TMetaNet: Topological Meta-Learning Framework for Dynamic Link Prediction
Authors: Hao Li
Hao Wan
Yuzhou Chen
Dongsheng Ye
Yulia Gel
Hao Jiang
Venue: ICML 2025 poster (2025)
Keywords: Dynamic graphs
Topological Data Analysis
Graph Neural Networks
Meta learning
Primary area: deep_learning->graph_neural_networks
TL;DR: TMetaNet is a topology-enhanced meta-learning framework that leverages Dowker Zigzag Persistence to adapt graph neural networks for robust and accurate link prediction on dynamic graphs.

Abstract: Dynamic graphs evolve continuously, presenting challenges for traditional graph learning due to their changing structures and temporal dependencies. Recent advancements have shown potential in addressing these challenges by developing suitable meta-learning-based dynamic graph neural network models. However, most meta-learning approaches for dynamic graphs rely on fixed weight update parameters, neglecting the essential intrinsic complex high-order topological information of dynamically evolving graphs. We have designed Dowker Zigzag Persistence (DZP), an efficient and stable dynamic graph persistent homology representation method based on Dowker complex and zigzag persistence, to capture the high-order features of dynamic graphs. Armed with the DZP ideas, we propose TMetaNet, a new meta-learning parameter update model based on dynamic topological features. By utilizing the distances between high-order topological features, TMetaNet enables more effective adaptation across snapshots. Experiments on real-world datasets demonstrate TMetaNet's state-of-the-art performance and resilience to graph noise, illustrating its high potential for meta-learning and dynamic graph analysis. Our code is available at https://github.com/Lihaogx/TMetaNet.

TMetaNet: Topological Meta-Learning Framework for Dynamic Link Prediction
Hao Li
Hao Wan
Yuzhou Chen
Dongsheng Ye Yulia Gel Hao Jiang'
Abstract
ited by their training strategies and evaluation approaches, rendering them inadequate for handling the unpredictabil- ity and rapid shifts in graph structure (You et al., 2022), This has led to growing interest in meta-learning, which provides a powerful framework for dynamic graph tasks by enabling models to quickly adapt to new changes with minimal data. Mitaa-earming-baar based approaches have demon- strated state-of-the- of-thheart performance in downstream tasks, such dynamic link prediction (You et al., 2022; Yang et al., 2022; Zhu et al., 2023b).
Dynamic graphs evolve continuously, presenting challenges for traditional graph learning due their changing structures and temporal dependen- cies. Recent advancements have shown poten- tial in addressing these challenges by develop- ing suitable meta-learning-basee based dynamic graph neural network models. However, most meta- learning approaches for dynamic graphs rely on fixed weight update parameters, neglecting the essential intrinsic complex high-order topologi- cal information of dynamically evolving graphs We have designed Dowker Zigzag Persistence (DZP), an efficient and stable dynamic graph per- sistent homology representation method based on Dowker complex and zigzag persistence, to cap- ture the high-order features of dynamic graphs Armed with the DZP ideas, we propose TMetaNet, meta-lcarning parameter update model based on dynamic topological features. By uti- lizing the distances between high-order topolog- ical features, TMetaNet enables more effective adaptation across snapshots. Experiments on real- world datasets demonstrate TMetaNet's state-of- the-art performance and resilience graph noise, illustrating its high potential for meta-learning and dynamic graph analysis. Our code is avail- able at https: //gtt/b// cww /iin//ihhggg////// aNet
The core idea of meta-learning lies in updating the model parameters across different tasks to enable adaptation to new tasks (Gharoun et al., 2024; Vettoruzzo et al., 2024). For a discrete-time dynamic graph GT = {G,, G2,,.. ..... .T) ROLAND (You et al., 2022) introduced a novel training strategy called live update, which treats the task of predict- ing new links from G, to G++1 1 as a separate task. This ap- proach helps avoid the large memory consumption typically associated with such tasks. Building on this, WinGNN (Zhu et al., 2023b) removes explicit time encoding and incorpo- rates temporal information into the graph neural network model through the self-adaptive aggregation of model pa- rameters across windows. Figures I (a) and (b) illustrate the main training processes of ROLAND and WinGNN.
However, dynamic graph snapshots are not merely separate tasks; they form part of a larger, continuously evolving system. In this context, the way model parameters change across tasks plays a crucial role determining the success of meta-learning. The aforementioned methods only account for the impact of time on parameter changes, neglecting the unique evolution of structures in dynamic graphs. Inspired by task-aware meta-learning (Chen & Zhang, 2021; Peng & Pan, 2023), it is crucial to find a representation of dynamic graphs that can accurately measure the differences between tasks and guide the parameter update process.
1. Introduction
The complexity of dynamic graph evolution, characterized by continuous changes in nodes and edges, makes it dif- ficult for existing graph learning algorithms to fully cap- ture the temporal relationships and dynamics within these graphs (Feng et al., 2024). Traditional methods are often lim-
Naturally, our attention turns persistent homology (PH), tool from computational topology used to extract and learn descriptors of the data's "shape" (Carlsson & Gabriels- son, 2020; Pun et al.. 2022). The persistent homology method constructs topological features of a graph's 'ss multi- scale information, capturing both low-order and high-order details. In recent years, PH has been widely combined with graph neural networks (GNNs) and applied to various down-
'Equal contribution 'Wuhan University, Wuhan, China University of California, Riverside, USA 'Hubei University of Automotive Technology, Shiyan, China 4 *Virginia Tech, Blacks- burg, USA. Correspondence to: Hao Jiang <jh @ih@whu.edu ww....du.nn>.
