Dynamic Link Prediction. We show the AUC-ROC for transductive dynamic link prediction with three negative sampling strategies Table 15. Since we cannot reproduce the same performance reported by FreeDyG [10], directly copy the results from their paper.

---

UCI
DyGFomme
Avenuge Precision
Mari-Mamba
0.9
0.925
0.900
TGN
GraphMiier
0.875
Mambaa
0.850
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0
0.1
0.2
0.3
0.4
0.5 0.6
noise Tile
moise rate
Can. Parl.
1.0
Average Precision
0.9775
GraphMixer
Dymi-
0.9750
DyGFermmr
0.8
DyGi-Mamba
0.9725
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.5
Dolse
moise rale
Figure 8: The robustness test on datasets UCI, Can. Parl.
Inductive Dynamic Link Prediction. We present the AP and AUC-ROC for inductive dynamic link prediction with three negative sampling strategies in Table 16 and Table 17.
Comparison of the effectiveness of Co-occurrence. DyGFormer designs co-occurrence module and has conducted ablation studies on it. DyG-Mamba focuses on detailed ablations of the modules propose. But also conduct additional ablation compare the effectiveness of co-occurrence encoding in table 12. From the results, DyG-Mamba still exhibits a strong ability to capture long-term dependencies, outperforming DyGFormer.
Table 12: Comparison of the effectiveness of Co-occurrence Frequency Encoding.
uci USLegis
UN Trade
DyG-Mamba
96.79:10.08
74.11
68.55110.16
DyG-Mamba w/o Co-occurrence
93.09+1.31
73.53+2.43
66.3240.55
DyGFormer
95.79
71.11
66.4611..29
DyGFormer w/o Co-oceurrence
83.05+0.338
70.59
61.93+1.79
Ablation Study with Co-occurrence and skip connection. We conduct ablation studies on three datasets. "w/o Co-occurrence' refers removing Co-occurrence Frequency Encoding from DyG- Mamba, while "w/o skip connection' denotes the removal of the skip connection in the SSM.
Table 13:
We conduct ablation studies three datasets. "w/o Co-occurrence' refers to removing Co-occurrence Frequency Encoding from DyG-Mamba, while "w/o skip connection' denotes removal of the skip connection in the SSM.
UCI
US Legis
UN Trade
DyG-Mamba w/o Co-occurrence
96.79 +-2 0.08
74.11 + 2.32
68.55 + 0.16
93.09 1.31
73.53
2.43
66.32 +-t 0.55
w/o kxip-connection
95.37 + 0.06
73.64 + 2.51
67.64 + 0.25
Performance with different sequence length. As shown in Table 14, DyG-Mamba achieve better performance with longer input sequences. However, to ensure a fair comparison, we follow the input sequence length as DyGFormer rather than incorporating additional historical information.
Hyperparameter Sensitivity, DyG-Mamba insensitive to hyperparameters, see Figure 9. There- fore, the hyperparameters listed in Table 6 are applied consistently across all experiments.
25

---

AP score across different sequence length. DyGFormer.
Table 14:
* denotes the sequence length used in
32
256
512
1024
2048
4096
uci
96.79:10.0
96.82:10.09
96.95-10.06
97.45:10.03 97.3
07.47
USLegis
73.99+1.52
74.11+2..2
74.17:2..14
74.34.2..17
74.47:+2.44
74.96+22.04
94
(A)
(B)
(C)
Enron
93
AP
92
91
8 16
32
128
0.5
1.5
2
2.5
#Mamba blocks
5SM dimension dam
Expanded factor
Can. Parl.
99
(D)
(E)
(F)
98
AP
97
96
128
0.5
1.5 2
2.5
#Mamba blocks$
SSM dimension dam
Expanded factor
Figure 9: We tune the parameters of DyG-Mamba, including the number of blocks, the SSM dimension, and the expansion factor. on two datasets: Enron (A-C), Parl (D-F). For each parameter, we adjust its value while keeping the others fixed at DyG- DyG-Mamba's default setting (the red line).
Table 15: AUC-ROC for transductive dynamic link prediction with random, historical, and inductive negative sampling strategies.
Dataacts
IODIE
DyRep
TOAT
TGN
CAWN
FligeBonkk
TCE
CimphMiis
PraDDyC
DycTFfemmer
DyG-Manise
Wikipetis
94.37
0.09
76
99.41
0.02
98.31
0.05
0.05
98.47
99.01
95.37
0.00
99 00.50:0
99.15
0.01
0U00
MOOC
K3.81
2.09
85.00
0.56
81.11
80.35
0.26
60,86
0:00
85.12
0.18
3995
37.91
0.55
90.55
0113
LastMM
10.69
1:66 71.16
1.89
71.59
0.18
78.61
254
85.92
0.00
83.77
0:0
66.06
7533
a 0.12
95-42
9508
0.10
61,99
[U02
Euron
67.96
2 0.52
84.89
3.00
68.89
1.10
35.32
0.99
90-45
2 1..14
87.05
0:00
75.74
0.72
$4.36
0.21
9401
2: 8.11
93.35
0.133
0.13
93.03
= 0.06
Social Evo.
92.05
2 0.45
90.76
0.1
94.76
0.16
95.39
0.17
87.34
2 0.08
$1.60
0:00
94.34
0.17
95.23
2 0.07
96.59
2: 0,04
96.20
0.00
0.01
96.39
= 0.01
md
UCI
90.44
2 0.49
68.77
2.34
75.53
0.74
92.03
2 1.13
93.87
1 0.08
77.30
0:00
87
1.36
91.11
2 0.67
95.00
0.21
94,89
2: 0.25
96.50
= 0.06
Can. Part
78.21
4 0.25
73.35
3.67
25.09
0.78
76.99
1.30
75.70
1: 3.27
64
0:0
72
3.23
85.17
3 0.53
N/
90.76
0.61
96.77
0.05
US Legi.
62.85
1.07
82.28
0.32
75.84
1.99
83.34
1..3
75.16
8 0.30
62.57
0.00
76.27
0.53
76.06
2 0.79
N/A
77001
0.55
78.27
= 2.80
UN Trad=
69.82
= 0.44
67.44
0.55
64.01
0.12
69. 10
1.67
68:54
0.18
66
0.00
64.72
0.00
65.52
3: 0.51
N/A
70.20
1.44
72.25
IUIT
UN Vate
68.53
2: 0.95
67.18
1.04
52.83
1.12
69.71
2.65
$3.09
3 0.22
62.97
0.00
51.85
L 0.36
$2.46
0.27
N/A
57.12
0.62
69.53
0.55
Connel
96.06
0.80
96.48
99.99
0.00
N/A
6.01
W0.38
Ang. Rank
5.39
N/A
1.15
Wikipodia
80:77
SS
MOOC
60.75
LasFM
75.22
47
Lnnon
75.39
74.89
3:55
61.85
65
79.59
0.00
67.95
15
0.52
78.00
Social Evo.
90.06
3:15 02.12
0.34
91.08
0.59
87.43
85.81
0:0
93.44
98.39
97
0.00
17
0.00
W8.8
0034
UCI
73.64
3:50
57.91
3.12
5888
71
7A8
57
0.15
69
0:0
72.25
77
2.00
$0.38
0.26
16.97
0:34
77.35
1.25
Can. Part
62.44
:: 1.11
70.16
1.70
70.86
054
23.23
3,88
72.05
2 3.54
63.04
0:00
69.5
3.70
7900
+ 1.01
N/A
97.61
1: 0.00
96,97
0,30
Legis.
67.47
3 6.40
91.44
1.18
73.47
5.25
33.53
4.55
78.62
4: 7.46
67.41
0:00
9399
3.71
35.17
1 0.70
N/A
90.77
2: 1.96
97.11
UN Trade
68.92
4: 1.40
64.36
1.40
60.37
0.68
61
3 5.41
63.09
1 0.74
86.61
0.00
61.45
1.00
65
2: 1.54
N/A
73.86
2: 1.13
15.26--0115
UN Vate
76.84
= 1.01
74.73
1.45
$3.95
3.15
73
5.20
51.27
0.33
80,62
0.00
52.20
52
+ 1.44
N/A
64.274
1.78
6487-431
Conncct
96.35
= 0.02
96.00
0.23
05.30
0.43
93.76
+ 1.29
85.06
à 0.32
92.
1T
0.00
93.34
0
13
-: 0.34
NIA
97.17
- 0.05
97.43 = (LLI
Avg. Rank
5.00
9.67
7.06
3.92
$.25
6,50
7.25
5.67
NIA
3.35
2.33
Wikipodia
70.06
0.78
0.96
70L95
81.73
0:00
78.69
Redit
83.51
0.15
0.31
84.5
0.00
56.23
0.51
82:32
MO0C
66.63
2.30
77.44
70.32
0.00
70.36
0.76
13.02
70.92
1.05
Social
00.01
93.07
1..5
$7.85
94
9K.77
UCI
64.14
1.26 54.25
201
60.80
58.05
0.26
5K00
000
75.39
05.96
61.36
255
Can. Part.
52.88
0.80
63.53
0.65
72.47
72.91
1.78
61.41
0:0
69.47
70
K/A
0.8%
96.99
066
Legs.
59.05
09.64
071
71
12
46
66
000
9255
1:91
84
NIA
5799
91.51
0.26
UN Trade
66.82
: 1.27
63.60
1.28
63
13
0.78
66.37
4 5.99
71.73
0.74
74.20
0.00
67.00
1.21
66.53
2 1.21
NAA
62.56
1.51
71.3
1.14
UN Vote
73.73
2: 1.61
72.80
2.15
$3.04
255
72.69
4 3.72
52.75
1 0.50
72.85
0:00
52
1.54
$1.89
2 0.14
ISA
$3.37
1.26
62.39
= 221
Comaci
94.47
-: 1.08
94,23
2 0.18
94.10
0,1
91.64
3 1.72
87.68
1 0.24
95.87
1 0.00
91
25
2 0.19
90.96
3 0.27
I/A
95.01
1 0.15
91.94
= 0.21
Agg. Rank
6.75
7.58
6:00
5,33
5.75
0.08
6.00
5.67
NIA
381
3.50
26

