Analytical Construction on Geometric Architectures: Transitioning from Static to Temporal Link Prediction
Authors: Yadong Sun
Xiaofeng Cao
Ivor Tsang
Heng Tao Shen
Venue: ICML 2025 poster (2025)
Keywords: Dynamic systems
Graph neural networks
Hyperbolic representation learning
Primary area: deep_learning->graph_neural_networks

Abstract: Static systems exhibit diverse structural properties, such as hierarchical, scale-free, and isotropic patterns, where different geometric spaces offer unique advantages. Methods combining multiple geometries have proven effective in capturing these characteristics. However, real-world systems often evolve dynamically, introducing significant challenges in modeling their temporal changes. To overcome this limitation, we propose a unified cross-geometric learning framework for dynamic systems, which synergistically integrates Euclidean and hyperbolic spaces, aligning embedding spaces with structural properties through fine-grained substructure modeling. Our framework further incorporates a temporal state aggregation mechanism and an evolution-driven optimization objective, enabling comprehensive and adaptive modeling of both nodal and relational dynamics over time. Extensive experiments on diverse real-world dynamic graph datasets highlight the superiority of our approach in capturing complex structural evolution, surpassing existing methods across multiple metrics.

Analytical Construction on Geometric Architectures: Transitioning from Static to Temporal Link Prediction
Yadong Sun
Xiaofeng Cao 12 Ivor W. Tsang" 34 Heng Tao Shen 25
Abstract
Single
Single Geometry Temporal Method
Geometry
Static systems exhibit diverse structural proper- ties, such as hierarchical, scale-free, and isotropic patterns, where different geometric spaces offer unique advantages. Methods combining multi- ple geometries have proven effective captur- ing these characteristics. However, real-world systems often evolve dynamically, introducing significant challenges modeling their temporal changes. To overcome this limitation, we propose unified cross- prossggeometric learning framework for dynamic systems, which synergistically integrates Euclidean and hyperbolic spaces, aligning embed- ding spaces with structural properties through fine- grained substructure modeling. Our framework further incorporates a temporal state aggregation mechanism and an evolution driven optimization objective, enabling comprehensive and adaptive modeling of both nodal and relational dynamics over time. Extensive experiments on diverse real- world dynamic graph datasets highlight the su- periority of our approach in capturing complex structural evolution, surpassing existing methods across multiple metrics.
Static Method
Im. Tu
Tr
Multi
Geometry!
Temporal
Method
Desent Seeretre
E:
Fuciidann Geameity
Correct Prediction
Hyperbolic Geometry
Incorrect Prediction
Ligo Nodes
Figure I: Dynamic graph requires continuous and inde- pendent modeling of snapshots at each timestamp, and the geometric properties of local structures may vary with link evolution. Existing methods typically apply single geometry analysis at static timestamps (0) or across time sequences (3). leading mismatches between geometric heterogene- ity and data, causing modeling distortions. Therefore, A temporal continuous multi geometry analysis method (3) that selects different geometric embeddings for distinct local structures in each snapshot is more universal and effective.
such as hierarchical structures, GNNs in hyperbolic space emerge as the preferred choice (Peng et al., 2021). Nev- ertheless, static GNNs are inherently limited in capturing the evolving nature of relationships in real-world systems, where interactions between entities-suu as in social in- teractions (Liao al., 2018; Huang et al., 2021), scientific collaborations (Yasunaga et al., 2019), and transportation networks (Yu et al., 2017; Ji et al., 2023)-are dynamic and continuously change over time, posing significant chal- lenges for static models (Han et al., 2021). Temporal link prediction aims to forecast the appearance of new links and the potential disappearance of existing links based on cur- rent and past graph states (Divakaran & Mohan, 2020). With the expansion of network scale and increasing complexity of link evolution, accurate temporal link prediction is crucial not only for enhancing the understanding of dynamic net- works but also for improving the performance and decision quality of intelligent systems (Zhang et al., 2023).
1. Introduction
In recent years, static graph neural networks (GNNs) in dif- ferent geometric spaces have achieved remarkable progress in the field of link prediction (Yang et al., 2022; Chen et al., 2023). For graphs with general structural features, GNNs in Euclidean space demonstrate superior performance (Wu et al., 2022), while for graphs with specialized structures,
"School of Artificial Intelligence, Jilin University, China. "Scoool of Computer Science and Technology, Tongji University, China. 'CFAR and IHPC, Agency for Science. Technology and Research (A*STAR) Singapore. *College of Computing and Data Science, NTU, Singapore School of Computer Science and Engi- necring, University of Electronic Science and Technology China. Correspondence to: Xiaofeng Cao <xiaofengeaao @ @eteeelluuu... llu.edd.cc>....
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Conventional GNN methods are designed for static graph

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Analytical Construction on Geometric Architectures
structures and cannot adapt the evolving nature of dy- namic graphs (Zhu et al., 2020; Xu et al., 2022a; Sun al., 2024). When links appear or disappear, static models strug- gle to dynamically adjust embeddings, resulting in limited suitability for dynamic graph tasks. While dynamic GNNs have been proposed to model temporal variations using a single geometric space (Hajiramezanali et al., 2019; Pareja et al., 2020: Bai et al., 2023), they neglect the geometric het- erogeneity across varying local structures (Gu et al., 2018; Bachmann et al., 2020), limiting their ability to capture di- verse characteristics in dynamic graphs (Ravasz & Barabási, 2003; Bronstein et al., 2017). Furthermore, the evolution of links over time alters local geometric properties, a critical aspect overlooked by existing methods. Thus, there a need for approaches that integrate geometric heterogeneity with temporal dynamics, enabling models to dynamically select appropriate geometric embedding and adjust them in response to link evolution (Zhou et al., 2023), thereby cap- turing the temporal evolution of graph data more effectively.
for each past timestamp, and aggregates them obtain the input hidden states for the current timestamp. Finally, link evolution loss function is used to capture fine-grained link dynamics by optimizing distances between node pairs corresponding to newly appeared and disappeared links.
The salient aspects of our contributions are as follows:
Geometric analysis over time series characterizes intri- cate structural evolution, capturing fine-grained topo- logical variations and revealing the temporal transfor- mation of geometric features.
Building upon this more refined consciousness, fine- grained modeling framework for dynamic graph fea- tures is proposed, capturing complex structural varia- tions through state aggregation across different geomet- ric domains. Additionally, a link evolution loss func- tion is designed to precisely characterize link changes.
Demonstrated the effectiveness of the dynamic geomet- ric analysis paradigm through comprehensive evalua- tion across datasets of varying scales and types.
Limitations: Drawing upon the preceding analysis, we have identified two fundamental limitations in the existing link prediction methods: 1) Limited Flexibility in Geometric Representation: Existing approaches often rely on fixed ge- ometric spaces, which may not fully accommodate the com- plex, hierarchical, or non- non-Euclidean local structures present in diverse temporal graph data. 2) Inadequate Consider- ation of Temporal Dynamics: Many current methods fail to effectively capture the intricate temporal dependencies and patterns in evolving links of dynamic graphs, leading to suboptimal predictive performance over time. Below, we present our observations regarding these limitations: 1) Current dynamic models predominantly embed graphs into a single geometric space, which overlooks the geometric diversity inherent in local structures. Hierarchical subgraphs are better suited to hyperbolic space, while non-hierarchical ones align more effectively with Euclidean space (Zhu et al., 2020; Gu al., 2019; Shang et al., 2024). 2) These models mainly focus macro-level changes, often overlooking micro-level link variations across timestamps, which leads to lack of mechanisms to fully capture these nuances.
2. Related Work
Geometric Graph Learning Graph-based models have gained prominence due to their ability to represent rela- tional data effectively (Wu et al., 2020; Cheng et al., 2023). After the introduction of GCN (Kipf & Welling, 2017), many message-passing passing based GNNs (Gao & Ji, 2019; Song et al., 2020; Wan et al., 2021; Huo et al., 2023) were pro- posed, achieving excellent performance across various graph learning tasks. GAT (Veličković et al., 2018) enhances node representation by using attention mechanisms to weigh neighboring nodes differently. GraphSAGE (Hamilton et al., 2017) efficiently handles large-scale graphs by sampling and aggregating neighboring node features. Graph Transformer (Yun et al., 2019) use self-attention to capture complex de- pendencies in graphs. These models are all designed in Euclidean space. To address issues related to hierarchical and complex graph structures, some models, such as HGCN (Chami et al., 2019), H2H-GCN (Dai al., 2021), HAT (Zhang al., 2021), HypFormer (Yang et al., 2024), designed in hyperbolic space, allowing for more effective representation of hierarchical and topological relationships within graphs. In recent years, combining the advantages of both geometries has become a new graph learning paradigm (Zhu et al., 2020; Xu et al., 2022b; Shang et al., 2024).
Schemes: To relieve the limitations mentioned above, we propose a framework conducts dynamic analysis of tempo- ral link prediction by integrating multiple geometric spaces. Specifically, integrate dynamic temporal information into geometric analysis, it continuously extracts k-hop ego- graphs centered on key nodes at each timestamp, adapting the geometric representation based on the evolving local structural patterns over time. It then aligns and optimizes hicrarchical embeddings from both non- non-Euclidean and Eu- clidean spaces, using multi geometric information as graph features for the current timestamp. Additionally, the frame- work maps the representations from previous timestamps to a high-dimensional space, computes attention coefficients
Temporal Link Prediction Temporal Link Prediction fo- cuses forecasting the appearance of future links or the disappearance of existing links in dynamic graphs over time. To efficiently model temporal links, many dynamic graph neural networks with sequential architectures have been proposed (Skarding et al., 2021). GCRN (Seo et al., 2018) combines graph convolution and recurrent layers (Shi
2