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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TMclaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
Topological Festures
Snapshot &
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Argnegation
Dynamic GNN with Parameters
Update
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Update
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Pt
Pt-1
Pt
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Pc+1
Embeddings
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(a) ROLAND
(b) WinGNN
(c) TMetaNet
Figure 1. The parameter update mechanisms based on meta-learning in ROLAND, WinGNN, and TMetaNet. (a) ROLAND, model parameters between adjacent time slices are updated using fixed meta-lcarinng weights. (b) WinGNN performs parameter updates between adjacent time slices with a fixed learning rate, incorporating window gradient aggregation mechanism to replace explicit time encoding, making the embedding E, invisible to the model Pa+1 (c) TMetaNet learns the learning rate for dynamic GNN model parameters using high-order topologiaal features from adjacent time slices. The dashed line represents the training method that simultaneously adapts ROLAND and WinGNN.
stream tasks such as graph classification, node classification, anomaly detection, and link prediction (Yan et al., 2021;
Dowker complex and zigzag persistence.
Chen et al., a021a; a; Li et al., 2024; Immonen et al., 2023;
2. We derive theoretical stability guarantees of the re- sulting DZP representations and show their utilities to yield a comprehensive description of the topological structure of discrete-time dynamic graphs, enabling a deeper understanding of the graph evolution and its underlying patterns over time.
Kerber & Russold, 2024; Ye al., 2025). The ability of PH to capture high-order information in a graph reflects the inherent structural differences between snapshots and naturally complements meta-learning-basee GNNs (Chen et al., 2022c). Our goal is to bridge the emerging concepts of PH and meta-learning for time-evolving graphs, using topological features to enhance the parameter update mecha- nism of dynamic GNNs and to further boost their expressive power. This brings two key challenges: (i) How to effee- tively generate stable persistent homology features for dynamic graphs? and (ii) How to efficiently utilize persis- tent homology features to guide the parameter updates of dynamic GNN models?
Armed with DZP, introduce the notion of topo- logical meta-lcarning dynamic graphs and develop new mechanism for parameter updates that inte- grates topological feature enhancement. The proposed TMetaNet framework allows the model to more effec- tively and accurately capture the evolving topology of dynamic graphs under broad range of uncertainties.
To address the above challenges, we propose the Dowker Zigzag Persistence (DZP), computationally efficient and stable dynamic graph persistent homology representation method based on Dowker complex and zigzag persistence. DZP effectively captures high-order structural information of dynamic graphs, using only a small subset of nodes, but without sacrificing the accuracy. Based on DZP, we design a new topological meta meta-learning framework TMetaNet, which updates the model parameters by leveraging the high-order features of the tasks as shown in Figure 1 (c). By incorpo- rating DYP into the meta-learning framework, it not only improves the model's ability capture deep structural dif- ferences between graph snapshots but also enhances the adaptability of dynamic GNNs to evolving graph topologies. Our main contributions are summarized as follows:
We illustrate the utility of TMetaNet in application to link prediction in directed and undirected dynamic graphs. Our extensive experiments indicate that TMetaNet outperforms state-of-the-arr benchmarks up to 74.70% and yields up to 31.1 % gains in robustness.
2. Related Work
Meta-learning for Dynamic GNNs has become one of the hottest research topics in recent years due to its potential to adapt the resulting model to new tasks with minimal retrain- ing in a dynamic environment. particular, meta-learning extracts prior knowledge from training tasks to apply to new tasks with limited data (Gharoun al., 2024). In turn, mroell-ggnsste meta-learning (MAML) is a meta-learning framework which is popular due to its versatility in broad range of tasks and scenarios (Finn et al., 2017). In static graphs, it is used with GNNs for few-shot predictions (Zhou et al., 2019; Bose et al., 2019; Chauhan et al., 2020; Yao et al., 2020). For dynamic graphs in spirit of MAMI.,
1. We propose a new Dowker Zigzag Persistence (DZP) approach which offers a computationally efficient and robust representation of the key topological properties of dynamic graphs, by harnessing the strengths of the
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
MetaDyGNN (Yang et al., 2022) extracts multi-level knowl- edge for temporal link prediction, while MetaRT (Zhu et al., 2023a) predicts time-aware knowledge triples. In discrete- time dynamic graphs, ROLAND (You et al., 2022) uses live update training to model tasks as predictions between snapshots, maintaining adaptability with -xxd-weight up- dates. WinGNN (Zhu et al., 2023b) predicts links without explicit time encoders using window gradient aggregation. TMetaNet focuses on the more common discrete-time dy- namic graphs, combining perspectives from meta-learning and graph neural networks to explore meta-learning param- eter update mechanisms enhanced by persistent homology features, bringing new insights to the field.
(Carlsson & Gabrielsson, 2020; Chazal & Michel, 2021;
Hensel al., 2021) (for more details the concept of persistent homology graphs, see Appendix A.)
Leveraging the mathematical the theory of quiver represen- tations, zigzag persistence (ZP) is a generalization of the conventional tools of persistent homology (PH), allowing, in contrast to PH, to consider topologiaal spaces which are con- nected via inclusions going into multiple directions (Carls- son & Silva, 2010; Tausz & Carlsson, 201 1). As such, ZP is particularly well suited to study topological properties of dynamic objects and has recently gained increasing atten- tion in various data analytics tasks involving time-varying processes (Chen et al., 2021b; 2022a; Myers et al., 2023b; Ma et al., 2025). In particular, for a discrete-time dynamic graph GT = {GI,..... Gr}. the zigzag filtration over graph snapshots takes the following form:
Zigzag Persistence is tool from computational topology, allowing us for more effective and reliable analysis of tem- poral evolution processes (Carlsson & Silva, 2010; Tausz & Carlsson, 2011; Carlsson et al., 2019). It has shown promising results in a range of downstream tasks, such as neuroscience (Mata et al., 2015; Chowdhury et al., 2018), swarming phenomena in biology (Kim et al., 2020), cy- bersecurity (Myers et al., 2023a), and power distribution planning (Chen et al., 2023). However, the application of zigzag persistence in deep learning (DL) remains largely unexplored. Some studies have transformed zigzag persis- tence into topological representations directly usable in DL, such as zigzag persistence image, zigzag filtration curves, and zigzag spaghetti (Chen et al., 202 2022b;a; Chen & Gel, 2025), as well as explored combination of zigzag with multi multi-persistence (Coskumuzer et al., 2024). However, zigzag persistence is limited due to its scalability restric- tions and the associated high computational costs (Dey & Hou, 2021: 2024). To further apply zigzag persistence to the representation of dynamic graphs, new approaches based on weaker complexes such as Dowker complex which effec- tively reduce computational complexity (Choi et al., 2024; Li al., 2024), constitute a promising research direction. To achieve higher scalability, in this paper, we propose TMetaNet which bridges the gap between meta-learnigg and persistent homology. Through the efficient and stable dynamic graph topological representation learning based on Dowker Zigzag Persistence (DZP), we provide a new param- eter update mechanism for dynamic graph meta-learning. This provides new insights into the link prediction problem in large-scale dynamic graphs.