---

Table 16: AP for inductive dynamic link prediction with random, historical, and inductive negative sampling strategies.
Dunsets
JODIE
DyRep
TGAT
TGN
CAWN
TCL
OraghMiser
FreeDyG
DyGFermer
DyG Mamba
Wikipedia
94,82 = 0.20 92.43
= 0.37 92
+ 0.07
92.83
= 0.00
98.
24
+ 0:03
96.22
0.117
9666
0.02
98.97
0.01
98.59
0.003
98.66
+ 0.00
Reddit
96.50 1:
96.09
: 0.11
97.09
1: 0.04
97.50
: 0.07
95.62
1 0.01
94.09
0.07
95.36
0.02
98.91
0.00
98.84
1: 0.00
98.91
8.01
MODC
79.63 2 1,92
81,07
1 0.46
85.50
1 0.19
89.04
2 1.7
81.42
4 0.24
80,50
0.22
$1.41
0.21
87.75
0,62
56.96
2: 0.43
89.98
4 0.46
LaiFM
51.61
+ 3.8
83.02
1.48
7863
+ 0.51
81445
+ 4:29
89
+ 0:07
13.53
166
82.11
0.4
04.89
0.01
94.23
+ 0.00
95.16
+ 0.05
Enron
80.72 1 1.39 74.55
3.95
67005
1.51
77.94
102
8633
1 0.51
76.14
0.79
75.88
0.48
89.89
0.17
89.76
10.30
90,97
1 0.01
91.96
90.14
79.54
0.18
91.35
91
94.76
91..44
UC7
79.86
2.05
36
91.19
94.85
0
53.92
54,02
55.80
0.69
51.30
0.66
55.91
0.52
NIA
57.70
0.10
US Legis
$4.93
57.28
51000
3.11
58.63
0.37
53.1
52.59
0.97
50.7
0.76
NIA
55.25
4.95
UN Trade
59.0
+ 0.77
37,002
= 0.09
61403
4 0.15
58.51
2 3.15
05
24
0.21
02.21
0.12
02:17
0.31
NIA
64.35
0.62
67.04
=: 0.20
UN Vote
56.64 0.96
54.62
2.22
52.24
+ 1,46
58,85
2.51
20
+
0.45
$1.50
0.97
S0.88
0.44
NIA
55.01
0.30
58.08
0..55
Contact
94.34 1: 1.45
92.18
0.41
95.87
0.11
93.82
1 0.99
89.55
1 0.50
91.11
0.12
90.59
0.05
NIA
98.03
1.0 0.02
98.10
1 0.02
Avg. Kank
5.83
6:53
6:21
4.87
4.92
6.71
6,00
NIA
2.50
1,33
Wikipodia
668.09 +
0.30
62.18
+
1.27
14
+
0.22
81.76
+ 0.32
67 27
+ 144
$2.20
2.18
N7.60
0.30
H2.78
0.30
71.42
+
73.68 + 4.06
Reddit
62.34
0.54
61.60
0.72
63477
1
0.36
64.85
1 0.85
63.67
1 004
60.83
0.25
64.50
0.26
66
0.41
65.37
1 0.60
66.74 1 0.13
MOUC
63:22
62.93
1.24
71.07
3.41
74.68
0.68
74.27
0.53
74.00
$1.63
0:35
$1.64 0.67
LasFM
00.39
0.47
27.28
76.35
79.32
0.33
Earon
62.08
41
1.31
62.1
1.16
60
67.11
0.62
72
23.001
67.07
75.12
1,43
Social Evo.
83.51 0.87 88.72
1.10
9397
1 0.54
96.66
166
70
1 0.38
94 10
0.31
94.01
0.47
96
96.82
0.16
95.45
1 0.30
hist
UCI
63.11
52.67
2 2.06
70.32
1 0.9
70.78
4 0.78
54.54
1 947
76.71
100
$1.06
0.49
$2.35
0.39
72.13
1.87
73.65
4 3,70
Can. Pul.
52.60
+ 0.88
52.28
- 0.31
56.72
+ 0.7
54,42
+ 0.77
57
14
+ 0:07
55.71
0.74
53.94
0.73
NIA
57.40
+ 0.85
91.59
+ 1,10
US Legis.
52.54
1 2.11
62.10
1.41
51
3.95
11
10
55
56
1.71
53.87
1.4
$2.00
1.02
NIA
56.31
3.16
55.90
1 0.74
UN Trade
55.46
=: 1.39
53.49
8 0.84
55
25
4 0.71
52.80
1 3.19
55
00
2 0.38
55.76
1:03
54.94
0.97
N/A
53.20
2: 1.07
56,80
= 0.27
UN Vote
61.04
1.
60.22
1.78
53.05
3.10
63.74
3.00
47
98
0.84
54.19
2.17
45.
09
0.43
N/A
$2.63
1.36
58.46
+ 0.91
Contact
90.12
2.34
89.22
94.15
0.45
88.13
1.50
74
0.80
90.44
0.17
89.91
0.36
NIA
93.56
1.
92.66
0.56
Avg. Bank
4:58
Wikipedia
68:70
0.39
20
87,51
1.4.3
Reddit
62.32
61.58
0.72
63
0.36
0.84
0.1
00:11
0.26
64.98
0.13
MODC
62.92
1.24
76.73
0,50
77.07
3.40
0.68
74.28
0.53
73.09
0.97
$1
0.31
80.82
0.30
$1.64
0.67
LasiEM
70.29
1 4.31
71,45
1.25
76
1 0.25
09
4.65
31
1 0.47
65.78
0.65
76.12
0.22
77.01
0.15
76.35
0.52
79.22
0.33
Earon
68:86
+ 3.71
62.08
2.27
61
+ 1.30
62.90
+ 1.16
50.72
+ 0.
67.11
0.62
72.37
1.38
72.85
0.81
67.07
+ 0.60
75.12
= 1.43
Social Evo.
88.51
1: 0.87
88.72
1.10
97
1 0.54
90.65
1 1662
89
1 0.59
94.10
0.32
94.01
1 0.47
96.91
0.12
96.82
1.0.17
95.45
0.30
ind
UCI
63.16
2 2.27
52,47
+ 2.0
70.49
1 0.9
70.75
2 0.79
64.54
4 0.47
76,65
0.99
$1.64
2: 0.49
82,06
0.58
72.13
1,86
73.65
2: 3.70
Can. Pur.
52.58
+ 0.86
52.24
+ 0.28
56.46
+ 0.50
54.18
+ 0.75
S7 06
0.08
$5.46
0.69
55.76
N/A
57.22
+ 0.53
91.59
+ 1,10
US Legis.
52.91
2.11 62.10
1.41
5183
3.95
1.10
55 56
1.71
53.87
1411
$2.03
1.02
NIA
56.31
1. 3,46
55,90
0.71
54
52.56
56,80
0.27
61.17
00.29
3.10
2.97
54.13
2.16
0.40
NIA
52.61
0.9
Contact
89.22
0.65
0.45
1.50
0.81
90.43
0.17
89.91
0.36
NIA
23.66
0.56
Avg. Rank
6.29
5.58
4.83
5.08
6.75
4:38
4,38
NIA
3.83
2.17
Table 17: AUC-ROC for inductive dynamic link prediction with random, historical, and inductive negative sampling strategies.
Damsett
IODIE
DyRup
TGAT
TGN
CAWN
TCL
OiighMixer
FreeDyG
DyGFermer
DyG- Mamba
Wikipedia
94.33
0.23
91.69
0.45
95.90
+: 01009
91.72
2: 0.03
95.03
4: 0.04
95.57
0.20
96.30
0.94
99.01
0.02
95.48
0.003
98.55
:: 0.01
Reddit
96.52
+ 0.13
96.05-
0.12
96
+
0.04
97.39
0.07
92.42
+ 0.00
93.80
0.07
94.07
0.05
08
-0.11
98.71
00001
98.91
- 0.01
MODC
83.16 1 1.30
84.03
0.49
96.81
1 0.17
91.24
0.99
$1.56
1 0.25
$1.43
0.19
82.77
0.24
87.01
0.74
$7.62
1..51
$9.98
0.46
LasEMM
81.13
3.39
82.24
= 1.51
7699
+ 0.29
82.61
+ 3.15
8782
+ 0.12
70,84
0.85
80.37
0.18
54.32
0.03
94.08
01.00
54.22
=: 0.01
Foron
81.96
+ 1.34
76.34
2 40
64663
+ 1.74
78.83
1111
07
+ 0.50
72.33
0.99
76.51
0.71
89.51
0.20
90.69
0.26
90.60
1:1
Social Evo.
93.70
0.29
91.18
0.49
93.41
93.43
0.59
54.73
0.27
93.71
0.18
94,09
0.07
96.41
95.29
53.41
0:00
UCT
78.00
84.29
93,01
92.63
92.00
0.23
55.83
56 12
NIA
0.09
US agis
52.12+
61.07
62.38
0.48
50.43
47
0.89
NIA
52.73
1.24
UN Trade
62.28 0.50
58.82
0.98
62.72
1
0.12
59
3.50
57.05
0.21
63.76
0.07
63.48
0.37
NIA
67.25
1.05
69.37
11 0.06
UN Vote
58.13 + 1,43
55.13
3.46
51
+
1.35
61.23
2.71
48
+
0.76
50.51
1.05
50.04
0.86
NIA
56.73
0.69
60:03
- 0.02
Cuntact
95.37
1
91.89
1 0.38
96.51
1 0.10
94.84
1 0.75
07
1 0.34
93.05
0.09
92.83
0.05
NIA
08. 30
100002
98..3
:1 8.01
Arg. Kank
3.67
6.33
6.25
4.25
5.17
6.92
6,08
NIA
1.25
1.67
Wikipedia
66186
0.53
57.54
= L.9
78.38
+ 0.20
75.75
+ 0.29
02
0A6
79.79
0.96
N2.87
0.21
82.08
0.32
68.33
a: 2.82
69.73
:: 0.48
Roddit
61.69
0.30
60.45
+ 0.37
64
+
0.27
64.55
0.50
64
94
0.21
61
0.26
64
0.13
66.79
0.31
64.81
+ 0.235
67.82
+ 0.30
MODC
61.48
1.64
61.23
1.29
74.08
0.27
77.69
3.55
7166
0.91
69.82
0.32
72.53
0.84
$1.32
0:37
80.77
82.74
1 0.75
5.62
88
70.07
0.37
karoa
61,50
2.50
57.84
2
62.68
1.09
64.6
08
20:09
03.77
75.35 1.00
Fvo.
88.53
87.93
1.00
91
92
83
0.24
93.28
0.60
93
96.94
06.91
96.03
0.34
hist
UCI
60.24 1.94 51.25
2.37
62.32
1
62.69
0.90
56
0.10
70.36
194
75.98
0.84
76.01
0.75
65.35
1.00
66.93
2.56
Can. Part.
51.62
= 1.00
32.38
0.46
58
+ 0.61
55.64
2 0.54
50.11
+ 0.4
57.
1.03
50.68
1.20
NIA
55.00
0.74
91.54
= 0.39
US Legis.
58.12 2.94
67.94
0.98
49
+ 4.88
64
6665
44
1:51
12
2.13
49
78
0.86
NIA
56.57
2 3.20
$6.15
0.15
UN Trade
58.73 1: 1.19
$7.90
1 1.35
59.74
1 0.59
35.61
3 3.55
50.95
1 0.50
62
0.97
59.38
1.17
NIA
58.46
1: 1.65
62,81
1 0.21
UN Vote
66.16
1.28
63.95
= 2.12
51.73
+
12
68.59
3.11
48.01
1:77
$4.66
2.11
45
49
0.42
N/A
53.85
2.00
62.69
+ 1.23
Contact
90.80
1 1.18
$8.85
0.68
93.76
1 0.41
88.84
1 1.39
74 79
0.37
90.37
0.16
90.04
0.29
NIA
94.
0.26
94.18
0.36
Arg. Kank
4.58
2.17
Wikipedia
161.87 0.53
57.54
78.88
0.20
75.76
0.20
0.65
79.79
60
83.17
0.31
68.33
0:48
Reddit
00.44
0.50
0.21
61.36
04
64.51
0.36
MODC
64.48
64.22
1.29
77.68
3.55
0.54
69.83
0.32
72.52
0.84
75.81
50.77
52.74
0.75
LasiMM
68.44
1. 3:26
68.79
1.08
69.89
0.28
66.99
5.61
57
68
0.24
55.83
888
70.07
0.20
71.42
0.33
70.75
0.37
72.52
0.31
Enron
63:32
=: 3.57
61.50
= 2.50
5783
4 2
62.68
= 1.08
62
+ 0.40
64.05
1.02
08.19
1.83
08.79
0.9
63.79
1: 0.42
75.35
= 1.06
Social Fvo.
88.53
0.55
87.93
1.05
98
+
0.72
92
1:22
83
+ 0.24
93.25
0.60
93.60
0:35
06.70
0.17
96.91
0.09
96.03
+ 0
ind
UCI
60.27
1 1.94
51.25
1
62
1 1.17
62.66
1 0.91
$6.39
1 0.11 70.42
1.93
15.97
0.85
73.41
0.85
65.58
1: 1.00
66.93
00051135 2.56
Can. Part.
51.61
+ 0.58
52.35
= 0.52
58.15
+ 062
55
+ 0.42
50.1
+ 0.47
56.88
0.93
5663
1.09
NIA
$8.51
01.73
91.54
= 0.39
US Legis
52.12
1 2.94
67.94
0.98
49 99
1
67
65
$441
1 131
52.12
2.13
49.38
0.86
NIA
56.57
1 3.20
55
: 0.15
UN Trade
58.71
0.59
3.39
0.79
61.01
0.93
59.71
NIA
57.28
62.81
0.21
51.78
4.14
68.58
3
48.04
51.85
45.57
NIA
53.87 2.001
62.69
1 1.23
Contact
1.18
88.87
93.76
0.40
88.85
1.39
74.79
0.38
90.37
0.16
9004
0.29
NIA
94.18
0.36
Avg. Rank
5.92
6.92
5.25
5.17
6.33
5.08
4.67
N/A
1.42
2.25
27