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Analytical Construction Geometric Architectures
al., 2015) for enhanced sequence prediction. DySAT (Sankar et al., 2020) captures node representations in evolv- ing graphs using dynamic self-altention. VGRNN (Haji- ramezanali et al., 2019) introduces hierarchical variational modeling to capture topology and attribute changes in dy- namic graphs. EvolveGCN (Pareja et al., 2020) improves dynamic graph learning by evolving GCN parameters with an RNN. Some methods model dynamic graphs in hyper- bolic space, such as HTGN (Yang et al., 2021) and HG- WaveNet (Bai et al., 2023), achieving promising results in capturing hierarchical and complex structures. While as dynamic graphs grow in complexity, the structural het- erogeneity of different local structures poses challenges for magle-geemetrie methods. Currently, there are no ap- proaches that utilize multiple geometries for this task.
H
B
Goodesics
B
The poincaré disk model B is given by projecting each point of hyperboloid model HI onto the hyperplane o.
Figure 2:
gº, which defined as:
B := { E R -clla
<1},
2
(4)
g X2 A
|
3. Preliminaries
where || denotes the Fuclidean norm, g² denotes the Euclidean metric, and the superscript indicates that the vector matrix is in the hyperbolic space modeled using
3.1. Problem Formulation
Given a dynamic graph G. = EV, E T. where T (f1, (2, is discrete timestamp set, V denotes the set of nodes and & S V V denotes the set of edges present at timestamp i. For temporal link prediction task, we aim to predict the states of links a future timestamp + E T. Predicting the appearance of an edge by estimating the probability that edge (u, v) exists at timestamp t
the Poincaré disk model.
Definition (Hyperbolic Operations) Given two points x; E B", the hyperbolic distance between them defined by
2 tanh (Vel Vc
de( y)
(5)
where Be denotes Möbius addition, given by
Dur (t- = Pr((u,,)) v) E Ect (gsll
(1)
x B. = (1 y) + c|ly|l?) x c||2=||
where Gst denotes the sequence (Gr, Ie..... }. Pre- dicting edge disappearance by estimating the probability that an edge (u, v) does not exist at time t+:
1 2c(x 3) +
y
(ar, y) denotes the Euclidean inner product of and y.
Delinition 3 (Tangent Space) The tangent space point x in hyperbolic space, denoted Tu approximates the local structure of the space to first order. This n-dimensional tangent space is isometric to Euclidean space R². The mapping between hyperbolic space and the tangent space facilitated by the exponential and logarithmic maps, defined as follows:
gabsent (tt) Pr((u, & Ect 9gel.
The overall objective is develop a temporal link predic- tion model f that provides probability distribution over the existence and non-existence of all potential edges vw,
f(ggt) + (t Jabbeeut (t+))un))VX
exp = x Be [aanh (ve
(7)
2
Vell
3.2. Hyperbolic Geometry
x D2 y
Hyperbolic geometry describes spaces characterized by con- stant negative curvature, where volume grows exponentially with the increase in spatial dimensions. A n-dimensional hyperbolic space a complete Riemannian manifold with constant negative curvature c, denoted (H)g,gF) where dd is the Riemannian metric. Hyperbolic space can be mod- eled using five isometric models (Beltrami, 1868; Cannon et al., 1997), in this paper, we adopt Poincaré disk model.
log (3) = dx(x, y)
Xe 11 B. all
(8)
where V T2 E B and X has same meaning in Eq. (4). To ensure consistency in error metrics across various directions, as the reference point
the origin in hyperbolic space
4. Methodology
Definition (Poincaré disk model) The Poincaré disk model B a manifold equipped with a Riemannian metric
In this section, we introduce the key components of the framework and its pipeline (Figure 3). We also detail the
3

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Analytical Construction Geometric Architectures
Link Prediction Loss
Optimization
Downstream Tasks
Temporal New Link Prediction
Fermi- Fermi-Dirac Decoder
Fewws Enewt
Link Evolution Loss
Temporal Link Prediction
GC-1
Ba 0
Encoder
Euclidean
Zo 2
Lanh
Geometric Non Nonlineat lineat Act votion Activatio
HoH Ego-Graph Generation
Graph Computation
Hyperbolicity Ego
Representations History Bank
Gc
Update Gate
High
Dimensional
Mapping
F2
exp6( Projection
Logh pojettion
Reset Gate
Temporal Attention
7
TSA
7,
Sro1
Hyperbolic Encoder
8 = 0
GRU Cell
Aggregation
X
fl
Dynamic Graphs
Dynamic Geometric Modeling
Temporal State Aggregator
B: Euclidean Geometrr
B: Hyperbolic Geometry
Nodes Euclidean Space
Nodes Hyperbolic Space
Ego Nodes
Figure 3: Architecture of our method. Given the graph G2 at timestamp f, the Dynamic Geometric Modeling (DGM) module calculates each local structure's 's hyperbolicity 5 via a k-hop ego-graph, selecting and optimizing the appropriate embedding space. The Temporal State Aggregator (TSA) layer then predicts the current hidden state by aggregating past representations using high-dimensional mapping and attention coefficients. Finally, a GRU cell processes the current representation, and link prediction is performed using the Fermi-Dirac decoder, with the model optimized via Link Evolution Loss (LEL) and traditional link prediction loss functions.
optimization process and complexity analysis of the full framework.
For node drs, its hyperbolicity d,, can be computed by:
5(1, U2 U3, ua)
6,, M1 M... EVO
max
(9)
D
4.1. Dynamic Geometric Modeling
where U1, U2, U3. Us denote node quadruplets from ego- subgraph Gr,, D denotes the maximum shortest path length between any two nodes among them. 8(u1, U2 ug uz) = l(u,, + (((g. ug)- -((u,, (g) +( M4). if shortest path lengths between node pairs satisfy E(u1, u2) + l(u3, us) 2 ((1,, uz) {(u2, u4) E(u1, ua) E(u2, uz).
To capture the geometric heterogeneity inherent in differ- ent local structures of dynamic graphs, Dynamic Geometric Modeling (DGM) module is designed to model various local structures of the graph at each timestamp into geometric spaces that best align with their inherent characteristics, thereby uncovering the latent geometric relationships and hierarchical features. This module enhances the representa- tion of complex structures and interactions, improving the ability of overall model to accurately predict and analyze dynamic link evolution over time.
As illustrated in (Adcock et al., 2013), structural features of ego-graphs become significantly apparent when the diameter of the quadruplets exceeds 2. When the diameter increases to 5, the resulting features become similar. Hence, to obtain the most accurate hyperbolicity values for local structures, we set the value to 4.
Definition 4 (k-Hop Ego-Graph of Centroid v) For a given node u E V, its corresponding k-hops ego-subgraph Go comprises all nodes E VV within a distance no greater than kt from u, along with their respective links.
For tree-like hierarchical local structures (ō,,1 RJ 0), we first map them into the hyperbolic space. Assuming that the fea- tures FE =00== of the local structure lie in the tangent space at the origin o of the Poincaré disk model, where the su- perscript E denotes that the features are Euclidean space. The features FO.E 200 are then mapped into the hyperbolic space via exponential map:
Let G. = -Vv,, &,) denote snapshot of the graph at timestamp t. According to the local subgraph preservation property (Huang& & Zitnik, 2020), the influence of neighbors on cen- troid decays exponentially with radius increasing. The local structural characteristics of a node can be captured by its k-hop ego-graph. Thus, we samples k-hop ego-graphs, and computes their hyperbolicities ōz, at
B S 00 = exp
X 8
(10)
1987) measures graph's 's tree-like structure; lower 5 values indi- cate higher hyperholicity, 5 representing tree.
'Gromov's 5-hyperbolicity (Adcock et al., 2013; Gromov,
4

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Analytical Construction Geometric Architectures
The next step involves calculating temporal attention scores, which assess the importance of each historical state. This is achieved through:
softmax (7( W at Zesa + batt) ))n)
(15)
where y denotes a sealar function applied to adjust the scale of the attention scores.
Lost Link
New Link
Timestamps
Figure 4: Link Evolution Loss adjusts the distribution of nodes based edges added and removed between the cur- rent and previous snapshots, dynamically capturing changes in relationships between nodes.
The predicted hidden state at the next timestamps is then derived by aggregating the historical representations weighted by their respective attention scores:
H.
OG Z
(16)
Then, we process the features using HGCN (Chami et al., 2019) to obtain the corresponding hyperbolic embeddings:
The TSA effectively harnesses the temporal dependencies within the data by emphasizing the most influential historical states. This integration of significant past information facili- tates enhanced prediction accuracy for future states while optimizing computational efficiency through advanced ten- sor operations.
FB 80 exp. log. (X) =00) Oe b".
(11)
where 8 denotes weight matrix, b² denotes bias vector.
For the remain local structures (6t, & 0), we use GCN (Kipf & Welling, 2017) to obtain corresponding Euclidean embeddings:
4.3. Link Evolution Loss
To model and analyze fine-grained link dynamics, we pro- pose the Link Evolution Loss (LEL) function, which to optimize the spatial arrangement of node embeddings by considering micro-level changes of edges. Specifically, LEL obtain edge sets, &new and E where &new &, and Enew _ 0. & S E and Ejoan 0. Let F, and F,-1 denote the cross-space embedding matrices at time t and f - 1, respectively.
F S WE,
XE 0 + b².
(12)
We approximate the optimal embedding space distribution nonlinear optimization function using multi-layer percep- tron, and ultimately obtain the final cross-geometric embed- ding for the current timestamp as:
X, = (W/- (log. (F.
F 60, 65,1300 + b¹), (13)
For nodes pairs (u, v) of new edges in G. the goal LEL is to bring the distributions of them closer if they are newly be connected. This is achieved by minimizing:
where 8 denotes geometric non-linear activation with different curvatures.
Lnew,
Σ
max (0, simmfu. fo.)l -T) (17)
4.2. Temporal State Aggregator
(u,v)ESmex
Temporal State Aggregator (TSA) is a more straightfor- ward and efficient network layer compared to existing methods, designed to forecast the hidden state at the sub- sequent timestamp by synthesizing temporal information from historical states.
where sim(.) denotes the cosine similarity function, fu,A and fu, denote embeddings of and v at timestamp 6, denotes penalty term, which is typically set to 0.5.
Conversely, for nodes pairs of lost edges, the objective is to increase distance between distributions nodes pairs that should longer connected:
sequence of representations (Zu, Zi...., 1, where each Z, denotes the represen- tations of dynamic graph timestamp I. TSA initially transforms these representations into higher- dingeso-dimen space using nonlinear mapping function to enhance the distinguishability between states at different timestamps:
Liost.t
Σ (H,
max (0, r [sim(fu.e (18)
The overall loss function combines these components and applies sigmoid function for normalization:
Z 1
W | Dia)::
(14)
where & is a nonlinear activation function such as the hyper- bolic tangent function, and Was and b are the transfor- mation matrix and bias vector, respectively.
CLEL = (Lnew + Lios))
(19)
where denotes the sigmoid function.
5