G1 + G + G2 + G3 Gx GRUGL (Vi U V 14+1,1B E& U Ea+1).
(1)
(2)
where each morphism + represents the inclusion of one graph into the union of itself with the next graph, while + denotes the reverse inclusion. This zigzag sequence allows for both the addition and potential removal of nodes and edges as the graph evolves. Based on the zigzag sequence of graphs with fixed scale parameter g, we obtain the corresponding zigzag complex sequence:
C(G
C(G2,E)
C(Ga, E)
&
L
C(G C(G2UG3.9)
Armed with this zigzag construction, we can track the evo- lution of various topological features.
Definition 3.1 (Zigzag Persistence Diagram). Given a zigzag filtration F = {G, Gran) and filtration scale parameter &, the k-dimensional Zigzag Persistence Diagram (F) is a multiset of points in the extended plane IP" where:
Dgm = { ((b,, d:) E RR by < d.
(3)
For the definition of dimension k, see A.2. Each point (b,, corresponds to a k-dimensional topological feature that is born in the complex Ci . {) and dies in C(Ga, E) at scale ç. If the feature appears in C(G,, UG {) or dis- appears in C(Ga, UGa, E), the corresponding coordinate is (b. + 1/2) or (d: + +1/1 1/2), respectively.
3. Background
Persistent homology is tool for analyzing multi-scale topo- logical structures in data. In the context of graph theory, persistent homology constructs a sequence of nested topo- logical spaces (such as simplicial complexes) and studies the evolution of their topological features (like connected components, cycles, etc.) as the scale parameter varies.
These birth and death times are represented as points in a Zigzag Persistence Diagram, which visualizes the per- sistence of topological features across different snapshots.
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TMclaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
By converting the zigzag persistence diagram into metric space, such as the zigzag persistence image (Chen et al., 2021b), the topological structure information of dynamic graphs can be applied to downstream machine learning tasks. Despite being a promising tool for studying a broad range of time varying objects, ZP remains substantially underex- plored in practice due to its often prohibitive computational costs.
in ([4 Wi) corresponds a subset of landmark nodes. A k-simplex is included in D(I,, Wa) if there exists least one witness node w E W, that is within distance from every landmark node in ø.
To select the suitable set L of landmark nodes for the Dowker complex, we leverage the s-nets algo- rithm (De Silva & Carlsson, 2004), which ensures compu- tational efficiency without loss of topological information, thereby addressing the key roadblocks inherent for applica- tions of ZP in dynamic graphs.
4. Dowker Zigzag Persistence
Although the inherent nature of zigzag persistence (ZP) appears as a perfect fit to capture the topological characteris- ties of dynamic graph evolution, computational complexity remains the major roadblock on the way of wider adoption of ZP in practice. For example, for 0- and 1-dimensional features on graphs, the currently best available algorithm to compute ZP has complexity of O(m log" (N) + m log((n) m)) and 0m log" (n)) respectively, where m is the length of the filtration and N is the number of nodes used to construct the complex in the graph (Dey & Hou, 2021; 2024).
Definition 4.2. E-Net. An E-net L. S Vi satisfies Ve E Ve, 31 E L, such that d(v, () I e. Additionally, the s-net is maximal, meaning that for any two distinct landmark nodes 4,17 E It, the distance between them exceeds e: W4,4 E Li, 1, + 13- + d(li) E. (See Appendix C. for the pseudocode of the e-nets algorithm.)
As a fundamental concept in computational topology, the e-net ensures that L, is a representative subset of nodes and accurately captures graph topology without unnecessary redundancy (De Silva & Carlsson, 2004). Once the E-net es- tablishes the set I, of landmarks, the remaining nodes in V; are designated as witnesses W, V, \ L, This partition fa- cilitates the subsequent construction of the Dowker complex by establishing bipartite relationship between landmark and witness nodes based on their mutual distances. (For more discussion of the s-net properties, please see Arafat et al. (2019)). Armed with this construction, we now turn to equip zigzag filtration with the Dowker complex to cap- ture the most essential topologiaal information in dynamic graphs.
One intuitive idea to address this fundamental problem is to use somehow only a subset of the available nodes, when computing ZP. However, can we do so, without sacrificing the topological information? Fortunately, the answer to this question is positive if we invoke the notion of a Dowker complex on graphs. Dowker complex belongs to the fam- ily of weaker complexes which also includes, for example, witness complex (Ghrist, 2014; Chazal et al., 2014; Chowd- hury& & Mémoli, 2018), and also may be viewed the witness complex counterpart on graphs (for more discussion on similarities and differences of witness and Dowker com- plexes see (Aksoy et al.. 2023)). Dowker complex has been recently successfully applied to such tasks as adversarial graph learning (Arafat et al., 2025), GNN pre-trainng (Liang et al., 2025), and dynamic link prediction (Li et al., 2024). (Note that there are two forms of the Dowker complexes, and the Dowker complex considered in this paper is not the same as the Dowker complex studied by (Liu et al., 2022).)
Definition 4.3. Dowker Zigzag Persistence (DZP). For a discrete-time dynamic graph sequence Gr = {G, )G with G, = (V,, Ex), based on Eq. 1, the evolution of the graph is captured through the bidirectional sequence of Dowker complexes:
e, - D(G E2 ~- D(G2, +
The ultimate idea is to assess shape of the graph based only on a substantially smaller subset of nodes, called landmarks, while using all other remaining nodes as witeessss which dictate appearances of simplices in the Dowker complex.