---

E Proofs for Theorems
E.1 Proofs of Theorem
Proof. Consider the following linear time-invariant (LTT) system on the interval [K [x], where
l 1--1 AtR for the i-th coordinate:
dh((s)
h(s) + BB ma, h(fx=1) = he-11
ds
where A diag( A1,.... Xn) and B₁ [R..I..... Bk T (also arranged diagonally for each
coordinate). Since Ag is diagonal with Re(),) O, the fundamental matrix solution for this system (i.e., the matrix exponential) is also diagonal and can be written
Aud diag e AiAla
An Alrio
Step 1: Expressing Ag. By definition of the matrix exponential for diagonal matrix:
A = ALdie = diag Ate,s ...., e Ale,
Step 2:
Expressing B
We compute the convolution term associated with the inhomogeneous part B₁ my. For diagonal system, the solution for each coordinate is:
Arei (.(Att,--7) myg dr.
N. 1g-14 +
Since ma, is constant w.r.t. and B is also constant in this interval, we can pull them out of the integral:
Ate.
hg diste hx-18 B my
cA((Ale,--)) dr.
Evaluating the integral:
Al.. Ar( dr = yolus (oldrn:-- du
Aex.
(etiAtti
A1
Hence,
hk.i A.AAki hE-18 +
1 (e> 1) Bk. ma,i.
This leads
B& diag Ar (ed1 - B ( - B
since each diagonal entry of B& matches the integral factor A; (edistens B Step 3: Final Form of hg Combining the above results yields the stated coordinate-wise update for hu.
h
A.A.E. h A (ct. - 1) BR,5 ma,i-
This confirms the forms of both Ag and B well the final expression for each coordinate hy.
E.2 Proofs of Theorem
Proof. Since DyG-Mamba has three core parameters, i.e., A, B and C that control its effectiveness. We define Theorem 2 to explain the main effects of the parameters B and C.
We first copy (8) from the paper, SSM-based node representation learning process, as follows
h A BRus,
N = Cyhes
(17)
28

---

h4 in (q.(1 can be further decomposed as follows:
hk
II
Biu +
II
A B +
Brug.
(18)
i=U
i=U
Then, considering A is one fixed parameter and A, = exp(A/ 2% in Eq. can be formulated as:
k-2
2/ = C2 II
A Buu + +.... C2
II
A 1- .... CR
(19)
1=0
1=0
= = At
Biu + ... (E Ar A) C. +.... Cy
where uy is the k-th input of M ie., = [e, :], and parameters B = Linnaa [M and C' Linearc(MM)) Thus, C₁ can be considered one query of k-th input us and B, can be considered the key of j-th input u, Then, C₁ can measure the similarity between uy and u, similar to the self-attention mechanism. According to the above- tnoov-mmnniooee proof, we can conclude that parameters and C can measure the similarity between current imput the previous ones and selectively copy the previous input.
E.3 Proofs of Theorem
Under spectral normalization constraints ([[ B la S 1, Woll2 S 1), the parameter perturbations induced by input noise AM satisfy:
Proof. Step I: Parameter perturbation bounds.
|AB| < |WBle||MM] < ||M,
| - |clle||MM]] s ||M,].
(20)
This follows directly from the Lipschitz continuity enforced by spectral normalization, where the spectral norm ||]]| precisely defines the maximum amplification factor of the linear transformation.
Perturbation propagation analysis. From the state update decomposition in Theorem 4.2, the output perturbation Am, can be expressed as:
Step 2:
k-j-1
Am
(IAC B [C
|m,ll
See Ac-i
(21)
==0
Using BBl C S 1 (spectral normalization) and | || & ||M (Step 1), we derive:
|mmil < AM (t)-
(22)
Here, the exponential term 11--1-1 Ae || S (llle-t))) arises from the eigenvalue constraint 7 max; Re():(A)) < 0.
Step 3: Stability via exponential decay. For y 0, the summation over exponentially decaying terms can be approximated as:
( cT(T-0 by] (1 (1-E?2)
20 (044-1))) (1) x
(23)
where T is the total sequence duration. This yields the final perturbation bound:
| S w| where x
(1 x2)
(24)
1>|
29

---

Impact Statement
This paper aims advance the field of Machine Learning by introducing Mamba-based framework for continuous-time dynamic graph modeling. We examine its potential impacts from two primary perspectives as follows. (i) dynamic graph modeling. With the rapid expansion of social and economic networks, dynamic graph modeling has emerged as prominent research topic in the machine learning community. In contrast to existing methods, our approach introduces novel Mamba-based framework, which incorporates irregular timespans control signals for continuous SSMs. This design enhances the model's ability effectively and efficiently capture long-term temporal dependency dynamic graphs. (ii) Time Information Effects. Despite significant advances in dynamic graph modeling, there is still lack of theoretical foundations regarding the influence of temporal information the evolution of dynamic graphs. Our work highlights the significant potential the timespan-basee -bsed memory forgetting mechanism to deepen the theoretical understanding of time in this domain. Overall, do not foresee any direct negative societal implications resulting from this research. Although more advanced modeling capabilities can be applied across a range of domains, believe that this methodology itself does not introduce any ethical societal concerns beyond those typically associated with improvements in machine learning.
30