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Analytical Construction Geometric Architectures
4.4. Overall Framework
dg and dy denote the dimensions of the Euclidean and hyperbolic embeddings, respectively.
Workflow:
With aforementioned key components, we can construct the complete framework, illustrated in Figure 3, the complete algorithmic procedure can be found in Ap- pendix C.I. At timestamp f, it inputs the node features of the graph snapshot G. into the DGM module, yielding the optimized cross-space feature embedding X,. Simultane- ously, the TSA layer computes the current hidden state input H. based on the representations from previous timestamps (Zn,... 1 It then feeds X, and H. into the GRU cell to obtain the node representation Z, at time t. Building upon this, it utilizes a Permi-Dirra decoder for link prediction, expressed by the following formula:
TSA: Transforming and aggregating previous states states involves a complexity of O -e), where T is the number of time steps and de is the dimensionality of the transformed state representations.
LEL: It adjusts node embeddings based on newly added and lost edges. The complexity of computing the link evo- lution loss is O + |Etoar)) [), where [Enewl and [tost] represent the number of new and lost edges, respectively.
The overall complexity depends on the graph size N, num- ber of timestamps T, embedding dimensions, and edge evo- lution. The computational complexity is:
1
PF-D(Z):2)) exp(d(zz. -0-)))) r)/s"
(20)
O N (d)+ de (Enovl + +|Etnat)): (23)
where and are syper-parameters, 2; and 2j denote the representations of the two nodes for which the edge is pre- dicted.
5. Experiments
In this section, we conducted comparative experiments, ef- ficiency analysis, hyperparameter analysis, and ablation studies to validate the effectiveness and advantages of our method. Due to space limitations, only key experimental details and results are presented here, with additional results available in Appendix D.
Optimization:
Our optimization objective is to maximize the prediction accuracy of future unobserved edges, as for- malized in Eq. 3. At timestamp I, the loss function for link prediction is given by:
LLPA = [e]
Σ
- log (Pr-D (1.)))
EE,
5.1. Experimental Setup
(21)
Σ
(1- log (PF- (21,,
Datasets. We conducted experiments on five real-world datasets of varying scales, including the academic coauthor networks DBLP (Hajiramezanali et al., 2019) and HepPh (Leskovec et al., 2005), the Ia-Enren employee email communication network (Rossi & Ahmed, 2015), the social media user communication network LFB (Viswanath et al., 2009), and the roll-call voting network in the United Nations General Assembly from 1946 2020 UNVote (Poursafaei et al., 2022). The statistics for these datasets provided in Table 3. For each dataset except UNVote and Ia Enron, we employed the same training and testing set splits as in prior work (Yang et al., 2021) across multiple snapshots. For UNVote and Ia-Enron datasets, we used a similar split ratio as applied to the other datasets. Additionally, we computed the global Gromov hyperbolicity 8 for each dataset, which indicates that these datasets exhibit certain degree of hyperbolicity.
In addition, the framework computes the link evolution loss fLEEL, which quantifies the microscopic changes in link structure between G. and G.-1. Consequently, the overall loss function at timestamp t is:
L = LIPA + LIF.L.--
(22)
By minimizing this combined loss function, the framework simultaneously optimizes both the accuracy of edge pre- dictions and the ability to capture evolving link patterns, thereby enhancing its overall performance in dynamic link prediction tasks.
4.5. Complexity Analysis
The computational complexity of our method can be sum- marized by three modules as follows:
Scale
Datasets
Links
# Entities
# Train / Test
ô
DGM: It calculating the hyperbolicity 5,,1 of k--hpp ego-
Large
UNVote
035, 742
201
72 /
0.5
HepPh
976, 097
9,746
8 8/3 3
1.0
graphs. The complexity of computing d.,, for each node involves evaluating node quadruplets, which is O N),
Medium
LPR
180, 011
45, 435
33 /3/3
2.0
la-Enron
50, 50,572 372
151
33 /
1.0
where N is the number of nodes. Additionally, mapping features to hyperbolic space and performing Euclidean or hyperbolic embeddings requires O(N (dg + dn)), where
Small
DRT.P
943
315
7
2.0
Table 3: Statisties of datasets.
6

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Analytical Construction Geometric Architectures
Datusets
DBLP
In-Enron
LFB
HepPP
UNVote
Metries
AUC
AP
AUC
AP
AUC
AP
AUC
AP
AUC
AP
Euclideun
EdgeBankk
19.00 1.,00
73.27
0.00
20.41
0,00
78.16
00.0
13.31 4: 0.00
59.65 1 0.00
15.68 =: 0.00
64.33 a: 0.00
22.11 4: 0.00
59.68 1: 0,00
GAR
17.54+00.3
74.21
0.49
85.16
0.78
87.06
= 0.21
63.07
+ 0.93
65.35 +: 0.90
09:44 = 0.56
73.61 + 0.58
56.98 + 0.19
00:40 + 0.10
GRUGCN
84,60 0.92
87.87
0.58
86.12 0.10
87.51
0.554
79.38
1.02
82.77
4 0.75
82.86 = 0.53
85.87 1: 0.23
60.08 1 0.27
60.75 4: 0,03
ExalveGCN
83.88 +.553
87.53
0.22
83.34
+ 1.33
83.64
1.33
76.85 + 0.85
80.87
+ 0.64
76.82 = 1.46
81.18 + 0.89
58.74 + 0.28
58.60
+ 0.15
DySAT
1.70
76.88
0.08
78.16
2:44
80.84
1.77
76.88
0.08
80.39 1: 0.14
81.02 = 0.25
$4.47
1: 0.23
59.43 1: 0.01
60.25
a: 0.05
Hyperbolic
HGCN
89.16++0.16
91.63 + 0:22
72.58 + 0.84
72.37
0.75
86.11 + 0.13
83.74 + 0.15
90.64 = 0.07
88.98 + 0.00
55.45 + 0.14
58.54 + 0.97
HAT
89.29 #..19
90.15 + 0.14
75.84
0.69
75.36
0.65
84.02 0.09
83.03 +: 0.15
90.52 = 0.04
89.53 + 0.04
55.33 + 0.08
58.47 + 0.05
HIGN-6
89.26 4.17
91.91
D.07
85.37
+: 0.79
88. 12 0.70
83.70 + 0.35
83.80 + 0.43
91.13 = 0.14
89.52 + 0.28
61.44 + 0.12
62.16 0.10
HIGN-C
88.56
91.05
85.1
0.32
87
0.41
82.98 0.11
79.95 1.26
0.15
61.27 1: 0.51
0.32
HGWaveeet
89.96 066---0.22 0.22
92.12
D5
N8.9 92. +: 0.71
90.70 0.03
N9.51 + 0.28
N6. 80 + 0.22
92.37 - 0.04
91.48 + 0.05
59.01 + 0.03
60.32 + 0.13
Ouess
94.13 0.46
0.47 92.52
94.17 0.20
94.35 0.50
93.31 0.32
89.58 0.41
97.19
0.03
94.68
0.05
61.68 0.31
62.43 + 0.06
Gain (%)
+4.17
+ -0040
+ +5.25
+ 33665
+3880
+,770
+,882
+3220
0,28
+.227
Table 1: AUC (1) and AP (1) scores (%) of temporal link prediction on real-world dynamic graphs. For all methods, the best results are in bold, the suboptimal results are underlined, results within one standard deviation of best results are in shaded cells. Partial results are from (Bai et al., 2023) under the same experimental setup. These annotations also apply to Table 2.
Datnsets
DBLP
In-Enron
LFB
HepPP
UNVo
Metrics
AUC
AP
AUC
AP
AUC
AP
AUC
AUC
Eacldden
EdgeBBank
0.00
52.83 : 0.00
14.42
0,00
61.69 = 0.00
11.36 + 0.00
56.67 : 0.00
14:24 = 0.00
61.33 : 0.00
9.55 : 0.00
54.51 + 0.00
GAE
69.55
66.11
0:1::000 0.42
61.18
1: 4.63
66.58 = 1.92
73.99 1 0.21
77.56 1: 0.23
73.15 2: 0.18
79.57 1: 0.93
42,10
1.84
$2.78 1: 1.06
GRUGON
75.60
78.55
: 1.05
64.39
1.74
67.91
= 1,28
77.69 1.03
81.07 + 0.77
81.97 = 0.49
84.78 4: 0.22
53.92 4: 3.78
61.19 1.26
EvalveGCN
0.86
77.11
0.44
67.45:22:4
68.94
= 0.90
74.19
1: 0.89
78.33
1: 0.66
74.79 = 1.61
79.04
:: 1.02
28.19
1 8.93
45.42 1: 5.16
DySAT
79.74:24435
83,47
3.01
63.80
2.13
66.38 1.45
74,97
0077:000 0.12
78.34 +: 0.07
79.01 = 0,26
82,53
4 0.25
68.42 +: 0.02
60.08 = 0.03
Hyppebbllcc
HGCN
81.20
83.28
0.23
59.74
1.05
59.33
2.15
81.04 1: 0.14
80.59 1: 0.13
89.64 =: 0.27
87.87 :: 0.11
46.21 4: 3.46
36.95 :: 1.28
HAT
0..29:20.15
$2.58 1: 0.08
61.34 +: 2.86
60.49 0.34
83.05
1 0.10
82,96
1: 0.18
89.63 = 0.05
88.31 + 0.04
47.13 1: 0.95
36.48 0.25
HTGN-8
0.554
84.06 4: 0.41
62.93 à: 1.43
70.93 = 0.76
82.21 4: 0.41
81.70 4: 0.46
90.11 = 0.14
88.18 4: 0.31
50.15 1: 0.52
$7.13 4: 3.60
HTGN-C
81.53
$3.11
0.22
61.73 1.54
70.40 0.08
81.67
0.14
78.36 1: 1.03
90.34 = 0,14
88:45 : 0.14
52,10 1: 0.50
56.00 1: 0.60
HGGeveeee
84.26 d: 0.46
$6.19 :: 0.33
69.19 4: 0.17
74.01 =: 0.03
88.59 4: 0.28
86.00 4: 0.30
91.45 =: 0.05
90.21 +: 0.04
44.11 4: 4.87
52.37 4: 5.68
Ours
93.95 1: 0.43
0.42 92,56
89.68 0.63
90.68 0.64
93.25
0.36
89.54 0.47
97.18 0.03
97.67 0.07
68.50 1: 1.47
71.83 1: 2.40
Gain (%)
1 19.69
+ 6.37
20.49
16.67
1 4.66
3.54
5.73
7.46
0.08
+ 10..4
Table 2: AUC (1) and AP (f) scores (%) of temporal new link prediction on real-world dynamic graphs
Baselines. To comprehensively validate the superiority of our framework, we conducted extensive experiments using a variety of competitive baseline methods. For Euclidean space-based methods, we selected static graph model GAE dy-
fined as predicting links in the next snapshot Gi+1 and the temporal new link prediction task aims to identify new links in G--1 that do not exist in G. All methods employ the Fermi-Dirac function (Eq, 20) to perform these two link prediction tasks on the test sets of various datasets, and use AUC (Area Under the Curve) and AP (Average Precision) as evaluation metrics. To avoid errors due to randomness, each AUC and AP score is the average of five experimental results, with the standard deviation reported.
(Kipf & Welling, 2016) and several namic graph models, including GRUGCN (Seo et al., 2018), EvolveGCN (Pareja et al., 2020), DySAT (Sankar et al., 2020), and EdgeBank (Yu et al., 2023), all of which exhibit strong performance in handling dynamic graph data. For hyperbolic space-based approaches, we chose representa- tive static and dynamic graph models. Static graph models include HGCN (Chami et al., 2019) and HAT (Zhang et al., 2021), which are effective in hyperbolic space graph embed- ding. Dynamic graph models include HTGN-B (Yang et al., 2022), HTGN-L (Yang et al., 2022), and HGWaveNet (Bai et al., 2023), which offer effective solutions for dynamic graphs in hyperbolic space. By evaluating these baselines, we aim to highlight the relative advantages of our method and demonstrate its efficacy in processing dynamic graph data across different geometric spaces.
5.2. Experimental Results
Temporal Link Prediction. The results of temporal link prediction experiments for our method and all baseline methods across different-scale datasets are shown in Ta- ble 1. Our method outperforms all baseline methods in both AUC and AP metrics across all datasets, with particularly notable improvements on Ia ron (AUC +5.25 %, AP +3.65%), LFB (AUC +3.80% AP +2.70%), and HepPh (AUC +4.82% AP +3.20%) These results demonstrate the significant advantage of our method in temporal link prediction, showing stability and reliability across datasets with various scales. Compared to the suboptimal results, our method improves AUC and AP by an average of 3.66% and 2.04%, respectively. While hyperbolic methods outperform
Tasks and Metrics. Our downstream tasks include tem- poral link prediction and temporal new link prediction. Specifically, given observed snapshots of a temporal graph g = (91...., }.), the temporal link prediction task is de-
7