Note that the Dowker Zigzag Persistence (DZP) does not require landmarks to remain fixed across different time snapshots. The varying landmarks across snapshots are naturally accommodated by zigzag persistence' interleav- ing mechanism. Through the canonical inclusion maps D(Gr,e, + D(G,UG+41, 6) + D(G++1, 6), topo- logical features persist when their corresponding simplices exist in both adjacent complexes or the intermediate union complex. This ensures consistency in recording topological features despite landmark variations.
Definition 4.1. Dowker Complex. For a snapshot G2 = (V, E.), the Dowker Complex D(G.) = D(L,, Wt) is a simplicial complex constructed from the landmark set L, and the witness set We as follows:
D(G.) to S Le Bw E W. such that Vi E a, d(l, w) < 8),
Furthermore, since s-net cardinality is bounded by n' where d is the diameter of the graph (Choi et al., 2024), the DZP reduces the complexity of com- puting 0- and 1-dimensional features on dynamic graphs
where E represents the maximum allowable distance be- tween landmark nodes and witness nodes. Each simplex a
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
to (m (n') + log(m)) and O( (on)))))) respec- tively, which demonstrates substantial improvement in com- putational costs.
networks. As illustrated in Figure TMetaNet consists of Topological Signature Generator and Topological Learn ing Rate Adaptor. In the Topological Signature Generator, we compute the Dowker Zigzag Persistence diagrams and the corresponding persistence images for discrete-time dy- namic graphs. In the Topological Learning Rate Adaptor, we employ convolutional neural networks to extract topological feature differences between adjacent time snapshots, and use fully connected layers to learn scalar as the learning rate. Finally, we apply this learning rate to the weight updates of the dynamic graph neural network, thereby achieving a lopology-enhanee
We now show that DZP enjoys the important stability guar- antees, ensuring robustness of DZP to uncertainties. To prove the stability of DZP, inspired by Ye et al. (2023), we first introduce the structural metric dg based on interleaved dynamic graphs which defined as follows:
Definition 4.4. Structural Metric of Graph Discrepancy. For two discrete-time dynamic graphs gx and gY, if they satisfy Definition B.7 of e-interleaved dynamic graphs, the structural metric of discrepancy between g* and G is given by:
meta-learning parameter update process which results in a stable oopology-enhancee meta- meta--earning- based dynamic graph neural network.
disv (R1) + disy (R2), dis (R1) + disw (R2),
5.1. Topological Signature Generator
dg(g", (0)),")== = 2 Ra
1 inf max
Topological Signature Generator is used to compute the Dowker Zigzag Persistence diagrams Dgm and the corresponding persistent image ZPIDZP (here we abbrevi- ate them as Dgm and ZPI respectively) for discrete-time dynamic graphs. For snapshot G2, first compute the Dowker complex (G,) under parameters En, and ön.
max C,
The structural metric dg quantifies dynamic graph differ- ences through node coverage discrepancy (disy). weight divergence (disw), and bidirectional co-distortion analy- sis. (For a more detailed discusson of dg (G*,'') see Appendix B.8.)
Dn = Dowker(La,n. We.., 8n); (In Won = E ---eeee((( (Gred(G:-w........ En).
Theorem 4.5. Dowker Zigzag Persistence Stability, For interleaved dynamic graphs Gx,g¹ ,g with structural metric dglG*, G)), ) with their respective Landmark sets L* and CY, witmess sets Wx and W given according to Defini- tion 4.2, their Dowker Zigzag Persistence diagrams satisfy:
Here, En and 6n are parameters for computing the e-net and Dowker complex D (G.), respectively. E-seed is the seed- bedd-based E-net generation algorithm. The input to E-seed is dynamic graph sequence with a fixed window length w. Through the expansion of the seed node set, it generates the e-net corresponding to G,, E-seed ensures the consistency of the g-net expansion, making the landmark node sets of adjacent time snapshots consistent. Algorithm details are provided in Appendix C.I. The construction of the Dowker complex D, (G.) is shown in Definition 4.1.
d (Dgmpzp(G*)) Dg DgmpDp2(" S K K. "d(lgx,9))
where constant K depends only on the homological dimen- sion k:
The proof of the theorem is given in Appendix B. This stabil- ity result ensures that the smaller differences among graphs tend to result in smaller differences among the zigzag persis- tence diagrams associated with DZP. The theorem provides theoretical foundations for stable meta-learning parameter updates based on higher-order graph structures, which are beneficial for more effective and robust dynamic link predic- tion. This unified framework ensures topological stability while enabling noise-resilient meta-learning updates. In particular, our experiments in 6.4 illustrate the robustness of DZP to noise, while the results in 6.3 indicate consistency of the DZP conclusions with respect to varying sizes of landmarks and witnesses on downstream tasks.
After obtaining Dowker complexes for each snapshot, for the snapshot sequence G. = (G.-ws.... G.}. and its corresponding Dowker complex sequence Dr(G.) = (D,, (G, D,, (G:)), we compute the zigzag persis- tence diagram for each Dowker complex sequence
Dgmm. (Gt) = (D,
where E denotes the function that computes zigzag persis- tence. By constructing the sequence as shown in Eq. 1, we compute the k-dimensional zigzag persistence diagram. Each persistence diagram is multiset of wwo-dimensional points (zt, y), representing the birth and death times of k- dimensional topological features. For each persistence dia- gram Dgm,, (9.), we compute its ZPI Zn, where Zn,, represents the k--dimensional ZPI under parameters En and 8n- See Appendix C.2 for more calculation details.
5. Topological Meta-Learning on Dynamic Graphs
In this section, we introduce the proposed TMetaNet, de- signed to enhance link prediction in dynamic graph neural
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TMctaNet: Topological Meta-Learning Framework for Dynamic Link Prediction
Topological Features Generator
Landwark Witness nodes
Window size
62
D(G,)
(G
G U 6
D(G,) u D(G.)