Reviews:
[{"review_id": "hCjMkoKz2p", "reviewer": "Reviewer_kbHS", "summary": "This paper proposes DyG-Mamba, a new approach for analyzing dynamic graphs. The method adapts state space models by using time gaps between events as control signals. It draws inspiration from Ebbinghaus' forgetting curve, which shows that people forget things quickly at first, then more slowly over time. The model works similarly by adjusting how much it forgets based on time intervals. It also learns to keep important past information while filtering out noise. The main benefit is speed and memory efficiency. DyG-Mamba achieves top results with linear complexity instead of quadratic, making it much faster than transformer-based methods. It also stays robust even when there's a lot of noise in the data. This efficiency makes it possible to handle much larger and longer graph sequences, which could be useful for social networks and recommendation systems.", "questions": "1. The paper only looks at adding edges, but real graphs are more complex - nodes appear and disappear, features change over time. How would you extend DyG-Mamba to handle these cases? Are there basic limitations with using SSMs for node changes?\n2. Your biggest dataset has 2.4M edges, but real networks like Facebook or Twitter are much larger. Do you have any evidence this actually works on bigger graphs? Even tests on fake large graphs would help. The linear complexity sounds good but the hidden constants might be huge.\n3. Why use spectral norm constraints of exactly 1.0? What happens if you change or remove them? The ablation study doesn't really test this. Some basic sensitivity tests would show if this choice actually matters.\n4. The case study is too simple. Can you show clearer examples of what the model chooses to remember versus forget? How does it handle when the data patterns change over time?\n5. Why no comparison with Graph Neural ODEs that also work with continuous time? Also, copying results from other papers instead of running the methods yourself makes it hard to trust the comparisons.\n\n**Main concerns:** The core idea seems reasonable but needs more proof. I'd want to see: (a) tests on realistic graph sizes, (b) better analysis of design choices, (c) clearer understanding of what the model actually does. Right now the scope feels too narrow to be convincing.", "limitations": "Yes.", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. This paper introduces the first use of state space models for continuous-time dynamic graphs, which is a solid contribution. The approach of using time gaps as control signals is clever and well-motivated by psychological research on memory. The three theorems provide theoretical backing for the design choices, and the linear complexity is a clear advantage over transformer methods.\n\n2. The experiments cover 12 datasets with multiple evaluation settings, which is thorough. The results are generally strong across different tasks and show good consistency. The efficiency gains are substantial - much faster training and lower memory usage compared to existing methods.\n\n3. The model handles noisy data well, maintaining performance even with 60% noise injection. This is important for real applications where data quality varies. The ablation studies help understand which components matter most.\n\n4. The paper is well-written overall. The connection between memory theories and the technical approach is explained clearly, and the figures help illustrate the main ideas.\n\nWeaknesses:\n\n1. The biggest limitation is that experiments only cover edge addition on relatively small graphs. Real applications involve much larger networks with node changes, feature evolution, and different graph types. The largest dataset has only 2.4M edges, which is small by modern standards.\n\n2. While the linear complexity sounds good, the actual tests only go up to sequence length 2048. It's unclear how the method performs on much longer sequences or whether memory requirements become problematic. The constant factors in the complexity could be large.\n\n3. Some baselines are dated, and newer graph ODE methods are missing. Results for some methods are copied from other papers rather than re-implemented, which raises questions about fair comparison. The evaluation doesn't include recent 2024 methods.\n\n4. The case study doesn't provide much insight into what the model actually learns. It's hard to understand when the method might fail or how it handles changes in data patterns over time. More analysis of the learned representations would be helpful.\n\n5. The paper lacks discussion of training stability, hyperparameter sensitivity, and deployment considerations. The evaluation is limited to link prediction and node classification, missing other important temporal tasks like anomaly detection or community tracking.", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you very much for your efforts and time to review our paper and give us many helpful comments! We have carefully addressed your comments and hope our responses help clarify our contributions.**\n\n*Q1 & Weak-1. The paper only looks at adding edges, but real graphs are more complex - nodes appear and disappear, features change over time...Are there basic limitations with using SSMs for node changes?*\n\nA1. We appreciate your comment. We would like to clarify that DyG-Mamba does include these complex settings and we will add these descriptions in our paper.\n- Node Changes: In continuous-time link prediction, nodes dynamically appear and disappear, since the model only observes the one-hop neighborhood within a recent time window (typically of length 32 to 1024) at each prediction step. As a result, the set of observed nodes is time-dependent and varies across prediction steps.\n- Feature Changes: In link prediction datasets, nodes always have different features at different timestamps. DyG-Mamba is capable of handling such time-varying node features.\n- Different Graph Types: In our experiments, (1) Wikipedia, Reddit, MOOC, LastFM are heterogeneous bipartite graphs; (2) Can. Parl., US Legis., UN Trade and UN Vote are dynamic snapshots; (3) Enron, Social Evo., UCI are continuous-time dynamic graphs.\n\nDyG-Mamba not only supports the settings described above, but is also robust to noisy real-world conditions (Figure 5). We hope this helps clarify our experimental design and alleviates your concerns.\n\n*Q2 & Weak-2. Your biggest dataset has 2.4M edges...Do you have any evidence this actually works on bigger graphs? Even tests on fake large graphs would help. The linear complexity sounds good but the hidden constants might be huge.*\n\nA2. Thank you for your suggestion, we add one large-scale benchmark tgbl-coin-v2 [1], which contains 638 K nodes and 22.8 M edges.  We hope the results alleviate your concerns regarding DyG-Mamba’s scalability on large-scale graphs.\n\n|Method|Performance|Epoch Running Time (s)| GPU Usage (M)|\n|-|-|-|-|\n|DyRep|45.20±4.60|178,719|48,336|\n|TGN|58.60±3.70|138,408|48,116|\n|GraphMixer|75.31±0.21|43,160|12,204|\n|DyGFormer|75.17±0.38|163,151|41,348|\n|DyG-Mamba|76.14±0.30|45,124|18,094|\n|\n\nWe provide a more detailed analysis of runtime and memory usage on the Can.Parl. dataset. We hope the table below helps alleviate your concerns regarding the hidden constant in DyG-Mamba's complexity.\n\n|Input Length|256|512|1024|2048|\n|-|-|-|-|-|\n|DyG-Former (AP)|74.62±0.37|84.94±0.08|91.69±0.06|97.35±0.06|\n|DyGFormer Running Time/epoch (s)|324.40|487.55|630.44|946.99|\n|DyGFormer Memory Usage (M)|1,986|5,176|21,548|40,692|\n|DyG-Mamba (AP)|78.35±0.27|94.83±0.07|98.19±0.06|98.37±0.07|\n|DyG-Mamba Running Time/epoch (s)|160.32|168.26|174.45|190.13|\n|DyG-Mamba Memory Usage (M)|1,282|2,234|5,230|11,728|\n|\n\nAs shown in the table, \n- Running time increases gradually, confirming the low constant factor in our linear-time implementation. \n- Memory usage grows linearly as expected, and remains within acceptable bounds even for long sequences.\n- Model performance (AP) remains stable across different lengths, indicating efficiency without compromising predictive power.\n\nWe believe these results provide strong empirical evidence that the hidden constant in DyG-Mamba’s complexity is small. We will add this analysis to our revised paper to clarify the practical benefits of our design.\n\n*Q3. Why use spectral norm constraints of exactly 1.0? What happens if you change or remove them? The ablation study doesn't really test this. Some basic sensitivity tests would show if this choice actually matters.*\n\nA3.We thank the reviewer for the insightful question. The spectral norm constraint of $\\leq$ 1 is a necessary theoretical requirement to ensure the robustness guarantee established in Theorem 3, as supported by the theoretical proof in Appendix E.3. Here, we provide additional ablations and promise to add this table in our paper:\n|Spectral Norm Bound|Can.Parl.|Enron|USLegis.|\n|-|-|-|-|\n| w/o constraints|96.65±0.13|92.06±0.12|73.24±0.75|\n| $\\leq$ 2|96.98±0.07| 92.68±0.11|73.59±0.67|\n| $\\leq$ 1.5|97.61±0.10|92.93±0.13|73.86±1.42|\n| DyG-Mamba ($\\leq$ 1)|98.37±0.07|93.22±0.03|74.11±2.23|\n|\n\nAs shown in the table, relaxing the spectral norm constraint leads to consistently lower performance across all datasets. In particular, removing the constraint entirely (\"w/o constraints\") results in the most noticeable drop, highlighting its necessity. Even modest relaxations (e.g., ≤2 or ≤1.5) yield inferior results compared to our default setting (≤1), suggesting that tighter control of the parameter magnitude contributes meaningfully to model stability and predictive performance. \nWe sincerely thank the reviewer for this suggestion and commit to including this ablation study and its analysis in the revised version of our paper to better justify the use of spectral norm constraints in DyG-Mamba.\n\n*Q4 & Weak-4. The case study is too simple. Can you show clearer examples of what the model chooses to remember versus forget? How does it handle when the data patterns change over time?*\n\nA4. We thank the reviewer for this suggestion. We will revise our case study with a longer input sequence and clearer visualization to illustrate what DyG-Mamba chooses to remember versus forget. Specifically, DyG-Mamba tends to retain neighbor nodes that (i) frequently reappear in the historical sequence or (ii) are semantically similar to the target node, as these provide more predictive signals. In contrast, neighbors that are rarely observed or semantically unrelated are gradually down-weighted. When data patterns change over time, DyG-Mamba adapts by adjusting the memory decay rate through the learnable, timespan-aware parameter Δt (Eq. 11). This enables the model to quickly suppress outdated interactions and prioritize recent, meaningful signals. For example, in a transaction network, if a user suddenly begins interacting with new counterparties, DyG-Mamba assigns lower weights to outdated neighbors and increasingly attends to newly frequent ones. This allows the model to dynamically respond to behavioral drift without manual resetting. Additionally, the input-dependent transition and projection matrices (B and C), regularized via spectral norm constraints (Eq. 13), enhance the model’s ability to filter out transient or noisy inputs.\nWe will incorporate this enhanced case study and detailed explanation in the revised paper to better showcase DyG-Mamba’s adaptive memory behavior.\n\n\n*Q5 & Weak-3. Why no comparison with Graph Neural ODEs that also work with continuous time? Also, copying results from other papers instead of running the methods yourself makes it hard to trust the comparisons.*\n\nA5. We thank the reviewer for highlighting the importance of comparing with Graph Neural ODE-based models. We have now included one SOTA graph neural ODE baseline CTAN [2] and two additional baselines [3,4] in our experiments. The AP performance is reported as follows: \n\n|Datasets|FreeDyG|CTAN|Ding et al.|DyG-Mamba|\n|-|-|-|-|-|\n|Wiki|99.23±0.01|98.92±0.01|99.00±0.01|99.06±0.01|\n|Reddit|98.99±0.00| 99.03±0.02|99.13±0.02|99.25±0.00|\n|MOOC|88.39±0.14| 88.58±0.13|89.12±0.15|90.17±0.19|\n|LastFM|90.28±0.30| 91.32±0.05|93.26±0.02|94.22±0.04|\n|Enron|91.53±0.09|91.88±0.04|92.56±0.04|93.22±0.03|\n|Social Evo.|94.54±0.01|94.21±0.01|94.12±0.01|94.75±0.01|\n|UCI| 95.88±0.12|95.31±0.08|95.35±0.09|96.79±0.08|\n|\n\nThese results show that DyG-Mamba consistently outperforms ODE-based methods across all datasets, highlighting its strong generalization in continuous-time dynamic graphs. Regarding the concern about copied results, we confirm that all baseline results were independently reproduced by using the open-source benchmarking toolkit Dynamic Graph Library [5]. We follow standardized training protocols, shared preprocessing, and consistent hyperparameters (from DyGLib load_configs.py) and fix random seeds to ensure fair comparisons. \n\n*Weak-5. The paper lacks discussion of training stability, hyperparameter sensitivity, and deployment considerations. The evaluation is limited...*\n\nA6. We thank the reviewer for pointing out this important aspect. We will add training stability results to provide a more comprehensive view of DyG-Mamba’s behavior during optimization. In particular, we include below the training loss across epochs on the Can.Parl. dataset:\n\n| Model|Epoch=20|Epoch=40|Epoch=60|Epoch=80| Epoch=100|\n|-|-|-|-|-|-|\n|Vanilla Mamba|0.2229|0.2026|0.1964|0.1922|0.1914|\n|DyG-Mamba|0.1691|0.1478|0.1480|0.1482|0.1480|\n|\n\nAs shown in the table, DyG-Mamba exhibits faster convergence and more stable loss reduction, completing training within 100 epochs. We observe similar trends across all other datasets, indicating the model’s training stability. In addition, hyperparameter sensitivity and deployment considerations have been discussed in the appendix. We will explicitly refer to these sections in the main paper for better visibility:\n- Appendix D.4 (Figure 9 and Table 11) presents hyperparameter sensitivity analysis.\n- Appendix D.2 (Tables 6, 7 and Figure 7) covers model size, memory usage, and runtime for deployment.\n- Limitation Section: We acknowledge that exploring more domains and tasks is an important direction for future work.\n\n\n**Could you kindly raise your score and champion our paper if you think all concerns are addressed ? Thank you again for your valuable comments and review.**\n\n[1] Huang, Shenyang, et al. Temporal graph benchmark for machine learning on temporal graphs. NeurIPS 2023.\\\n[2] Long Range Propagation on Continuous-Time Dynamic Graphs. ICML, 2024.\\\n[3] FreeDyG: Frequency Enhanced Continuous-Time Dynamic Graph Model for Link Prediction. ICLR-2024.\\\n[4] Efficiently Modeling Long-Term Temporal Dependency on Continuous-Time Dynamic Graphs with State Space Models. TMLR, 2025.\\\n[5] Towards Better Dynamic Graph Learning: New Architecture and Unified Library. NeurIPS, 2023."}, {"review_id": "skA9JHF7VG", "reviewer": "Reviewer_ps9a", "summary": "This paper introduces a Mamba-based approach to continuous-time dynamic graphs, with (with mainly) experiments on dynamic link prediction. It is the first state-space model applied to the continuous-graph domain, and the idea is very promising.", "questions": "The performance increase appears marginal; why is this the case? More discussion should be added to explore this phenomenon.\n\nCan't also the DyG-Former learn long-term dependencies by using a larger patch size?\n\nHow does the model behave when applied to high-frequency temporal networks?\n\nSome work in the continuous-time graph literature, for example, the Piecewise-Velocity Model for Learning Continuous-Time Dynamic Node Representations [1], is missing from the related-work section. Could it be connected to the proposed SSM (especially as an first-order SSM )?\n\n[1] Piecewise-Velocity Model for Learning Continuous-Time Dynamic Node Representations.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n\nThe paper provides a linear-time SSM model, and the idea of applying such methods to continuous-time graphs is both novel and very promising. The design choice of including two SSM blocks, considering both node-level and edge-level information, is also interesting.\n\nThere is solid theoretical depth to the paper, and the connection with Ebbinghaus’ review cycle makes the approach principled and well-motivated.\n\nResults on link-prediction experiments are quite good, and the node-classification challenge is also discussed.\n\nImportantly, the paper introduces an extensive experimental set-up, which demonstrates that the DyG-Mamba structure significantly improves computational and memory efficiency compared with the Transformer.\n\nThe paper studies multiple negative-sampling strategies, adding value and completeness.\n\nThe case study offers valuable insights regarding the comparison with the DyG-Former.\n\n**Weaknesses**\n\nThe central idea is that, for dynamic graphs, especially those in the arduous continuous-time category, a model should learn by examining the long history of the sequence, which can now be extended because the model’s memory requirements do not need extra aggregation. Looking at Figure 2, panel (A) shows at best a marginal, or not even substantial, performance increase, while the message in panel (B) is much clearer. It is unclear what happens with the other networks; one could argue that, for some datasets, relying on a long history is unnecessary. As I see it, the model is not able to distinguish between these two cases.\n\nThe introduction of Δt, connected to the Ebbinghaus curve, is an elegant way to model the decay of historical impact with all its attractive theoretical properties. This choice, however, has the drawback that the decay does not rely on the actual content of the sequence: if an important signal is temporally distant, it could be disregarded. This raises the question, or necessity, of additional gating functions.\n\nAlthough this limitation is acknowledged in the paper, the networks included are limited in both size and domain, making some of the main claims about long-history modelling not easily generalisable from the current experimental set-up.", "was_revised": true, "final_justification": "The paper makes a novel contribution by extending state-space models to the challenging setting of continuous-time dynamic networks, a problem of broad interest to the graph learning community.