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Analytical Construction Geometric Architectures
GRUGON
HCWayahee
CDCN
CRUCON
CDON
Traning Time (molsposh)
%0'
Inference Time (inespoch))
100
AUC Scores (%)
70"
10"
20'
80
10"
HepPh
LFB
10"-
HesPh
LFB
16
2
Embedding Dimansion
Figure 5: Efficiency comparison on HepPh and LFB.
Figure 7: The influence of embedding dimension on
HepPh.
AUC
New AUG
New
HapPh
LFB
96
Tasks
Temporal Link Prediction
Temporal New Link Prediction
Scores (16)
Metries
AUC
AP
AUC
AP
Full
93.31 +: 0.32
89.58 + 0.41
93.25 + 0.36
89.54
0.47
84
w/o DGM
90.89 +: 0.08
87.82 +: 0.14
90.48 0.10
87.07 0.06
88
(%%)
-2442
-1766
2.77
-2.47
wo TSA
92.10 + 0.02
88.53 + 0.05
92.16 0.05
88.38 0.11
80
86
A
1.21
-.05
-.009
1.16
w/o
91.77 +:
88.05 0.04
88.40 0.02
86.05 0.05
A())
1.54
1,53
4.85
3.49
Figure 6: The influence of k on HepPh and LFB.
Table 4: Ablation study.
Euclidean methods, they still fall short of our method. This because our method effectively captures the geometric heterogeneity between different local structures in graph, thereby reducing representation distortion.
computations only for specific local structures.
Ablation Study. To validate the effectiveness and analyze the importance of key components in method, we de- signed three ablation strategies on HepPh: (i) excluding DGM module (denoted as w/o DGM), using Euclidean em- bedding; (ii) excluding the TSA layer (denoted as w/o TSA), using the output representations from the previous times- tamp as hidden state of next timestamp: (iii) excluding LEL (denoted w/o LEL). The results of these ablation exper- iments conducted on LFB presented in Table 4. The results indicate that the DGM module plays the most critical role, while the link evolution loss has the greatest impact on temporal new link prediction. Additionally, the temporal state aggregator layer also contributes to the improvement of model performance.
The results of tempo- ral new link prediction experiments for all methods across different-seat datasets are shown in Table 2. These re- sults demonstrate that our method outperforms all baseline methods on both AUC and AP metrics across all datasets. Notably, it achieves 20.49% increase in AUC and 16.67% increase in AP on the Ia Ta-Enron dataset, 9.69% increase in AUC on the DBLP dataset, and 10.64% increase in AP on the UNVote dataset. On average, it improves AUC by 8.13% and AP by 8.94% compared to the suboptimal results. These substantial improvements validate the effec- tiveness of our proposed link evolution loss in capturing dynamic changes at fine-grained link level, optimizing the embedding distribution better align with the potential future distribution, and thus enhancing the performance of temporal new link prediction.
Temporal New Link Prediction.
Hyperparameter Analysis. We study the impact of two hyperparameters in method:
1) k of the k-hop ego-graph. We varied the value of k in ego-graphs of the DGM module, ranging from to 6. We conducted temporal link prediction and temporal new link prediction experiments on the HepPh and LFB datasets, respectively, with the results shown in Figure 6. The results indicate that when is less than 4, both AUC and AP values are suboptimal. When k is 4 greater, these metrics ap- proach optimal values and stabilize. This suggests that k value of 4 provides a comprehensive description of the local subgraph structure, consistent with the findings (Adcock et al., 2013). Note that when the value of & is too large, our method generally collapses into using only Euclidean space
Efficiency Comparison. We compared the training and inference times Euclidean SOTA GRUGCN, hyperbolic SOTA HGWaveNet and our method on HepPh and LFB datasets. As illustrated in Figure 5, our method achieves the highest computational efficiency. This is because, like HGWaveNet, which employs an expansion operation for historical information and embeds the graph data into the hyperbolic space at a coarse granularity, our method utilizes a straightforward aggregation layer to integrate his- torical information while leveraging complex hyperbolic
8

---

Analytical Construction Geometric Architectures
Link Prediction Task
New Link Prediction Task
Link Prediction Task
New Link Prediction Task
DBLP
la-Enron
DBLP
la-Enron
DBLP
la-Enron
DBLP
la-Enron
AUC
AP
AUC
AP
AUC
AP
AUC
AP
AUC
AP
AUCC
AP
AUC
AP
|ACC
AP
Clean
93.11
92.32
91.43
89.17
91.23
90.14
89,59
87.16
Euclidean
96.31
93.86
91.77
88.93
96.65
97.11
92.73
89.12
Spherical
96.60
93.49
92.40
88.62
96.30
96.72
92.95
88.97
5% (N)
90.60
89.69
86.26
79.91
88.63
86.90
85.33
79.17
10% (N)
88.49
89,50
83.24
76.87
86.82
86.89
81.69
74.83
Our
97.19
94.68
93.31
89.58
97.18
97.67
93.25
89.54
20% (N)
82.68
84.86
81.78
77.38
79.65
80.49
79.52
73.77
5%
(M)
92.16
89.79
87.91
81.17
89.57
86.26
88.26
81.84
Table 6: Link prediction and new link prediction perfor- mance under different geometric embeddings.
10%
91.91
89.15
87.61
79.59
89.37
84.66
88.57
81.06
20%
(M)
92.25
88.17
89.19
80.43
89.36
83.30
89.06
80.80
Table 5:
Performance under varying levels of noisy edges and missing edges on DBLP and Ia TE-Enron Datasets.
space of constant curvature +1.0, while retaining the Euclidean component unchanged. Specifically, the entire graph is embedded either in Euclidean space (c- = 0.0) or in spherical space, and experiments are re-run on HepPh and LFB. Performance is evaluated in AUC and AP, and results are presented in Table The results indicate that the spherical embedding does not improve upon either the Eu- clidean baseline our combined hyperbolic and Euclidean approach. We attribute this to the fact that spherical geome- try is most effective when the data exhibit strong periodicity or closed-loop structures. Neither HepPh nor LFB pos- sesses such global symmetries, so spherical embeddings fail to capture additional relational nuances. Thus, embedding geometry must align with graph structure.
embeddings (as most graphs are not purely tree-like), result- ing in performance similar to the ablation setting without DGM, while still maintaining decent performance.
2) Embedding dimension. We varied the embedding di- mensions to 2, 4, 8, and 16, and performed temporal link prediction experiments with our method, Fuclidean SOTA GRUGCN, and hyperbolic SOTA HGWaveNet on the HepPh and LFB datasets. The results, illustrated in Fig- ure 7, demonstrate that across different embedding dimen- sions, our method consistently outperforms both Euclidean and hyperbolic SOTA methods. Moreover, even when the embedding dimension is reduced to just 2, our method does not experience a significant drop in accuracy, unlike other methods. It is able to maintain relatively high precision of approximately 95% which demonstrates the strong capabil- ity of our approach in accurate modeling. This suggests that our method can effectively capture geometric information, even in lower-dimensional representation spaces. Optimal performance is achieved when embedding dimension is 16.
6. Conclusion
In this paper, we propose a multi geometric dynamic frame- work to overcome the limitations of single geometric static or dynamic graph models. By combining Euclidean and hy- perbolic geometries, our method effectively captures vary- ing local structures. Additionally, the link evolution loss enhances predictive accuracy by modeling the intricate dy- namics of link appearance and disappearance over time. Ex- periments show that our method significantly outperforms existing methods, including both Euclidean and hyperbolic models, offering a new paradigm for temporal link predic- tion through multi geometric perspective.
Robustness Analysis. To assess the robustness of our frame- work under more challenging conditions, we conducted addi- tional experiments on DBLP and Ta Ta-Enran Enron datasets with two types of graph corruption: (1) Noisy edges (N), where we randomly inserted 5 %. 10 %, and 20 of non-existing edges into the graph; (2) Missing edges (M), where we ran- domly deleted 5 %, 10 %, and 20 % of the original edges. The results, presented in Table 5, show that our method maintains high prediction accuracy across all corruption levels. Even when 20 % of edges are added or removed, the performance only degrades marginally. This demonstrates that our framework exhibits strong resistance to both noise and partial missingness the underlying graph structure, confirming its robustness in imperfect real-world settings.
A Acknowledgments
This work was supported by National Natural Science Foun- dation of China, Grant Number: 62476109, 62206108, the Natural Science Foundation of Jilin Province, Grant Num- ber: 20240101373/0 and Jilin Province Budgetary Capital Construction Fund Plan, Grant Number: 2024C008-5.
Impact Statement
Extension to Spherical Geometry. Hyperbolic space is complete, simply connected Riemannian manifold of con- stant negative curvature. To investigate whether positive curvature might benefit link prediction task, we replaced the hyperbolic component of our method with a spherical
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.