Snapshots (m Gm]
Dowker Zigzag Persistent Homology
Images ZP!
Topological Learning Rate Adaptor
ZPh-1
ZP4
CNN Layer
G--1
2P4-1
Learning Rate
ZPI
Ge
Update
Dynumic GNN with Parameters Pr-1
Dynamic GNN with Parameters Pt
Figure The overview of TMetaNet.
5.2. Topological Learning Rate Adaptor
topology-enhanced meta-learning parameter update process.
For adjacent snapshots, e.g., G; and Ge+1, the Topologi- cal Signature Generator computes their corresponding ZPI sequences Zn.+(Si (G.) and (n+x(G++1). Subsequently, the Topological Learning Rate Adaptor extracts the difference between adjacent ZPIs through convolutional neural net- work, and uses this as input to learn a scalar through fully connected layers, which serves as the learning rate for the dynamic GNN model. For adjacent ZPI sequences Zn, (G.) and & & (1)+1), we then compute and obtaín their differ- ence through convolutional neural network (CNN) and apply it as input to learn scalar through fully connected layers, which serves the learning rate for the dynamic GNN model. The specific formulas are as follows:
W+++1 w; 17 V (ww:),
(4)
where w, represents the weights of the dynamic graph neu- ral network at time f, r the scalar learned through the topological learning rate adaptor, and Vw.((w; is the gra- dient the loss function L with respect weights wc.
5.3. Training and Evaluation
For dynamic graph representation learning and use cases, ROLAND and WinGNN propose two different training strategies. ROLAND's live update training strategy divides each snapshot into training, validation, and test sets, and evaluates the model on each time slice. WinGNN, on the other hand, arranges snapshots in chronological order, with the first 70% as the training set and the last 30% as the test set. From meta-learning perspective, ROLAND's training, validation, and test tasks belong to the same distribution, while WinGNN's training and tasks belong different distributions. ROLAND focuses more model evaluation across the entire time scale, while WinGNN emphasizes pre- diction for a future time period. Appendix C.3 provides an illustration of different task splitting strategies. Compared to ROLAND and WinGNN, TMetaNet focuses on model parameter updates between adjacent time snapshots and
DeZr, Zn,K(G(41) Zn.x(Ge)) FC(o
where AZn, represents the difference between adjacent ZPI sequences, CNN represents the convolutional neural network layer (in TMetaNet, use simple CNN network with residual connections), is the non-linear activation function, FC represents the fully connected layer, and r represents the learned scalar learning rate.
After obtaining the learning rate, we apply it to the weight updates of the dynamic graph neural network, resulting
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TMclaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
can be applied to both training and evaluation approaches mentioned above, well as their corresponding metrics.
more dynamics, where topological signals are stronger and better leveraged by TMetaNet, while more homogeneous cases such as Reddit-Body under the ROLAND setting, the gains are less significant. Other metrics are presented in Appendix D.5.
6. Experiments
Datasets and Baselines We conduct experiments on six
On ALPHA data, compared to VR complex-based zigzag persistence, our method reduces the computational overhead by 46% on average per snapshot during complex construc- tion. When dealing with extremely large-scale graphs, we can sample from snapshots or remove nodes according to degree centrality from low to high to obtain subgraphs that preserve global higher-order features, and then calculate the learning rates.
public datasets, which are widely used benchmarks for eval- uating the performance of dynamic link prediction, i.e. (1) Bitcoin-OTC (OTC) and Bitcoin-Alpha (Alpha) are trust networks from transactions different Bitcoin platforms (Kumar et al., 201 8b; 2016). (2) Reddit-Body (Body) and Reddit- (Title) datasets are from the REDDIT platform, representing hyperlink networks in post titles and bodies, respectively (Kumar et al., 201 (3) UCI-Message (UCI) consists of private messages between users (Panzarasa et al., 2009). (4) ETH-Yocoin derived from the Yocoin transac- tion network on Ethereum blocks (Li al., 2020). Detailed dataset information can be found in the Appendix D.I. We compare TMetaNet with the following baseline methods: (1)
6.2. Ablation Studies
In this section, we conduct ablation experiments on the Bitcoin-Alpha and UCI-Message datasets to verify the ef- fectiveness of the model parameter update process guided by topological features in TMetaNet. We use three different parameter update settings. (1) Topo: Referring to the pa- rameter update process based on DZP images. (2) Random: Replacing DZP images with a random image of the same dimensions as the original image. (3) Dist: Replacing each DZP image with an image having the same statistical char- acteristics as the original image. The experimental results are shown in the Table 2.
GCN-L and GCN-G (Manessi et al., 2020). (2) EvolveGON (Pareja et al., 2020). (3) WinGNN (Zhu et al., 2023b). (4)
ROLAND (You et al., 2022). (5) DeGNN (Zheng et al.,
2023). Details of these methods are provided in the Ap-
pendix D.2. The experimental settings can be found in Appendix D.3.
6.1. Link Prediction Performance
In this section, we present the performance of TMetaNet on dynamic link prediction tasks under different settings. The live update setting refers to the training strategy used in ROLAND, while the WinGNN setting refers to the train- ing strategy used in WinGNN. Evaluation metrics include accuracy (ACC) and mean reciprocal rank (MRR), AUC, Recall@1 1, Recall@3, and @ecall@ 10. Specific training settings are also provided in the Appendix D.3. For the analysis of running time, please refer to Appendix D.4. The results for accuracy and MRR metrics are shown in Table 1. The proposed TMetaNet demonstrates consistent superior- ity across diverse experimental settings and datasets. Our topological meta-learning framework achieves statistically significant improvements (p < 0.05) in different settings. On average, TeMetNeet achieves 3.29% ACC improvement and 5.52% MRR improvement in the ROLAND setting, while showing even more pronounced performance in the WinGNN setting with 3.27% and 24.08% improvements respectively. This enhanced performance in the WinGNN setting may be attributed to the model's task of predicting future links, where TMetaNet's dynamic learning rate adap- tation mechanism demonstrates better adaptability to such prediction tasks. These results collectively validate that topological persistence features provide effective structural signatures for meta-learning in dynamic graph scenarios. The performance gap is especially evident in datasets with
The experimental results demonstrate that the Topo setting, using the zig7ag persistence diagram, consistently outper- forms both Random and Dist across all metrics. Specifically, Topo achieves higher accuracy and MRR, showing the clear advantage of leveraging topological features over random replacements. Furthermore, while Topo outperforms Dist in both accuracy and MRR, the improvement is more pro- nounced when compared the Random setting. This indi- cates that vectors following the same distribution also bring some performance improvement.