\n\nThe theoretical development is rigorous, and the authors present a comprehensive experimental setup that demonstrates the model’s robustness even under high-noise conditions.\n\nThe rebuttal clarified several key points; the additional details provided there should be integrated into the final manuscript.\n\nI also encourage the authors to include a controlled study on synthetic networks. Such an experiment would allow them to vary interaction frequency and noise systematically, highlighting the scenarios in which the framework excels.", "rebuttal": "**Thank you very much for your valuable and constructive comments. We sincerely appreciate your efforts to help improve the quality of our work.**\n\n*Q1. The performance increase appears marginal...More discussion should be added to explore this phenomenon.*\n\nA1. Thank you for your insightful question. We agree that the performance gain of DyG-Mamba appears relatively modest on certain datasets. This is because those datasets are dominated by high-frequency and short-range interaction patterns, which limit the benefit of long-term modeling. For example, in Wikipedia, user-page edit interactions often occur in rapid bursts within seconds. In such settings, recent interactions already provide sufficient predictive signals, making it difficult for long-term modeling to deliver substantial improvement. As a result, short-term models already capture most useful signals, leaving limited room for DyG-Mamba’s long-term memory to contribute further.\n\nIn contrast, DyG-Mamba shows more substantial improvements on datasets such as Enron, UCI and Can. Parl., which exhibit  longer-range and irregular interaction patterns. For example, Enron captures corporate email communications over several months, where interactions are less frequent and often separated by long time gaps. In these settings, modeling long-term and irregular dynamics is crucial. DyG-Mamba is specifically designed to address these challenges, allowing it to capture informative long-term signals while suppressing noisy or outdated ones.\n\nWe hope this explanation clarifies the performance variations across datasets and provides better insight into the strengths of DyG-Mamba. We will include this analysis in our paper to help readers interpret the results more effectively.\n\n*Q2. Can't DyG-Former...by using a larger patch size?*\n\nA2. Thank you for your thoughtful question. We provide additional experiments to investigate how patch size affects DyG-Former’s performance. Specifically, we fixed the input sequence length to 512 and varied the patch size on LastFM. The results are shown below:\n\n|Patch Size|8|16|32|64|128|\n|-|-|-|-|-|-|\n| DyGFormer (AP)|92.14±0.11|93.00±0.12|92.03±0.14|91.26±0.16|90.33±0.13|\n|\n\nAs shown in the table, performance consistently degrades as patch size increases. We attribute this to several factors:\n- **Temporal Coarsening**: Larger patches aggregate multiple events into a single unit, which blurs fine-grained temporal patterns essential for modeling continuous-time dynamics.\n- **Uniform Attention Assumption**: DyGFormer assigns equal importance to all patches and lacks mechanisms to adaptively focus on temporally irregular or informative regions. This limits its ability to capture the most relevant historical signals, especially in real-world scenarios with irregular timespans and noisy interactions.\n- **Sensitivity to Noise**: Larger patches are more likely to include irrelevant or noisy events, which further undermines performance.\n\nWe appreciate your suggestion and will include this discussion in the appendix to help readers better understand the limitations of the baseline DyGFormer.\n\n*Q3. How does the model behave when applied to high-frequency temporal networks?*\n\nA3. Thank you for your insightful question. In our experiments, Wikipedia, Reddit and Contact are high-frequency temporal networks, where interactions occur at second-level intervals (Table 5). Although DyG-Mamba consistently outperforms baselines on these datasets, the performance gains over baselines tend to be modest. This is because most predictive signals are embedded in recent interactions, and while long-range dependencies still contribute, their marginal benefit is reduced due to the temporal density and redundancy of recent events. To further investigate this, we conducted an ablation study by varying the historical window size while keeping all other settings fixed. The results (AP) are summarized below:\n\n|Dataset|32|64|128|256|512|1024|2048| \n|-|-|-|-|-|-|-|-|\n|Wikipedia|98.98±0.02|99.06±0.01|99.08±0.01|99.10±0.02|99.10±0.01|99.12±0.02|99.16±0.03|\n|Reddit|99.11±0.01|99.25±0.00|99.25±0.01|99.28±0.01|99.28±0.01|99.30±0.02|99.33±0.02|\n|\n\nWe find that increasing the sequence length yields only marginal improvements, indicating that while DyG-Mamba effectively leverages long-term information, the nature of high-frequency graphs limits the additional benefits from extended temporal modeling. We appreciate the reviewer’s question and will include this empirical analysis in our paper to help readers better understand DyG-Mamba's behavior in high-frequency dynamic graphs.\n\n*Q4. Some work...is missing from the related-work.* \n\nA4. We thank the reviewer for pointing out this relevant work and we will include it in the final version: PIVEM [1] learns dynamic node embeddings by approximating temporal evolution by piecewise linear interpolation, based on a latent distance model with piecewise constant and node-specific velocities. It can be viewed as a special case of a first-order SSM, where the hidden state corresponds to node velocity and evolves linearly over time.\n\n*Weakness-1. The central idea...to distinguish between these two cases.*\n\nA5. Thank you for your comment. The effectiveness of long-term modeling indeed varies by dataset. In high-frequency graphs like Wikipedia or Reddit, where interactions occur within seconds, useful information is concentrated in the short-term context. As shown in Table 1, increasing the input sequence length provides only marginal improvement because distant history adds little additional signal. In contrast, datasets such as Enron and Can. Parl. contain sparse, irregular interactions with longer temporal gaps. DyG-Mamba benefits more in these settings by leveraging its timespan-aware decay and memory update mechanisms to capture long-term dependencies while filtering out temporal noise.\n\nTable 1 Performance (AP) of DyG-Mamba with varying input length.\n|Dataset|32|64|128|256|512|1024|\n|-|-|-|-|-|-|-|\n|Wikipedia|98.98±0.02|99.06±0.01|99.08±0.01|99.10±0.02|99.10±0.01|99.12±0.02|\n|Reddit|99.11±0.01|99.25±0.00|99.25±0.01|99.28±0.01|99.28±0.01|99.30±0.02|\n|Enron| 86.28±0.06|88.43±0.05|89.95±0.05|91.19±0.04|93.22±0.03|93.58±0.06|\n|Can. Parl.| 76.22±0.57|77.12±0.28|77.96±0.35|78.35±0.27|94.83±0.07|98.19±0.06|\n|\n\nWe also compared DyG-Mamba with GraphMixer, a strong baseline that focuses on short-term input sequence settings. In Table 2, DyG-Mamba achieves notable performance gains, which shows that DyG-Mamba’s time-aware decay mechanism and noise-resistant memory updates provide value even on high-frequency graphs.\n\nTable 2  Performance comparison on high-frequency temporal graphs.\n|Dataset|Wikipedia|Reddit|Contact|LastFM| \n|-|-|-|-|-|\n|GraphMixer|97.25±0.03|97.31±0.01|91.92±0.03|75.61±0.24| \n|DyG-Mamba|99.06±0.01|99.25±0.00|98.37±0.01|94.22±0.04| \n|\n\nDyG-Mamba dynamically adjusts attention based on temporal distance and relevance, which allows it to extract meaningful history only when needed. We will revise the paper to clarify the dataset-dependent impact of long-term modeling and highlight DyG-Mamba’s adaptability to different temporal patterns.\n\n*Weakness-2. The introduction of Δt...raises the question or necessity of additional gating functions.*\n\nA6. Thank you for highlighting this important point. We fully agree that a purely time-based decay may overlook semantically important yet temporal distant events. To address this, DyG-Mamba incorporates a form of content-aware gating via its input-dependent parameterization of the core matrices B and C (Theorem 4.2). These matrices function similarly to Query and Key in Transformer, dynamically controlling how much a historical input contributes based on its semantic alignment with the current context. Specifically, while the exponential decay term exp(f(Δt)) reduces the weight of distant events, the actual impact of a past event is jointly determined by its content via the interaction term C and B in Eq.(12).  When a temporally distant input is semantically important, it can still have a strong impact. We appreciate this suggestion, and to avoid misunderstanding, we will revise the manuscript to explicitly highlight how our architecture achieves this implicit gating behavior.\n\n*Weakness-3. Although this limitation...from the current experimental set-up.*\n\nA7. Thank you for your valuable suggestion. We acknowledge the importance of evaluating model generalizability across larger and more diverse networks. To this end, we conduct additional experiments on the large-scale tgbl-coin-v2 [1], which consists of 638K nodes and 22.8M temporal edges. tgbl-coin-v2 includes heterogeneous edge types, with transactions from five stablecoins and one wrapped token categorized into two types to reflect different modes of fund transfer [2]. The performance of DyG-Mamba and strong baselines on tgbl-coin-v2 is summarized below:\n|Method|Performance|Epoch Running Time (s)| GPU Usage (M)|\n|-|-|-|-|\n|DyRep| 45.20±4.60|178,719|48,336|\n|TGN|58.60±3.70|138,408|48,116|\n|GraphMixer|75.31±0.21|43,160|12,204|\n|DyGFormer|75.17±0.38|163,151|41,348|\n|DyG-Mamba|76.14±0.30|45,124|18,094|\n|\n\nThese results show that DyG-Mamba achieves the best performance (MRR) on large-scale datasets, further supporting its ability to handle long-term temporal dependencies. We believe this extended evaluation helps address concerns about the scale and domain diversity. We will incorporate this result and its discussion into the revised version to strengthen the empirical validation of our method.\n\n**We sincerely hope our detailed responses and additional experiments have addressed your concerns and clarified the strengths of our work. If you find our revisions satisfactory, we would greatly appreciate it if you could consider raising your score and supporting our paper.**\n\n\n[1] Temporal graph benchmark for machine learning on temporal graphs. NeurIPS, 2023. \\\n[2] Labeled graph datasets for UTXO and account-based blockchains. NeurIPS, 2022."}, {"review_id": "WHfHA0GGQE", "reviewer": "Reviewer_NsaC", "summary": "DyG-Mamba reframes continuous-time dynamic graph modeling as long-sequence modeling with a selective SSM. It treats the irregular timespan between events as a control signal, redefining the SSM’s step size and transition matrix so forgetting follows an Ebbinghaus-style curve. To counteract over-forgetting and noise, it makes the projection parameters input-dependent and applies spectral-norm constraints, letting the model selectively “review” important history.  This design yields linear-time and memory complexity, enabling full-length sequences without pooling.  Across 12 benchmarks in link prediction and node classification, DyG-Mamba attains state-of-the-art accuracy, is up to 8.9× faster, and uses 77 % less GPU memory than Transformer baselines, while remaining robust to up to 60 % noisy edges.", "questions": "None", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**: \n\n1. The authors reformulate dynamic-graph modeling as a continuous SSM and give formal guarantees on stability, forgetting behaviour (Theorem 4.1), and noise attenuation (Theorem 4.3).\n\n2. The experiments are comprehensive, demonstrating convincing performance compared with various baseline methods.\n\n3. Given the ability of the SSM backbone that scales linearly with sequence length, the proposed method opens the door for much more efficient implementation for many downstream tasks.\n\n**Weakness**:\n\n1. The choice of the specific learnable function for Δt in Equation (11) seems to be important to the method. The paper does not provide a detailed justification or ablation for this specific mathematical form versus other possible monotonically increasing functions. Was this form empirically chosen, and how sensitive is the model to this choice?\n\n2. The paper lacks a detailed discussion on how the theoretical bound in Theorem 4.3 is related to the empirical evaluation in Section 5.4.\n\n3. Experiments remain on modest-scale graphs; it is unclear whether the linear scan kernel remains GPU-efficient on million-edge streams or with heterogeneous edge types.\n\n4. My overall feeling of the paper is to use Mamba architecture on DyGFormer and address the same problem that DyGFormer worked on. Most of the advantages the method exhibits are inherent in the Mamba architecture. Although it is good to see that the authors made improvements on SSM to tailor it for the DGL task, it is still hard to find highlights in the paper.", "was_revised": true, "final_justification": "Most of my concerns have been addressed. I still believe that DyG-Mamba is more akin to a mamba extension of DyGFormer, and the scope of this work should be improved from this perspective. Given the new results and extra effort the authors have made, I'd like to increase the rating for the paper.", "rebuttal": "**We sincerely thank you for your constructive and thoughtful feedback. We truly appreciate your recognition of our theoretical contributions and comprehensive experiments. We will address each of your concerns in detail and revise the manuscript accordingly.**\n\n*Q1. The choice of the specific learnable function for Δt in Equation (11) seems to be important to the method. The paper does not provide a detailed justification or ablation for this specific mathematical form versus other possible monotonically increasing functions. Was this form empirically chosen, and how sensitive is the model to this choice?*\n\nA1. We thank you for raising this insightful question. The design of the learnable function for Δt is inspired by the Ebbinghaus forgetting curve [1,2]. Specifically, our formulation Eq.(11) mirrors the mathematical form of the Ebbinghaus forgetting curve as proposed in [3,4]: R=exp⁡(−t/S), where R denotes memory retention, t is the time interval, and S is a decay constant. In our design, we reinterpret the retention function R as part of the timespan-dependent scaling factor to ensure that larger timespans induce stronger decay, aligning the model behavior with human memory dynamics. Following your suggestion, we provide several alternative monotonic functions and conduct ablation studies to validate our design. These results will be included in our paper:\n- (1) Linear Variant: Δt = $w ⊙ (\\frac{t_{k+1}-t_{k}}{\\tau-t_{1}})$.\n- (2) Logarithmic Variant: Δt = $w_1$ ⊙ log$(1+ w_{2} ⊙ \\frac{t_{k+1}-t_{k}}{\\tau-t_{1}})$.\n- (3) Sigmoid Variant: Δt = $w_{1}$ ⊙ Sigmoid$(w_{2} ⊙ \\frac{t_{k+1}-t_{k}}{\\tau-t_{1}})$.\n- (4) Exponential Variant: Δt = $w_{1}$ ⊙ exp$(w_{2} ⊙ \\frac{t_{k+1}-t_{k}}{\\tau-t_{1}})$.\n- (5) w/o $(\\tau-t_{1})$: remove normalization item in Eq.(11).\n- (6) w/o timespan: replaces timespans with input samples u(t) as control signals, i.e., Δt =SiLU(Linear(u(t))).\n\n|Variants|Can. Parl.|Enron|USLegis.|\n|-|-|-|-|\n| Linear|94.64±0.18|91.13±0.06|70.28±1.84|\n| Log|97.18±0.12|92.35±0.05|72.46±2.34|\n| Sigmoid|95.84±0.13|92.26±0.04|72.15±2.36|\n| Exp| 94.68±0.11|90.84±0.06|71.23±1.48|\n| w/o ($\\tau-t_{1}$)|97.32±0.20|92.46±0.16|72.45±0.84|\n| w/o timespan|96.90±0.18|92.14±0.12|72.26±0.76|\n| DyG-Mamba|98.37±0.07|93.22±0.03|74.11±2.23|\n|\n\nThese results demonstrate that our DyG-Mamba achieves superior performance. Compared to the Linear and Log variants, our decay function better models diminishing memory influence with a smoother and bounded curve. The Sigmoid variant suffers from early saturation, limiting its ability to differentiate mid- and long-range intervals. The Exp variant over-amplifies long timespans, making it sensitive to noise. Moreover, removing normalization or timespan leads to performance degradation, confirming the necessity of our full design. We hope this response addresses your concerns. We appreciate your insightful suggestions and will incorporate this ablation study and its analysis into the revised paper to help readers better understand the motivation and effectiveness behind our Δt design.\n\n[1] Über das gedächtnis, Duncker & Humbolt, 1885.\\\n[2] Replication and analysis of Ebbinghaus’ forgetting curve. PLoS ONE, 2015.\\\n[3] Two components of long-term memory. Acta neurobiologiae experimentalis, 2010.\\\n[4] The psychology and neuroscience of forgetting. Annu. Rev. Psychol., 2004.\n\n*Q2. The paper lacks a detailed discussion on how the theoretical bound in Theorem 4.3 is related to the empirical evaluation in Section 5.4.*\n\nA2. Thank you for your insightful question. We will explicitly strengthen the discussion linking Theorem 4.3 to the empirical robustness results in Section 5.4 to help readers better understand its practical implications. Specifically, Theorem 4.3 provides a theoretical robustness guarantee by bounding the output perturbation under input noise. Section 5.4 empirically supports this claim by injecting random noisy edges into the input historical sequences and evaluating model performance. DyG-Mamba shows only minor performance degradation compared to baselines, demonstrating its strong resistance to noise. To further strengthen this connection, we include two additional experiments:\n- (1) Lipschitz Bound Tightness: We examine whether the theoretical bound in Theorem 4.3 accurately captures the empirical output deviation under input perturbations. Using the Wikipedia dataset, we report the input/output perturbation, the bound, and whether it holds. These results confirm that the empirical behavior of DyG-Mamba adheres to the theoretical bounds.