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Analytical Construction Geometric Architectures
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Analytical Construction Geometric Architectures
A. Notations
The descriptions of all the symbols used in the paper are provided in Table 7. In the order of appearance the paper.
Symbols
Descriptions
Symbols
Descriptions
Discrete dynamic graph denoted by snapshots.
x
Embedding matrix in hyperbolic space modeled by poincaré disk model timestamp Embedding matrix in Euclidean space at timestamp t.
Entity set of discrete dynamic graph G.
Link set of discrete dynamic graph G.
x
T
Timestamp of discrete dynamic graph G.
Current timestamp.
F #00
Feature matrix of the tree-like ego-graph centered entity at timestamp t. Feature matrix of the non-tree-like ego-graph cen- tered of entity at timestamp t.
G.
Snapshot of dynamic graph at timestamp t.
&
Link set of snapshot Gr.
F &
++
Future timestamps relative to timestamp f.
Pr(-)
Probability estimation function.
Geometric non-linear activation with different cur- vatures Nonlinear activation function such as hyperbolic tangent function.
Bav
Predicted I label indicating the existence of the link (u, v).
(()
Buv
gabsent
Predicted 0-1 label indicating the non-existence of the link (u, v).
(-)
Scalar function with default setting of [(x)
By
N N-dimensional Poincaré disk model with curvature
Q
Attention vector corresponding to the set of past timestamps.
ght
Riemannian metric of manifold H. Riemannian metric of manifold B. Euclidean norm.
Enew Eioss |
Link set satisfying &new S & and &new O & Link set satisfying ELoss S E and Ensa NE
gB
ge
Absolute value function.
Ta B
Tangent space at a point æ in manifold B Hyperbolicity of entity i's ego-graph timestamp t.
Penalty term, which is typically set to 0.5.
65,5
L(U1, ug)
Shortest path length between i1 and Ug-
Table Notations.
B. Background of Geometry
B.1. Euclidean Geometry
Euclidean geometry is a fundamental branch mathematics that examines the properties and relationships of shapes and spaces with zero curvature. This geometry is based on a flat, infinite plane where parallel lines never converge, and sum of the angles in triangle always equals 180 degrees. It forms the basis for most classical and contemporary mathematical modeling and neural network operations.
In Euclidean space, volume grows polynomially with the growth of radius, and the majority of neural network models are designed to operate within this space. Operations such as convolution, pooling, and activation functions are performed using basic arithmetic operations like addition, subtraction, multiplication, and division.
B.2. Hyperbolic Geometry
Hyperbolic geometry represents Ene--uullden geometric model with constant negative curvature, contrasting sharply with the zero curvature of Euclidean space. The negative curvature in hyperbolic space leads exponential growth of volume with respect to the growth of radius, which contrasts with the polynomial growth in Euclidean space. This characteristic allows hyperbolic models to efficiently handle data with hierarchical or complex structures, which are difficult to represent in Euclidean space.
Hyperbolic space can be modeled using several isomorphic representations, including the Lorentz model, Poincaré disk model, Poincaré half-space model, and Klein model. In our study, we employ the Poincaré disk model due to its advantageous
13

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Analytical Construction Geometric Architectures
Parameters
DBLP
Ia-Enron
LFB
HepPh
UNVote
UNLegis
Layers of Encoder
3
Snapshots of Test
3
3
3
6
Learning Rate
0.0005
0.001
0.005
0.001
0.01
0.01
Weight Decay
0.0000005
0.0000005
0.0000005
0.0000005
0.0000005
0.0
Hidden Layer Dimension
16
16
16
16
16
16
Dropout Rate
0.0
0.0
0.0
0.0
0.1
0.0
Initial Curvature Setting
None
None
None
None
None
None
in Fermi-Diras Decoder
2.0
2.0
2.0
2.0
2.0
in Fermi-Dirac Decoder
1.0
1.0
1.0
1.0
1.0
1.0
Table 8: Parameter settings.
properties for graph-based data. The Poincaré disk model IB" is defined as:
1eth
fa R ||-||| < 1},
where || denotes the Euclidean norm.
In this model, the distance between points x and y is given by:
d(x, v) arcosh (1+2
| yl
x )[ 11
Key operations in hyperbolic geometry include:
Möbius Addition @::
(1 + 2(z, + |1>/13) + (1 1|x
x y
1 2 v) |
Möbius Scalar Multiplication @:
tanh (rartan(()aa) (| ) if E B",
1. 8 x
0,
if x = 0.
Möbius Vector Multiplication M (x)):
M (x) = tanh
[|Maal] ||||||
actann((1=:l))
Mx
||aa||
These operations leverage the hyperbolic space's properties to handle complex relationships and structures, which are not easily manageable in Euclidean space. Our use of the Poincaré disk model aims to capture these intricate structures effectively in our experiments.
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Analytical Construction Geometric Architectures
Tasks
Temporal Link Prediction
Temporal New Link Prediction
Metries
AUC
AP
AUC
AP
EdgeBank
65.78
16.30 0.00
50.04 + 0.00
19.05
Euclidean
GAE
50.62 1:62::0.33
51.81 0.09
34.17 = 1.59
39,97
GRUGCN
53.26 0.15
0.17 56.39
33.08 + 0.25
38.89 0.45
EvolveGCN
$7.55
58.62 =
39.27 = 0.14
41.48 0.38
DySAT
53.95 1.28
56.69 1 0.11
34,24 1: 0.39
39.76 0.17
Hyperbolic
HGCN
51.61
$2.83
0,45
37.36 3.85
42.14 2.75
HAT
53.70 1.24
54.87 1.08
39.31 4.61
43.79 0.39
HTGN-B
52.89 0.50 0.50
52.88 0.37
40.51 = 0.01
43.89
HTGN-2
57.87 + 0.71
56.89
0.38
39.91 + 1.52
42.09 1.22
HG HGWaveNet
55.63 + 0.22
56.99 = 0.51
32.55 = 0.86
38.42 0.25
Ours
80.07 0.47
71.93 0.35
74.56 + 0.41
68.59 + 0.44
Gain
(%
22.2
13.31
24.52
24.7
Table 9: AUC (1) and AP (f) scores (%) of temporal link prediction and temporal new link prediction on USLegi graphs. For all methods, the best results are in bold, the suboptimal results are underlined.
C. Experimental Details
Algorithm Goose-Geomttric Dynamic Link Prediction Framework
Input:
Graph snapshots {9r)T-o where G. (Vi, Et) Parameter: Initial hidden state Hg, number of hops A: for ego-graph
Output: Final graph representation Hr
for each timestamp 0 to T do
2: Dynamic Geometric Modeling:
3: for each node v, V/ do
4: Compute k-hop ego-graph (
5: Calculate hyperbolicity dz,, of (
6:
if',, 2 then
7:
Map features F, into hyperbolic space
8:
Generate hyperbolic embedding F
9:
else
10:
Generate Euclidean embedding FE 1,t end
11:
end for
Aggregate cross- cooss-eeometric embeddings form X,
14:
Temporal State Aggregator:
15:
Map historical representations (Zo, Zl.... Z,-13 } to higher-dimensional space Compute attention scores a; for 0 to
16:
17:
Predict current hidden state A, Σ i=
Z
18:
Identify edge sets Enew and Exss
19:
Compute loss L, based on Enew and Elost
Optimization:
21:
Update model parameters using gradient descent:
22:
0 - 0 Vo(Cum.1 + LIP,))
23: end for
24: return Final graph representation Hy
C.1. Algorithm
To clearly illustrate the workflow of our proposed method, we provide a detailed description of its core steps here. Initially, for each timestamp, we extract the k-hop ego graph for each node from the given graph snapshot and compute hyperbolicity
15