6.3. Sensitivity Analysis
In this section, conduct experiments using the UCI- Message dataset to analyze the sensitivity of the DZP pa- rameters. The specific parameters include the combination of parameters E and which are used to extract the Dowker complex when constructing higher-order features, and the window length w which is used to construct the zigzag persistence.
As shown in Figure 3, each line represents parameter com- bination [(s, d)] with window length on the horizontal axis. In the ROLAND setting, performance drops signifi- cantly as the window size increases. The (2, 2) configura- tion achieves the highest MRR when w 5, but shows a sharp decline when using the full-window setup, suggesting
7

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TMclaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
Table Dynamic link prediction performance of different models under different experimental settings. The hest results are highlighted in bold, and the second-best results are underlined. Each experiment is repeated five times, and the mean and standard deviation are ompitig %). indicates statistical significance (p-value 0.05) between TMetaNet and the second seenddbesttte result).
LIVE UPDATE SETTING (ROLAND)
DATASET
METRIC|
GCN-L
GCN-G
EGC-O
EGC-H
ROLAND
DEGNN
TMETANET
IMPR.
ACC
73.81 0.72
66.90 1.12 71.34 1.89
70.86 1.78 83.18 1 2.77
76.7 11.33
$86.84 1 1.02
8.14% 11.8%
ALPHA
MRR
9.87 = 0,45 9.00 1.32 5.27 = 0.47 1.87 + 1.42 15.23 1 0.47
12.5 1 1.03
17.68 1 0.55
ACC
77.66 1.11 70.98
2.98 72.33 1.09
66.58 1.76
84.77
4: 1.06
76.9
1.33
*85.89 +: 1.22
1.32%
OTC
MRR
17.134 0.72 16.98 + 1.32 10.27 + 0.47 6.87 + 1.42 17.59 + 0.75 15.0 + 1.21
18.06 + 1.22
2.67%
ACC
189.19 1.76 82.19 1.76 76.17 1.26 72.17 0.97 91.63 + 0.09
89.7
3.61 3.61
89.59 + 1.17
BODY
MRR
(33.19 0.76 34.99 + 0.76 10.27 + 0.47 6.87 + 1.42 36.75 + 0.42 26.7 3.50
34.93 + 1.07
ACC
889.66 1.76 90.39 + 0.59 80.48 = 0.49 82.98
0.65 93.58 = 0.15
*93.96 + 0.02
0.41% %
TITLE
MRR
(33.52 0.76 30.25 + 1.76 22.27 + 0.47 18.87 + 1.42 40.25 + 1.15 40.1 + 4.82
42.72 + 1.01
6.14%
ACC
73.52 - 1.02 74.98 1: 2.22 76.12 75.12
1.76 80.13 + 1.14
76.4
1.12
#80.88 1 0.08
0.94%
UCT
MRR
9.38 = 0.09 9.121 0.37 7.86 = 1.02 7.65 1 0.46 10.39 1 1.40
9.2 1 4.32
$10.99 1 0.92
5.77%
ACC
78.05
1.76 78.12 1 1.76 72.12 1 2.76 77.02
1.48 77.01
1: 2.58
62.6
1 1.73
#85.10 1 1.46
8.93%
ETH
MRR
30.98 1 0.62 31.12 1 0.60 32.12 1 0.73 27.12 1 0.85 35.68 1 1.30
33.0 1 2.62
38.08 1 1.57
6.73%
WINGNN SETTING
DATASET
METRIC|
GCN-L
GCN-G
EGC-O
EGC-H
WINGNN
DEGNN
TMETANET
IMPR.
ACC
59.722 1.98
57.12 + 1.78 58.91 + 1.98 53.12 0.79 83.15 + 0.51 81,48 4 2.87
$8.992 + 1.84
8.14%
ALPHA
MRR
5.12 0.79 4.12 : 0.79
6.12 = 0.79 3.12 0.79
34.59 = 3.36 32.36 + 0.90
#38.93 = 3.06
12.55%
ACC
51..43 1.22
50.78+ 2.89
50.12 1.57 50.12 + 0.97 85.70 + 1.25 81.87 0.08
#90.43 + 1.17 0.08
5.52%
OTC
MRR
11.43 1.25
10.78 1.34
10.12 1.25 12.12 + 0.98 37.92 = 2.16 29.85 + 0.59
*39.98 + 2.16
5.43%
ACC
70.42+
69.78+
1.32
77.34 3.76 74.12 97.43 + 1.40
OOM
*98.26 + 1.29
0.85%
BODY
MRR
3.56 1.32 1.32
4.76+ 2.01
5.43 = 2.32
5.74 2.32 16.56 + 4.54
0OM
28.93 + 2.06
74.70%
ACC
70.3440.56
72.01
77.47+1.27
83.45+1.87
98.38 + 0.09
OOM
*99.63 + 0.07
1.27%
TITLE
MRR
2.56
2.76# 2.01
4.23 = 2.32
3.88 188:33 3.32 30.57 = 5.09
0OM
34.96 = 2.06
14.36%
ACC
50..5 1 1.24 50.82 1 3.01
55.43 1 1.98
56.12 1.98
1.98 85.05 1 2.69 75.11 + 1.02
$86.37 37.37 1 5.63
1.55%
UCI
MRR
7 6.82 + 3.01
8.12 10.43
1.98 20.94 = 3.78 25.31 1 + 1.02
#25.31 + 1.02
20.87%
ACC
66098 + 1.37 61.78 + 2.97 66.43 +: 5.37 67.12 4: 4.09 95.62 + 1.92
OOM OOM
97.83 1 1.53
2.31 %
ETH
MRR
9.98 = 3.32 11.78 + 1.92 16.43 + 3.91 17.12 + 4.07 66.97 = 2.12
78.07 = 1.09
16.57%
that longer temporal aggregation dilutes informative signals and introduces noise. the WinGNN setting, the perfor- mance is generally more stable and less sensitive window size. The (1, 1) configuration again achieves the highest peak MRR at w 10, while the (2, configuration per- forms comparably well with smaller variance. Interestingly, even with large windows, most configurations maintain ac- ceptable MRR values, indicating the WinGNN setting is more tolerant to extended temporal aggregation.