\n\n| Noise Ratio | Input Perturbation | Output Perturbation | Theoretical Bound | Within Bound? | Performance |\n|-|-|-|-|-|-|\n|10%|0.13|0.14|0.16|Yes|0.987|\n|20%|0.27|0.30|0.33|Yes|0.985 |\n|30%|0.39|0.44|0.48|Yes|0.982 |\n|40%|0.48|0.54|0.60|Yes|0.973 |\n|50%|0.57|0.65|0.71|Yes|0.968 |\n|60%|0.66|0.74|0.82|Yes|0.958 |\n|\n\n- Ablation Study: We further demonstrate that Theorem 4.3 plays a critical role in enhancing the model's robustness. Here, we use Can. Parl. dataset as a case study.\n\n| Baselines/Noise Ratio (%)|0|10| 20| 30| 40| 50|\n|-|-|-|-|-|-|-|\n| w/o Theorem 4.3| 96.65±0.13|96.25±0.11|95.75±0.10| 95.25±0.08|94.83±0.08|93.21±0.11|\n| DyG-Mamba|98.37±0.07|98.32±0.06|98.05±0.05|97.95±0.05|97.90±0.07|97.74±0.06|\n|\n\nIn summary, Theorem 4.3 explains DyG-Mamba’s strong noise robustness. Our new experiments confirm the bound’s tightness and show that removing it leads to clear performance drops. We will add these results in our paper to better highlight the practical value of Theorem 4.3.\n\n\n*Q3. Experiments...it is unclear whether the linear scan kernel remains GPU-efficient on million-edge streams or with heterogeneous edge types.*\n\nA3. Thank you for your valuable suggestion. To further verify the scalability and efficiency of DyG-Mamba on million-edge dynamic graphs with heterogeneous structures, we conduct additional experiments on the large-scale tgbl-coin-v2 [1], which consists of 638K nodes and 22.8M temporal edges. tgbl-coin-v2 includes heterogeneous edge types, with transactions from five stablecoins and one wrapped token categorized into two types to reflect different modes of fund transfer [2]. This setting enables us to evaluate DyG-Mamba’s capacity to handle both large-scale and heterogeneous temporal interactions. For a fair comparison, we follow the training details as listed in [3] and adopt the best-performing hyperparameters for all baselines. The performance (MRR) of DyG-Mamba and strong baselines on tgbl-coin-v2 can be summarized below:\n\n|Method|Performance|Epoch Running Time (s)|GPU Usage (M)|\n|-|-|-|-|\n|DyRep|45.20±4.60|178,719|48,336|\n|TGN|58.60±3.70|138,408|48,116|\n|GraphMixer|75.31±0.21|43,160|12,204|\n|DyGFormer|75.17±0.38|163,151|41,348|\n|DyG-Mamba|76.14±0.30|45,124|18,094|\n|\n\nThese results show that DyG-Mamba achieves the best performance on the heterogeneous tgbl-coin-v2 dataset, highlighting its strong modeling capability under multi-type temporal interactions. In addition, compared to DyGFormer—another strong baseline designed for long-range dependency modeling—DyG-Mamba exhibits significantly lower runtime and memory consumption while delivering superior performance. We will include these results and the corresponding discussion in the revised version to further illustrate DyG-Mamba’s scalability and practical efficiency.\n\n[1] Temporal graph benchmark for machine learning on temporal graphs. NeurIPS, 2023.\\\n[2] Labeled graph datasets for UTXO and account-based blockchains. NeurIPS, 2022.\\\n[3] An Empirical Evaluation of Temporal Graph Benchmark. arXiv, 2023. \n\n*Q4. My overall feeling...it is still hard to find highlights in the paper.*\n\nA4. Thank you for raising this thoughtful question.  We fully understand your concerns and would like to explain that DyG-Mamba is not merely an architecture replacement of DyGFormer. Instead, DyG-Mamba introduces (1) a new inductive bias by the Ebbinghaus forgetting curve to guide memory updates; (2) a novel timespan function for effectively modeling irregular temporal patterns; (3) a linearly scalable architecture with theoretically grounded noise attenuation and empirically validated robustness. \nWe fully agree that Mamba’s core strengths—such as its linear-time complexity—form a strong foundation for DyG-Mamba. However, the main contribution lies in adapting and extending the SSM to address the unique challenges of dynamic graph learning. The \"highlights\" of our paper could be summarized as follows: \n- **Ebbinghaus-Inspired Timespan Function for Irregular Temporal Modeling**: We extend the continuous SSM to dynamic graphs by incorporating irregular timespans as key control signals. Inspired by the Ebbinghaus forgetting curve, we design a learnable timespan function that redefines Δt and guides memory decay through initialization of the transition matrix A. This enables effective modeling of irregular temporal patterns that vanilla SSMs like Mamba do not address.\n- **Robust Memory Control via Input-Dependent Parameterization**: Inspired by Ebbinghaus review cycles, we make the core SSM parameters B and C input-dependent and regulate their capacity using spectral norm constraints. This enables DyG-Mamba to selectively revisit relevant history while suppressing noisy or irrelevant signals, enhancing robustness in noisy environments. \n- **Addressing DGL-Specific Challenges**: RNN-based methods suffer from gradient issues, while Transformer-based ones face quadratic complexity. DyG-Mamba overcomes both with a linear-time architecture enhanced by timespan-aware mechanisms. Through selective memory updates and input-adaptive parameterization, DyG-Mamba mitigates the impact of noisy or irrelevant historical signals, improving stability in real-world dynamic graphs.\n\n**We sincerely hope that our explanation helps alleviate your concerns and provides a clearer understanding of our work. Could you kindly raise your score and champion our paper if you think all concerns are addressed ?**"}, {"review_id": "Ax0IARIvEF", "reviewer": "Reviewer_Cc3i", "summary": "This paper introduces DyG-Mamba, a novel continuous state space model (SSM) tailored for dynamic graph modeling. By drawing inspiration from Ebbinghaus forgetting and review cycles, the model introduces a timespan-informed mechanism to dynamically forget or reinforce historical information. DyG-Mamba redefines the core SSM parameters—Δt, A, B, and C—to handle irregular timespans and filter noisy data, thereby aiming to improve the effectiveness, robustness, and efficiency of dynamic graph modeling. The authors evaluate DyG-Mamba on 12 benchmark datasets across tasks like link prediction and node classification, reporting SOTA performance in accuracy, speed, memory, and robustness.", "questions": "1. What does the performance look like on smaller graphs with short sequences? Does DyG-Mamba still justify its complexity?\n2. The paper reports linear complexity but test on sequences ≤2,048.  Can DyG-Mamba handle billion-edge graphs (e.g., social networks)?  If not, what are the bottlenecks?", "limitations": "N/A", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "#### Strengths:\n\n1. The paper proposes a timespan-informed continuous SSM for dynamic graph modeling, integrating inspiration from psychological memory theories, which is an intuitive and creative angle. \n2. DyG-Mamba redefines core SSM parameters (∆t, A, B, C) to enable time-aware forgetting and input-conditioned reviewing, aiming to improve robustness and generalization — a technically meaningful contribution. \n3. The model demonstrates strong empirical performance across 12 datasets, with results exceeding or matching state-of-the-art baselines under various negative sampling schemes. \n\n#### Weaknesses:\n1. The metaphorical explanations (e.g., Ebbinghaus forgetting/review curves) are somewhat overemphasized and can feel stretched when mapped to a technical context.\n2. The comparison with recent SSM-based dynamic graph models (e.g., DG-Mamba) is missing, even though such models target similar goals. \n3. While the model is described as lightweight, the added architectural complexity (learnable ∆t function, spectral constraints, dynamic B and C) may increase training difficulty without clear discussion of stability or convergence behavior.", "was_revised": true, "final_justification": "I wish to express my sincere appreciation for the authors' thorough and substantive rebuttal.\n\nThe supplementary experiments on small-scale datasets (Table 1) convincingly establish DyG-Mamba’s adaptability beyond long sequences, while scalability validation on the tgbl-coin-v2 benchmark (22.8M edges) substantiates linear efficiency claims. Furthermore, the theoretical refinements to metaphorical descriptions and expanded comparisons against contemporaneous works (e.g., Ding et al.) significantly strengthen the paper’s technical framing.\n\nConditional upon seamless integration of these new results, expanded discussions, and comparative analyses into the final manuscript, I endorse this work for acceptance.", "rebuttal": "**Thank you very much for your valuable and constructive comments. We sincerely appreciate your efforts to help improve the quality of our work. We have carefully responded to your concerns and hope that our revisions will help alleviate them.**\n\n*Q1. What does the performance look like on smaller graphs with short sequences...*\n\nA1. Thank you for your thoughtful question. We appreciate your concern regarding the applicability of DyG-Mamba to smaller graphs with shorter temporal sequences. To address this, we conducted additional experiments and provided detailed empirical comparisons, which will be included in our paper to enhance clarity. Specifically, we evaluate DyG-Mamba on three relatively small-scale datasets—UCI, Can. Parl., and US Legis.—under short input sequence lengths of 32 and 64. We compare DyG-Mamba with two strong baselines GraphMixer and DyGFormer as follows:\n\nTable 1. Performance (AP) on smaller graphs with short input sequences.\n\n|Model|UCI|LastFM|Enron|\n|-|-|-|-|\n|GraphMixer-32|93.25±0.18|75.60±0.25|82.23±0.03| \n|GraphMixer-64|93.12±0.22|75.61±0.28|82.21±0.04|\n|DyGFormer-32|95.79±0.17|89.12±0.15|83.51±0.06|\n|DyGFormer-64|95.76±0.15|90.86±0.14|83.74±0.05|\n|DyG-Mamba-32|96.79±0.08|89.22±0.17|86.28±0.06|\n|DyG-Mamba-64|96.83±0.05|91.12±0.13|88.43±0.05|\n|\n\nAs shown in the table, **DyG-Mamba consistently outperforms both baselines, even on small graphs with short input sequences.** This suggests that our model effectively adapts to different input sequence lengths, not just long-term modeling. We believe this is because short sequences can still contain irregular timestamps and noise, which often degrade the performance of baselines. In contrast, DyG-Mamba leverages timespan-aware memory updates and selective noise filtering to address these challenges. We hope this explanation helps alleviate your concerns.\n\n*Q2. The paper...Can DyG-Mamba handle billion-edge graphs...*\n\nA2. Thank you for raising this important point. Although we noted the dataset scale as a limitation in our paper, we fully agree with your suggestion that demonstrating performance on larger graphs is essential to support our scalability claims. To further verify the scalability and efficiency of DyG-Mamba on million-edge temporal graphs, we conducted additional experiments on tgbl-coin-v2 [1], which contains 638K nodes and 22.8M temporal edges. We strictly followed the standardized training pipeline and hyperparameter setup as proposed in [2] to ensure fair comparison. The results are shown in Table below:\n\n|Method|Performance|Epoch Running Time (s)| GPU Usage (M)|\n|-|-|-|-|\n|DyRep| 45.20±4.60|178,719|48,336|\n|TGN|58.60±3.70|138,408|48,116|\n|GraphMixer|75.31±0.21|43,160|12,204|\n|DyGFormer|75.17±0.38|163,151|41,348|\n|DyG-Mamba|76.14±0.30|45,124|18,094|\n|\n\nAs the table illustrates, **DyG-Mamba not only achieves superior predictive performance but also outperforms the best-performing baseline in training efficiency, further validating its scalability on real-world large-scale dynamic graphs.** We sincerely apologize for not including experiments on billion-edge graphs. This is mainly due to the current lack of publicly available benchmark datasets of that scale in the continuous-time dynamic graph domain. We acknowledge this as a limitation and will explicitly add this point to the revised version of our paper.\n\n[1] Shenyang, Huang et al. Temporal graph benchmark for machine learning on temporal graphs. NeurIPS 2023.\\\n[2] Le Yu. An Empirical Evaluation of Temporal Graph Benchmark. ArXiv, 2023.\n\n*Weakness-1: The metaphorical explanations are somewhat overemphasized and can feel stretched when mapped to a technical context.*\n\nA3. Thank you for your thoughtful suggestion. We understand your concern regarding the potentially overemphasized metaphorical explanations. Our intention was to use these metaphors as intuitive motivation to help readers better understand the rationale behind our temporal modeling design. Following your suggestion, we will revise the original descriptions to be more concise and technically focused, ensuring that the metaphorical elements serve only as high-level intuition without overshadowing our core technical contributions. Specifically, **we will streamline and modify Lines 152-157 regarding the Ebbinghaus forgetting curve as follows**:\n- The Ebbinghaus forgetting curve can be formulated as R=exp⁡(−t/S) [1,2], where R denotes memory retention, t is the time interval, and S is a decay constant. Inspired by this formulation, we reinterpret R as a timespan-dependent decay coefficient, allowing the model to apply temporal decay to historical states proportionally to the elapsed timespan.\n\nIn addition, we will revise the metaphorical explanations in Introduction to make them more concise and technically focused. We sincerely hope that the revisions we have committed to will help resolve your concerns.\n\n[1] Wozniak, P. et al. Two components of long-term memory. Acta neurobiologiae experimentalis, 2010.\\\n[2] Wixted, T. The psychology and neuroscience of forgetting. Annu. Rev. Psychol., 2004.\n\n*Weakness-2: The comparison with recent SSM-based dynamic graph models (e.g., DG-Mamba) is missing, even though such models target similar goals.*\n\nA4. We sincerely thank the reviewer for pointing out the missing comparison with DG-Mamba [1]. We would like to clarify that while DG-Mamba is indeed a recent SSM-based model for dynamic graphs, it is specifically designed for discrete-time dynamic graphs where the input is segmented into predefined snapshots with fixed or uniform time intervals. In contrast, our work targets continuous-time dynamic graphs, where events occur at irregular time intervals, and thus demands a fundamentally different temporal modeling approach. Although we cited DG-Mamba in Lines 85–86 of the original manuscript, we acknowledge that our explanation may have been too brief and could have caused confusion. To address this, **we plan to revise Lines 85–86 in the paper to clearly state the scope and limitations of DG-Mamba and why it is not directly comparable in our experiments as follows**：\n- DG-Mamba [1] and GraphSSM [2] extend Mamba to discrete-time dynamic graphs by modeling snapshot sequences with fixed time intervals. However, these methods are not applicable to continuous-time dynamic graphs with irregular timestamps, and thus are not directly comparable to our setting.\n\nTo further alleviate your concerns, we additionally include a recent concurrent work, Ding et al. [3], which adapts Mamba for continuous-time dynamic graph modeling, as a new baseline for comparison. Ding et al. propose a heuristic framework that stacks node-level and time-level Mamba blocks to capture edge-specific temporal patterns for link prediction. We reproduce this baseline following the settings and best-tuned hyperparameters reported in their paper, and report the performance (AP) in the table below. And our method consistently outperforms Ding et al.[3] across multiple datasets, demonstrating the effectiveness of our proposed Mamba adaptation specifically tailored for dynamic graphs.\n\n|Benchmarks|LastFM|Enron|MOOC|Reddit|Wikipedia|UCI|Social Evo.|\n|-|-|-|-|-|-|-|-|\n|Ding et al. [3]|93.26±0.02|92.56±0.04|89.12±0.15|99.13±0.02|99.00±0.01|95.35±0.09|94.12±0.01|\n|DyG-Mamba|94.22±0.04|93.22±0.03|90.17±0.19|99.25±0.00|99.06±0.01|96.79±0.08|94.75±0.01|\n|\n\nWe sincerely hope that the additional comparisons and corresponding revisions will help alleviate your concerns regarding the selection of baselines.\n\n[1] Haonan Yuan et al., DG-Mamba: Robust and Efficient Dynamic Graph Structure Learning with Selective State Space Models. AAAI, 2024.\\\n[2] Jintang Li et al., State space models on temporal graphs: A first-principles study. NeurIPS, 2024.\\\n[3] Zifeng Ding et al., Efficiently Modeling Long-Term Temporal Dependency on Continuous-Time Dynamic Graphs with State Space Models. TMLR, 2025.\n\n*Weakness-3: While the model is described as lightweight, the added architectural complexity (learnable ∆t function, spectral constraints, dynamic B and C) may increase training difficulty without clear discussion of stability or convergence behavior.*\n\nA5. We appreciate your insightful comment regarding training difficulty. To address this, we will include clarifying discussion in our revised version. These task-specific designs are lightweight and stable in practice, as detailed below.\n- Learnable ∆t Function：We implement ∆t as an element-wise exponential decay function, realized by a scalar-wise MLP. This function is differentiable, non-recurrent, and incurs negligible computational overhead. \n- Spectral Norm Constraint: This serves as a regularization to prevent the B and C matrices from exploding, without introducing any additional parameters. As proven in Theorem 3, this design ensures bounded output under input noise, thereby enhancing both robustness and convergence stability.\n- Input-Dependent B and C: We re-define B and C as input-dependent parameters using shallow, non-recurrent linear layers. This structure introduces minimal additional parameters and computational cost while enabling the model to selectively attend to relevant inputs. To further support our training efficiency, we show smooth loss curves and consistent convergence behavior on Can.Parl. dataset as follows:\n|Model|Epoch=20|Epoch=40|Epoch=60|Epoch=80|Epoch=100|\n|-|-|-|-|-|-|\n| Vanilla Mamba|0.2229|0.2026|0.1964|0.1922|0.1914|\n|DyG-Mamba|0.1691|0.1478|0.1480|0.1482|0.1480|\n|\n\nAs shown in the table, our method converges within 100 epochs, demonstrating fast and stable training behavior. We observe similar convergence patterns across all other datasets as well.  We hope these additional details clearly address your concerns regarding Weakness-3, particularly the model's training stability.\n\n**We sincerely hope our response has helped address your concerns. If you feel that all issues have been resolved, we would truly appreciate your support in the final evaluation.**"}]