---

Analytical Construction Geometric Architectures
determine the appropriate embedding space. Subsequently, by leveraging cross-geometric embeddings and temporal state aggregation, generate the node representations for the current timestamp. Finally, optimize the link evolution loss function to enhance the accuracy of dynamic link prediction. The detailed pseudocode is provided in Algorithm 1.
GRUGCN
HGWayeNet 10
CDGN
Training Time (ms/epoch)
10
Inference Time (ms/epoch)
10
10
10
10
10
10
10
10
10'
0-
la_ Enron
USLegis
la Bn_Enon
USLegis
Figure 8: Efficiency Comparison Ia-Enron and USLegis.
GRUGCN
HGWayeNet 10"
CDGN
Training Time (ms/epach)
10
Inference Time (msiepoch)
10
10
10
10
10
2
10
10
UNVote
DBLP
UNVote
DBLP
Figure Efficiency Comparison on UNVote and DBLP.
C.2. Data Preprocessing
Partial datasets are from (Yang et al., 2021), and thus the same dataset preprocessing methods are employed. The specific details are as follows:
LFB (Viswanath et al., 2009) social network graph of Facebook Wall posts where each entity is a user and each link the interaction related their wall posts. We take the activates over the last three years in the dataset as 36 snapshots. The FB dataset is associated with a large number of users but very sparse connections.
HepPh (Leskovec al., 2005) is citation network related high energy physics phenomenology, which is collected from the e-print arXiv website. Each entity represents paper, and an link represents one paper citing another. The data covers papers in the period between January 1993 to April 2003 (124 months in total). It is directed graph network, but we learn and predict as it was an undirected graph. According to the real physical meaning, we use three months of data per snapshot and use the last 36 months as the full dataset in our work.
DBLP (Hajiramezanali et al., 2019) is an academic cooperation network, including the academic cooperation of 315 researchers from 2000 2009, Each entity on the graph represents an author, and an link denotes co-authorship relation. We split the dataset by year and obtain 10 snapshots.
The remaining datasets are processed using similar preprocessing methods. The details are as follows:
UNVote (Poursafaei et al., 2022) is a dataset of roll-call votes in the United Nations General Assembly from 1946 to 2020. Each entity represents nation. two nations both voted "yes" for an item, then the link weight between them is incremented by one. We split the dataset by year and obtain 78 snapshots.
16

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Analytical Construction Geometric Architectures
GRUGCN
HGWaveNet
CDGN
64
AUC Scores (%)
62
60
58
56
16
4
Figure 10: The influence of embedding dimension on UNVote.
GRUGCN
HGWaveNet
CDGN
100
90
AUC Scores (%)
80
70
16
8
2
Figure 11: The influence of embedding dimension on LFB.
Ta-Enron (Rossi & Ahmed, 2015) is a dataset of edges that represent emails sent from one employee to another. There are 50572 links, and each of them contains timestamp information. Links refer to 151 unique entity IDs in total. The dataset comprises link data over months. However, since the records for the first and last months are incomplete, they are merged with the second month and the second-to-last month, respectively. Consequently, the dataset is partitioned into 36 snapshots based on the months.
In the appendix, we additionally present the comparative experimental results of our method on new dataset. The details of this dataset are as follows:
USLegis (Poursafaei et al., 2022) a senate co-sponsorship graph which documents social interactions between legislators from the US Senate. The link weights specify the number of times two congress persons have co-sponsored bill in a given congress. The dataset contains 12 months of data, comprising 225 entities and 60,396 links. It is divided into 12 snapshots based on the months, and split into training and sets with 9:3 ratio.
C.3. Environments
The hardware environment consists of an Intel Core 17-13700KF CPU with 16 cores and 24 threads, rumning at 3.40GHz, paired with an NVIDIA GeForee RTX 4070Ti GPU that has 12GB of VRAM and 7680 CUDA cores. The system equipped with 16GB of RAM and operates Windows 11. Programming is performed using Python 3.10, with PyTorch 1.13.1 and torch geometric 2.2.0 for deep learning, CUDA 11 11.7 for GPU acceleration, and package management handled by Anaconda 3.0. For processing large datasets like UNVote and HepPh, high-performance server is utilized, featuring 4 Intel Xeon Gold 5220 CPUs, each with 18 cores and 36 threads, clocked at 2.20GHz. This server also includes NVIDIA Quadro RTX 6000 GPUs, each with 24GB of VRAM and 4608 CUDA cores, and is equipped with 500GB of RAM. The server runs on Ubuntu 18.04.6.
17

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Analytical Construction Geometric Architectures
GRUGCN
HGWaveNet
CDGN
100
90
AUC Soores (%)
80
70
50-
16
2
Figure 12: The influence of embedding dimension on Ia-Enron.
GRUGCN
HGWaveNet
CDGN
100
90
AUC Scores (%)
80
70
60
50
16
8
2
Figure 13: The influence of embedding dimension on DBLP,
C.4. Parameter Settings
Our method primarily depends on the following hyperparameters: hidden layer dimension, number of test set snapshots, learning rate, weight decay, number of encoder layers, dropout rate, initial curvature setting (where None indicates that it is trainable), and the parameters r and for the Fermi-Dirac decoder. The parameter settings used for different datasets are detailed in Table 8.
D. Supplementary Experiments
D.I. Further Performance Comparison
We conducted experiments on new dataset, comparing the performance of various hyperbolic and Euclidean dynamic graph models with our method both temporal link prediction and temporal new link prediction tasks. The results, as shown in Table 9, demonstrate that our method consistently outperforms all baseline methods. Specifically, it achieves an AUC score improvement of 22.2% and an AP score improvement of 13.31 % on the temporal link prediction task, while on the temporal new link prediction task, the AUC score increases by 24.52% and the AP score increases by 24.7%.
D.2. Further Efficiency Comparison
Due to space limitations, only subset of the dataset efficiency comparisons is presented in the main paper. We include the remaining efficiency comparisons for other datasets here. Efficiency comparisons on the Ia-Enron and USLegis datasets are shown in Figure 8, while comparisons on the UNVote and DBLP datasets are illustrated in Figure 9. The results indicate that our method consistently demonstrates the best efficiency on USLeg is datasets.
18

---

Analytical Construction Geometric Architectures
GRUGCN
HGWaveNet
CDGN
100
90
AUC Scores (%)
80
70
50
40-
16
2
Figure 14: The influence of embedding dimension on USLegis.
D.3. Further Hyperparameter Analysis
Due space constraints, the main paper only presents the analysis of the hyperparameter embedding dimension on a subset of datasets. We provide the remaining analyses here. The analysis of embedding dimension the UNVote dataset is shown in Figure 10, on the LFB dataset in Figure 11, on the la-Enron dataset in Figure 12, on the DBLP dataset in Figure 13, and on the USLegis dataset in Figure 14. The results indicate that the embedding dimension has minimal impact on the performance of our mehtod, with the best performance observed at a dimension of 16. Additionally, it outperforms both the hyperbolic SOTA method HGWaveNet and the Euclidean SOTA method GRUGCN across all dimension values.
19