2.2
12-
MRR
10
5
20
Pull windows
Window Size
0.26
MRR
0.24
These findings highlight the importance of maintaining fine- grained temporal resolution and prioritizing local structural relationships when extracting topological features in dy- namic graph scenarios.
0.20
windows
Window Size
6.4. Noise Robustness
Figure Impact of hyperparameters of zigzag persistence. The top figure shows the results under the ROLAND setting. while the bottom figure corresponds to WinGNN setting.
We also conduct noise robustness experiments under differ- ent settings on the Reddit- Title dataset by using two types
8

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TMclaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
7. Conclusion
Table Ablation studies the Alpha and UCT datasets.
SETTINGS METRIC
ALPHA
UCI
We have presented Dowker Zigzag Persistence (DZP), which systematically captures higher-order structural fea- tures in evolving graphs. To bridge topological analysis with deep learning, we have developed TMetaNet based on DZP, a novel lopology-enhance
LIVE UPDATE SETTING (ROLAND)
ACC
*86.84 1.02
*80888 + 0.08
TOPO
MRR
*17.68 + 0.55
*10.99 = 0.92
meta-leaming framework for dynamic graph neural networks that integrates persistent homology with meta-learning. Comprehensive experiments on six real-world datasets have demonstrated TMetaNet's superior performance and robustness. Our techniques pro- vide a new reliable and computationally efficient pathway for extracting higher-order topologiaal features from dy- namic graphs, bringing new insights into adaptive parameter updates for meta-learning.
ACC
83.07 + 2.44
80.04 + 1.26
RANDOM
MRR
15.85 0.63
10.32 + 0.36
ACC
84.24 + 1.42
79.96 + 0.94
DIST
MRR
15.55 + 0.72
10.88 = 0.87
WINGNN SETTING
ACC
*89.92
1.84
#86.37 = 5.63
TOPO
MRR
#38.93 3.06
*25.31 = 1.02
ACC
82.14 + 1.76
83.83 + 4.13
Acknowledgements
RANDOM
MRR
33.98 + 3.12
21.33 +: 4.92
This work was supported by the National Key R&D Pro- gram of China under Grant Nos. 2024YFE01 15900 and 2024YFB2908001. Chen and Gel have received no re- search support from any non-US-based entity. Chen and Gel are grateful for the travel support from the Institute of the Mathematical Sciences (IMS) of the National University of Singapore (NUS) to attend IMS-NTU joint workshop on Biomolecular Topology: Modelling and Data Analysis which has sparked some of the presented ideas.
ACC
88.63 + 1.23
84.81 + 2.31
DIST
MRR
38.02 = 2.92
24.20 + 2.72
of noise injection methods, i.e., evasion attack, which con- taminates only the test set, and poisoning attack, which contaminates both the training and test sets. Both methods involve selecting certain proportion of nodes to flip edges (deleting existing edges and adding non-existing edges) with contamination ratios of 5%, 10%, 20%. and 30% respec- tively. From Figure 4, we can see that T'MetaNet exhibits better stability than WinGNN under both poisoning and eva- sion attacks, demonstrating that Dowker Zigzag Persistence has good robustness when facing noise.
Impact Statement
We do not anticipate any negative impact from our topolog- ical meta-learning framework. Conversely, we posit that integrating persistent homology with adaptive meta-learning mechanisms establishes new foundations for reliable, ex- plainable, and robust analysis of evolving network systems. Specifically, Dowker Zigzag Persistence enables principled extraction of multiscale topological signatures that capture essential structural dynamics while filtering transient noise- a critical capability for high-stakes applications like finan- cial fraud detection and critical infrastructure monitoring. The meta-learning adaptation mechanism further enhances operational safety by dynamically adjusting model sensitiv- ity to topological shifts, preventing overreaction to spurious patterns.
30-
10.2%
'MMataNe ROLAND
3.7%
9.2%%
295
32.
4.2%
1.8%
0.5
0.30
0,20
0.30
P.5
D.10
2.20
p.30
3.24
4.7%
0.20-
51.5%
0.18
0.18
Figure 4. Robustness analysis of TMetNNet under different settings, where e_10 represents 10% edge evasion attack. The top figure shows the results under the ROLAND setting, while the bottom figure corresponds to WinGNN setting.
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TMclaNet:
Topological Meta-Lcarning Framework for Dynamic Link Prediction
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
A. Persistent Homology on Graphs
Specifically, given a graph G EV, the process of computing persistent homology can be described in the following two steps. 1) Construct the complex. Based on a filtration function f(G), we construct a sequence of nested subgraphs
G S G -... S Gh, and further construct the corresponding sequence of complexes Co S C₁ S .... C C(G²). 2) Compute the homology groups. Based the sequence of complexes, we compute their corresponding homology groups HC,C,,, p,p 2 O, where denotes the dimension, and 3 E Hp(C,) represents the p-dimensional topological features. As the sequence of complexes evolves, the homology groups reflect the evolution of topological features. Finally, it is represented in the form of a persistence diagram, PD(C) = {bb,, da) 3 E Hy(C)), where b3, da represent the birth and death times of the topologiaal features, respectively.
common complex on graphs is the Vietoris-Rips Complex, defined as follows:
Definition A.1 (Vietoris-Rips Complex on Graphs), Given graph G = (V, with graph distance dG, the Vietoris-Ripp complex at scale 8 2 0 the simplicial complex defined by:
VR&(G) = fo s v Vu,eE a, du(u, 8) s 6}
(5)
where de (u, v) denotes the shortest path distance between nodes and v. A k-simplex is included if and only if the diameter of its (k + 1) vertices (maximum pairwise distance) is at most
When faced with dynamic graphs, the Vietoris-Rips complex needs to be computed for all nodes at each time step, which leads to high computational complexity.