Author rebuttal:

Meta-review: This paper introduces DyG-Mamba, the first state-space model applied to continuous-time dynamic graphs, which reframes dynamic graph modelling as long-sequence modelling with a selective state-space model (SSM). The approach treats irregular time intervals between events as control signals that redefine the SSM's step size and transition matrix to implement Ebbinghaus-style forgetting curves, while using input-dependent projection parameters and spectral-norm constraints to selectively "review" important historical information and counteract over-forgetting and noise. DyG-Mamba achieves linear-time and memory complexity, enabling processing of full-length sequences without pooling, and demonstrates state-of-the-art performance across 12 benchmark datasets on tasks like dynamic link prediction and node classification, excelling in accuracy, speed, memory efficiency, and robustness.

Reviewers felt that the paper makes a novel and theoretically rigorous contribution by extending state-space models to continuous-time dynamic graphs, with solid theoretical grounding that connects Ebbinghaus' review cycles to principled forgetting mechanisms and provides formal guarantees on stability, forgetting behaviour, and noise attenuation. The approach redefines core SSM parameters to enable time-aware forgetting and input-conditioned reviewing, resulting in a technically meaningful framework that scales linearly with sequence length and opens doors for efficient implementation across downstream tasks. Experimentally, DyG-Mamba demonstrates superior performance across 12 benchmarks in link prediction and node classification, achieving state-of-the-art accuracy while being an order of magnitude faster and using a quarter of the GPU memory compared to Transformer baselines, with robustness to up to 60% noisy edges and comprehensive evaluation under multiple negative sampling strategies.