Reviews:
[{"review_id": "jVSJZppHWI", "reviewer": "Reviewer_Tf9y", "summary": "The paper proposes a unified cross-geometric learning framework that integrates both Euclidean and hyperbolic spaces to address the dynamic nature of graph data. The key idea is to analyze the evolving local structures via k‑hop ego-graphs and select the embedding space that best matches the local geometric characteristics. TSA is introduced to fuse historical states, while a novel LEL is used to finely capture the appearance and disappearance of links over time. Extensive experiments on multiple real-world datasets demonstrate improved performance over state-of-the-art methods in both temporal link prediction and new link prediction tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the time and effort in reviewing our paper. We take all comments seriously and try our best to address every raised concerns. Please feel free to ask any follow-up questions.\n\n---\n# No Supplementary Material\nOur appendices include notations, geometric background, experimental details, supplementary experimental results. A zipped code package is available in the Supplementary Material.\n\n---\n# W4\nOur work has several key innovations. Geometrically, we propose a novel cross-geometric dynamic modeling method by analyzing geometric properties from a local graph perspective in snapshots, advancing the theoretical boundaries of cross-geometric dynamic link prediction. Algorithmically, we design a dynamic link prediction framework with a Temporal State Aggregator layer for efficient historical information extraction. Optimizing from a microscopic viewpoint, we designed a novel Link Evolution Loss function. Practically, our framework achieves state-of-the-art results across various types and scales of continuous graph data.\n\n---\n# W5 &  Q3\nSince GRU has inherent memory capabilities, the TSA module primarily serves as an auxiliary enhancement. Its design was iteratively refined through extensive experiments. We analyzed and simplified corresponding components in SOTA methods, finding that an attention-based MLP retained the ability to extract key historical information without performance loss while significantly reducing computational complexity, which guided the final design. The table below shows that TSA offers comparable MRR(%) with other modules but with the least time consumption(ms).\n|||HepPh|||LFB||\n|-|-|-|-|-|-|-|\n||MRR|training|inference|MRR|training|inference|\n|transformer|99.61|60.79|47.26|99.17|19.03|13.62|\n|GRUGCN(counterpart)|98.29|19.66|1.104|98.12|2.273|1.087|\n|HGWaveNet(counterpart)|99.49|1493|11.04|99.32|1021|9.791|\n|Our(TSA)|99.98|17.28|1.047|99.92|1.001|0.999|\n\n---\n# W6\nWe discussed the hyperparameter k (line 430, Figure 6), where k ≥ 4 stabilizes model performance. Figure 7 shows that varying embedding dimensions has minimal impact on the results.\n\nScalability：\n- Task: Our method can be extended to link prediction on continuous-time dynamic graphs (CTDGs) (Please refer to our responses to **Reviewer yHfU's C1**).\n- Framework: Each component of our model can be extended and upgraded. The results of component replacement are shown below:\n||HepPh||LFB||\n|-|-|-|-|-|\n| |AUC|AP|AUC|AP|\n|GRU->LSTM|96.82|94.34|92.45|88.96|\n|GCN->GAT|97.47|94.87|93.02|89.15|\n|TSA->Transformer|97.93|95.23|93.86|89.89|\n|Original|97.19|94.68|93.31|89.58|\n\n---\n# W7\nThe legends for Figures 1 and 3 are in the dashed box at the bottom. We apologize for any confusion and will adjust the legend size and provide further explanations if revised.\n\n---\n# Q1\nIn Figure 6 of the paper, we present the AUC and AP values for different tasks on HepPh and LFB datasets as k varies. The results show that when k ≥ 4, the results stabilize, indicating that the geometric features of the local subgraph can be fully described in most cases when k is greater than or equal to 4, which is consistent with the description in work [1].\n\n---\n# Q2\nWe designed experiments to test the model performance under the two scenarios you suggested. The results demonstrate that our method exhibits strong noise resistance and robustness even when some edges are missing. The experimental design is as follows:\n- Noisy(N): Randomly add 5%, 10%, and 20% of non-existing edges to the graph.\n- Partially Missing(M): Randomly remove 5%, 10%, and 20% of the existing edges.\n||DBLP||Ia-Enron| |DBLP(new)||Ia-Enron(new)||\n|-|-|-|-|-|-|-|-|-|\n||AUC|AP|AUC|AP|AUC|AP|AUC|AP|\n|Original|93.11|92.32|91.43|89.17|91.23|90.14|89.59|87.16|\n|5%N|90.60|89.69|86.26|79.91|88.63|86.90|85.33|79.17|\n|10%N|88.49|89.50|83.24|76.87|86.82|86.89|81.69|74.83|\n|20%N|82.68|84.86|81.78|77.38|79.65|80.49|79.52|73.77|\n|5%M|92.16|89.79|87.91|81.17|89.57|86.26|88.26|81.84|\n|10%M|91.91|89.15|87.61|79.59|89.37|84.66|88.57|81.06|\n|20%M|92.25|88.17|89.19|80.43|89.36|83.30|89.06|80.80|\n\n---\n# Q4\nFigure 5 in paper shows the average epoch training and inference time for the Euclidean SOTA method GRUGCN, the hyperbolic SOTA method HGWaveNet, and our method. Following your suggestion, we measured runtime on the largest dataset in [2][3] Contact (2,426,280 dynamic edges) and a million-scale UNVote dataset (1,035,742 dynamic edges). The results (in ms) indicate that our method computational complexity scales linearly while maintaining the least time overhead.\n||UNVote| |Contact||\n|-|-|-|-|-|\n||training|inference|training|inference|\n|GRUGCN|726.2|0.538|392.4|3.294|\n|HGWaveNet|7490|16.11|3529|57.41|\n|Our|604.4|0.192|280.9|2.129|\n\n---\nReferences:\n\n[1] Tree-like structure in large social and information networks, ICDM 2013\n\n[2] Towards Better Dynamic Graph Learning: NewArchitecture and Unified Library, NeurIPS 2023\n\n[3] Towards Better Evaluation for Dynamic Link Prediction, NeurIPS 2022", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "No", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "The paper is well-situated within the current literature on geometric deep learning and dynamic graph neural networks.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n1. The paper introduces a novel way to integrate multiple geometric spaces for dynamic link prediction.\n2. Extensive experiments and ablation studies provide strong empirical evidence.\n3. The framework is comprehensive, covering both spatial and temporal aspects.\nWeaknesses:\n1. The theoretical contributions are incremental, relying mostly on standard concepts.\n2. Some components (like TSA) seem more heuristic than rigorously justified.\n3. Additional discussion on hyperparameter sensitivity and potential limitations on scalability would be valuable.\n4. Clarity in some figures could be improved(Figures 1 and 3 are cluttered and lack clear legends).", "other_comments_or_suggestions": "Refer to weaknesses.", "questions_for_authors": "1. Can you elaborate on how sensitive the performance is to the choice of k in the k‑hop ego-graph, particularly in graphs with diverse structural properties?\n2. How robust is the proposed framework when the underlying graph data is noisy or partially missing? Have you evaluated the model’s performance under such conditions?\n3. The TSA layer seems to be a key innovation; could you provide further justification or intuition behind its design and whether it generalizes to other temporal prediction tasks?\n4. Could you provide more empirical runtime analysis on significantly larger graphs to validate the computational complexity claims, beyond the datasets used?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YKnzUecSYD", "reviewer": "Reviewer_yHfU", "summary": "This paper introduces a novel framework for dynamic graph learning that integrates both Euclidean and hyperbolic geometric spaces. The authors propose a unified cross-geometric learning framework designed to capture the evolving nature of real-world systems, which are often dynamic. This framework includes a temporal state aggregation mechanism and an evolution-driven optimization objective to model both nodal and relational dynamics over time.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the time and effort in reviewing our paper. We hope that our response can resolve your concerns. Please feel free to ask any follow-up questions.\n\n---\n# W1\nWe fully agree that MRR is an important metric for evaluating model performance in dynamic link prediction tasks. Since the original papers of our baseline methods report AUC and AP, we initially adopted these metrics for direct comparison. Following your suggestion, we computed MRR by ranking each positive example within its candidate set (i.e., counting the number of negative examples with higher scores plus one) and averaging the reciprocal of these ranks, integrating it into our evaluation framework. The table below presents the experimental results using MRR (%), where our method achieves the highest scores across all datasets.\n||DBLP|Ia-Enron |LFB|HepPh|UNVote|\n|-|-|-|-|-|-|\n|GAE|58.93|85.47|97.54     | 97.83     | 5.52     |\n|GRUGCN|60.15| 86.08     | 98.12     | 98.29     | 5.79     |\n|EvolveGCN|60.82| 86.73     | 98.67     | 98.85     | 6.03     |\n|DySAT|61.07| 87.01     | 98.83     | 99.04     | 6.11     |\n|HGCN|60.53| 86.52     | 98.45     | 98.66     | 5.94     |\n|HAT|60.74| 86.81     | 98.53     | 98.74     | 6.02     |\n|HTGNB|61.26| 87.21     | 98.91     | 99.12     | 6.19     |\n|HTGNL|61.52| 87.49     | 99.08     | 99.27     | 6.41     |\n|HGWaveNet| 61.81| 87.77     | 99.32     | 99.49     | 6.63     |\n|**Our** |**62.44**|**88.04**| **99.92** | **99.98** | **6.76** |\n\n---\n# C1\n\nOur method focuses on discrete-time dynamic graphs (DTDGs), which differ from continuous-time dynamic graphs (CTDGs) in data storage and processing. CTDGs have diverse representations \\[1\\], leading to various models. We explore extending our method to CTDG link prediction in two ways:\n- CTDGs are a special case of DTDGs, with fine-grained timestamps approximating a CTDG. For example, in \\[1\\]\\[2\\], the _UCI_ and _Contact_ datasets have 58,911 and 8,065 timestamps, respectively. A sliding window approach segments event streams, adapting CTDGs to a snapshot-based framework. Our method, applied to these datasets with fixed timestamps (e.g., daily/monthly), consistently outperforms baselines in AUC, AP, and MRR.\n\n||| Contact ||| UCI|||Contact(new)|||UCI(new)||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|| AUC| AP | MRR| AUC| AP| MRR| AUC| AP| MRR| AUC| AP| MRR|\n|GAE| 67.45| 64.23|8.76| 69.34| 59.89|2.98| 65.32| 62.12| 9.23| 60.98| 51.87| 3.98     |\n|GRUGCN| 70.12| 67.45     | 9.87| 72.45| 62.98| 3.27| 68.45| 65.56| 10.32| 63.89| 53.32     | 4.76     |\n|EvolveGCN | 71.23| 68.12| 10.23| 73.12| 63.78|3.19| 69.56     | 66.12| 10.78     | 64.56     | 54.12     | 4.12     |\n|DySAT| 71.45| 68.23| 10.45 | 73.78  | 64.12|3.02| 70.12| 66.78| 10.89     | 65.12     | 55.67     | 4.23     |\n|HGCN| 70.32| 66.78| 10.12| 72.67 | 63.34| 3.56| 68.98| 64.87        | 10.56     | 64.23     | 54.76     | 4.98     |\n|HAT| 70.89| 67.12| 10.34| 72.23| 63.89| 3.62| 69.23| 64.23        | 10.67     | 64.98     | 54.23     | 4.10     |\n|HTGNB| 72.12| 67.23| 10.78| 74.23| 64.05| 3.23| 70.56     | 65.32        | 11.12     | 64.87     | 55.20     | 4.45     |\n|HTGNL| 72.78| 68.45| 10.98| 74.89| 64.18| 3.34     | 70.12     | 66.45        | 11.34     | 65.12     | 55.45     | 4.67     |\n|HGWaveNet | 73.12| 68.67| 10.12| 74.12| 64.12| 3.45| 71.05     | 66.78| 11.45     | 65.45| 55.67     | 4.78     |\n|**Our**   | **73.87** | **69.56** | **11.00** | **75.23** | **64.87** | **3.64** | **71.92** | **67.33**    | **11.47** | **66.06** | **56.26** | **5.32** |\n\n- If the CTDG format remains unchanged, the model must incorporate temporal information, such as decay functions or Neural ODEs. This is a key area for improvement.\n\n\n---\n# C2\nYes, another common geometric space is the spherical space, which has a constant positive curvature. We replace the hyperbolic space with the spherical space, and the experimental results on two datasets are shown in the table below.