The main weaknesses identified include several key issues. First, reviewers found insufficient justification for the specific learnable function form for $\Delta$t in Equation 11. They raised concerns about sensitivity to this choice and noted the lack of detailed ablation studies. The paper also failed to connect theoretical bounds with empirical evaluation.

Reviewers also noted that experiments were limited to modest-scale graphs. This raised questions about GPU efficiency on million-edge streams or with heterogeneous edge types. Critics viewed the approach as primarily applying Mamba architecture to DyG-Former's problem domain. They argued it lacked significant novel highlights beyond inherent Mamba advantages.

Additional concerns included the model's inability to distinguish when long history is unnecessary versus beneficial. The content-agnostic decay could potentially disregard temporally distant important signals. Reviewers also criticized the overemphasis on metaphorical explanations that felt stretched in technical contexts.

In response, the authors provided ablation studies demonstrating superior performance of their $\Delta$t function choice over alternatives, confirmed Lipschitz bound tightness through additional experiments, and addressed content-aware gating concerns by highlighting input-dependent parameterisation that functions similarly to Transformer Query-Key mechanisms. They also presented large-scale experiments on heterogeneous networks showing best MRR performance, refined theoretical descriptions to reduce metaphorical overemphasis, and expanded comparisons with contemporaneous works, while supplementary small-scale experiments demonstrated adaptability beyond long sequences.

Among the majority of reviewers their was support for acceptance and I endorse this view.
Decision: Accept (poster)