\n|||HepPh|||LFB|||HepPh(new)|||LFB(new)||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||AUC|AP|MRR|AUC|AP|MRR|AUC|AP|MRR|AUC|AP|MRR|\n|Complete Euclidean|96.31|93.86|99.90|91.77|88.93|99.91|96.65|97.11|99.90|92.73|89.12|99.85|\n|Complete Spherical|96.60|93.49|92.85|92.40|88.62|99.85|96.30|96.72|99.85|92.95|88.97|99.89|\n|Our(Hyperbolic+Euclidean)|**97.19**|**94.68**|**99.98**|**93.31**|**89.58**|**99.92**|**97.18**|**97.67**|**99.97**|**93.25**|**89.54**|**99.91**|\n\nThe experimental results show that the spherical space does not provide any gain. This may be because spherical space is suitable for graphs with periodic or closed structures, and if the data lacks periodicity or global symmetry, spherical embeddings may not effectively capture the relationships between nodes.\n\n---\nReferences:\n\n[1] Foundations and modeling of dynamic networks using dynamic graph neural networks A survey\n\n[2] Towards Better Dynamic Graph Learning: NewArchitecture and Unified Library, NeurIPS 2023\n\n[3] Towards Better Evaluation for Dynamic Link Prediction, NeurIPS 2022", "claims_and_evidence": "Yes, the claims are supported by empirical evidence in Section 5.", "methods_and_evaluation_criteria": "The paper proposed a cross-geometric framework for learning on dynamic graphs. The evaluation metrics AUC and AP are standard for papers that treat link prediction as a binary classification. However, for more robust evaluation, the author should consider using the MRR metric which treats the task as a ranking problem and ask the model to rank the positive edge above many negative edges. This is shown to be more challenging in recent work[1][2]. \n\n[1]. You J, Du T, Leskovec J. ROLAND: graph learning framework for dynamic graphs. InProceedings of the 28th ACM SIGKDD conference on knowledge discovery and data mining 2022 Aug 14 (pp. 2358-2366).\n\n[2]. Huang S, Poursafaei F, Danovitch J, Fey M, Hu W, Rossi E, Leskovec J, Bronstein M, Rabusseau G, Rabbany R. Temporal graph benchmark for machine learning on temporal graphs. Advances in Neural Information Processing Systems. 2023 Dec 15;36:2056-73.", "theoretical_claims": "The paper focuses on empirical validation rather than making theoretical claims.", "experimental_designs_or_analyses": "The experiment design and analysis is sound. The authors conduct comparative experiments, efficiency analysis, hyperparameter analysis, and ablation studies to validate the effectiveness and advantages of their method.", "relation_to_broader_scientific_literature": "Yes, using geometric models for temporal graphs is interesting and the authors showed better performance than prior work with similar hyperbolic geometric such as HTGN.", "essential_references_not_discussed": "No missing essential references.", "other_strengths_and_weaknesses": "Strength: \n- the paper is well-written\n- the methodology is clearly presented and supported by empirical evidence \n\nWeakness:\n- MRR metric can be used for evaluation instead", "other_comments_or_suggestions": "Comment:\n- how would you extend this to continuous time dynamic graph where instead of graph snapshots, a stream of links are provided? \n- can other geometries be used instead of hyperbolic?", "questions_for_authors": "None", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Imk2X46JKn", "reviewer": "Reviewer_4etC", "summary": "This paper proposes using multiple geometries to analyze the problem of dynamic graphs. The authors propose to check for hyperbolicity locally and use hyperbolic geometry to update the node embeddings. They propose a framework consisting of two primary modules - Dynamic Geometric Modeling, Temporal State Aggregator. They also propose a loss function to train this model. They perform experiments on known datasets to test their framework.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the time and effort in reviewing our paper. We take all comments seriously and try our best to address every raised concerns. Please feel free to ask any follow-up questions.\n\n---\n# S1\nThe typos on line 652 will be corrected in the next version, and we will carefully proofread the entire text again.\n\n---\n# Q1\n\nThe calculation of local hyperbolicity is not related to the number of nodes but depends on the links to determine the hierarchical properties of the graph. When we compute the hyperbolicity of local subgraphs, we observe the link structure between nodes to decide whether to adopt a hyperbolic embedding. For example, in a graph containing 100 nodes, if the connections between nodes form a grid-like structure, the graph is more suited to Euclidean space. On the other hand, if the connections resemble a full tree, the graph is better suited for hyperbolic space.\n\nLocal subgraphs, defined as k-hop subgraphs centered around a node, vary in size depending on k. We analyzed the hyperbolicity distribution in 100 randomly selected local subgraphs from the HepPh dataset.\n\n|                   | # Nodes \\[0-3000\\] | # Nodes \\(3000-6000\\] | # Nodes \\(6000-9746\\] |\n| ----------------- | :----------: | :-------------: |:-------------: |\n| Hyperbolicity δ=0 | 6          | 10            | 18            |\n| Hyperbolicity δ≠0 | 9          | 31            | 26            |\n\nThe results show that hyperbolicity varies independently of the node count.\n\n---\n# Q2\nReasons for using the tangent space:\n\n- The tangent space serves as a local linear approximation of the hyperbolic space. It enables the use of Euclidean space operators while preserving the hyperbolic characteristics, ensuring that feature vectors remain within the manifold. This approach reduces the embedding loss caused by geometric heterogeneity, thereby enhancing the model's accuracy.\n- Our framework combines Euclidean and hyperbolic spaces, with the tangent space serving as an intermediate space that allows for flexible transformation, facilitating alignment operations within the framework.\n\nAs you pointed out, if the feature vectors do not lie in the tangent space, the exponentiation might require an additional isomorphism operation. However, in the Poincaré model, since the feature vectors naturally reside in the tangent space, the exponential map can be directly applied without the need for an extra isomorphism operation.\n\n---\n# Q3\nWhen the curvature of each layer in the HGCN is fixed to a constant value, its forward propagation process may indeed degrade into the situation you mentioned.\n\nOur framework uses adaptive curvature, so the exponential and logarithmic mappings between layers cannot be canceled. These operations, based on the geometric properties of hyperbolic space, ensure node embeddings stay within the tangent space of the Lorentz model, helping the model capture hierarchical features and reduce modeling losses caused by geometric heterogeneity.\n\nIf operations were conducted in Euclidean space with the exponential mapping applied only at the final layer, the model would reduce to a single-layer hyperbolic network. While simpler, this approach overlooks the unique properties of hyperbolic space, particularly its negative curvature, which is beneficial for modeling hierarchical data. This would likely reduce the model’s ability to capture complex structures.\n\n---\n# Q4\nYes, theoretically, a graph can exhibit positive curvature locally.\n\nWhen considering embedding into hyperbolic space, we rely on the consensus in the community that hyperbolic space is better suited for graph data with hierarchical structures. \nSimilarly, when considering embedding a graph into spherical space, it is crucial to assess whether the graph’s structural characteristics align with those of spherical space. However, there is currently no widely accepted standard in the community to determine if a graph is suitable for embedding in spherical space.\n\nTo validate whether the spherical space is superior, we completely embed the graph in a spherical (curvature=1) or Euclidean (curvature=0) space and re-conducted experiments. The results show that spherical space method did not outperform other method on HepPh and LFB datasets. This may be due to the fact that spherical space typically exhibits periodic or closed structures, and when the graph data lacks periodicity or global symmetry, the spherical embedding may not effectively capture the relationships between nodes.\n\n||| HepPh||| LFB||| HepPh(new) ||| LFB(new)  ||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n| |AUC|AP|MRR|AUC|AP|MRR|AUC|AP|MRR|AUC|AP|MRR|\n|Complete Euclidean|96.31|93.86|99.90|91.77|88.93|99.91|96.65|97.11|99.90|92.73|89.12|99.85|\n|Complete Spherical|96.60|93.49|92.85|92.40|88.62|99.85|96.30|96.72|99.85|92.95|88.97|99.89|\n|Our(Hyperbolic+Euclidean)|**97.19**|**94.68**|**99.98**|**93.31**|**89.58**|**99.92**|**97.18**|**97.67**|**99.97**|**93.25**|**89.54**|**99.91**|", "claims_and_evidence": "The claims made in the paper are sufficiently supported.", "methods_and_evaluation_criteria": "The proposed methods seem to tackle the problem that the authors have outlined in the paper.", "theoretical_claims": "The are no specific theoretical claims as such. The time complexity analysis seems correct.", "experimental_designs_or_analyses": "The authors evaluate their framework on standard datasets known for temporal link prediction.", "relation_to_broader_scientific_literature": "The paper proposes a way to approach dynamic graphs using ideas from hyperbolic geometry. The idea that the local geometric curvature information (flat or negatively curved) should be utilized while updating the embeddings seems novel.", "essential_references_not_discussed": "I don't think so.", "other_strengths_and_weaknesses": "Strengths:\n\nThe paper is well-written and is easy to follow.\n\nI like the fact that the authors have provided a table with notations in the appendix.\n\nWeaknesses:\n\nPlease refer to the questions.", "other_comments_or_suggestions": "There is a small typo on line 652 of the paper (An extra \"s\").", "questions_for_authors": "On an average how many nodes do you find that have a high local hyperbolicity? Because, if I understand correctly, only then do you embed the node features into hyperbolic space and extract the non-Euclidean information?\n\nAny intuition about why the feature vectors of the local structures lie in the tangent space? I understand that the tangent space is isomorphic. However, in order to exponentiate, the vectors should necessarily lie in the tangent space, if not then the exponentiation will need to be composed with the isomorphism.\n\nIn equation 10 and 11, essentially, Euclidean features are not changing. Exponential and logarithmic operations provide a faithful way to move between hyperbolic space and the tangent space. In which case, can this unnecessary confusion can be avoided by removing equation all together and just using the Euclidean features to multiply with the weight matrix and then exponentiating it once and for all to send it to the Poincaré disk?\n\nCan a graph be positively curved locally? In which case you might need to use spherical geometry?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This submission proposes novel methods for analysing dynamic graphs based on _geometrical information_. Essentially, the paper suggests  checking for local hyperbolicity, then mapping features into a hyperbolic space. This is accomplished using the interplay of two modules, viz. a  "Dynamic Geometric Modeling" module, and a "Temporal State Aggregator." The paper demonstrates the utility of this framework on a variety of graph-learning tasks.

After the rebuttal, reviewers were unanimous in their support of the paper, citing in particular the clarity of the writing and comprehensive method description as its strength. I agree with their assessment and am convinced that this submission would make a suitable addition to the conference programme. I trust the authors to implement the changes they promised during the rebuttal and revise their paper accordingly for a camera-ready version.
Decision: Accept (poster)