Definition A.2 (Homological Dimension). The homological dimension k of a simplicial complex C' is defined as:
k = dim HC) sup(n EN (C) + 0}
(6)
where (C) denotes the n-th homology group of C.
B. Proof of Dowker Zigzag Persistence Stability
This section provides a detailed proof of the stability of Dowker Zigzag Persistence (DZP) in discrete-time dynamic graphs. The proof is based on the Theorem in (Ye al., 2023), showing that DZP remains stable under controlled perturbations of the underlying graph structure. Below we restate essential definitions and theorems, then show step-by-step arguments.
B.1. Foundational Definitions
Definition B.1 (Discrete- Time Dynamic Graph). A discrete-time dynamic graph G = {Go; GI,.., GT), where G. (V,, Et), consists of:
Time-indexed node sets [Vo)?= where V22 C V, V is the vertex set of G
Weight matrices {we V x V/
R+1T 1
satisfying temporal consistency:
E (1....,T-1}, V. #
This definition captures evolving graph structures with temporal connectivity through overlapping node sets across discrete time steps. real-world networks, this is common phenomenon.
Definition B.2 (Tripod). tripod R between G and gY consists of:
Intermediate set W with surjections:
X
m1 W
12 Y
Temporal consistency: AL E (1...., T).
(V. = オミ (V,
(8)
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
Tripods establish temporal correspondence through shared intermediate nodes while maintaining alignment of node existence across time steps.
Lemma B.3 (Tripod Composition). Given tripods gx R1 g" and gr R2, G there exists composite tripod R = R1 such that:
gx Rg gz
(9)
Proof. Construct the fiber product:
W f(w1, W2) E W1 W2 m2 ( (w1) ( (w2)
(w2))
with projections:
R (w1, W2) = (1) (w))
ft (w1, = 22 (w2)
(2)
Temporal consistency follows from:
(xf))-((²>) = (m (V??)
This lemma establishes the compositionality of tripods, which is crucial for building complex temporal graph alignments. The composite tripod R is constructed through fiber product that ensures the intermediate graph GY serves bridge between G. and g². The temporal consistency is preserved through the alignment of nodes in the intermediate graph gY, where (T22 (
and (xi (=()))-'(((()) establish the correspondence between the temporal nodes of G and G This composition operation allows to chain multiple tripods together, enabling the alignment of temporal graphs across multiple domains while maintaining temporal consistency.
Definition B.4 (Bottleneck Distance for Dowker Zigzag Persistence). Let Dgm pzp (GX) and Dgm (g denote the Dowker Zigzag persistence diagrams (or barcodes) arising from two dynamic graphs G (G ]t=1 and G [G) Y
1
We define the bottleneck distance d3 (Dgmpzr((" ), Dgm (2 as follows:
dp (Dgg Dgmpzr( = inf sup || (x=))))>|b> J DZP((E))
(10)
where we view each zigzag persistence diagram as a multiset of points (or equivalently, interval endpoints) in a suitable metric space (often the extended plane), and μ ranges over all bijections (or partial matchings) between the points of D and (ZP J.The distance || || typically denotes the 8 metric on the plane when diagrams are depicted as (birth, death) points, but other equivalent definitions exist. In simpler terms, this quantity measures the smallest value p such that can pair off the points of the two diagrams (with unpaired points matched to the diagonal if needed) that each pair lies within distance p.
B.2. Discrete e-Smoothing
Definition B.5 (Diserete e-Smoothing). For E N+ and E (1,..., T}, define:
min(T,
S.V. :=
U
V
max (1)
S,i1(x, = 2e +
Σ
w (x,,)'))) (awrev))
r=
where 1 is the indicator function for temporal existence.
The smoothing operator S. aggregates node sets within temporal windows and averages edge weights with existence validation, ensuring meaningful comparisons across snapshots.
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
Lemma B.6 (Smoothing Bijectivity). For I ( (max smoothing radius), there exists identity tripod R2 satisfying:
gX R1, S.gx
S.g* R4 g
Proof. Take W X with identity maps = idx. Temporal consistency holds because:
(1) Vt(() and wx S S.w
Bijectivity is preserved through the indicator function in Definition B.5.
B.3. Discrete c-Interleaving
Definition B.7 (Discrete e-Interleaved Dynamic Graphs). Two discrete-time dynamic graphs G = (VEX we Ji=1 and g = =(V,, wf J are e-interleaved if there exist:
Tripods R1 R2 satisfying Definition B.2
Smoothing operator S, from Definition B.5
such that:
GX mi S.
gy Re S.
with discrete distortion measures:
V
MT (Ra) + VV (2())))
disv (It)
max 151ST
| + V
(R)
sup ISET (R,,)ER,
ww((x) (z) - w/ (v)l
B.4. Discrete Structural Metric
Definition B.8 (Discrete Dynamic Graph Structural Metric), For two e-interleaved dynamic graphs G.X ,gY. their structural distance is:
disv (R1) + (isv((2)) dis (R:) RR) + dis R2),
1 inf max R.
d ,
(1I)
max Ct(R1, R2)
where co-distortion measure:
sup SSa (x, (x()) - Sauf (st (o)()),))) x NEVY
C, R2) max
(12)
Sww (y. S. (ve(2),
sup
,EVY
.EV
with Φi π2 _
u. 1/2 TT2
being the induced mappings.
B.5. e-Net Interleaving Preservation
Theorem B.9 (e-Net Interleaving Inheritance). Let GX , be e-interleaved dynamic graphs with e-nets L LYY the E-nets satisfy:
2 6max
(13)
where 6max max, diam(S.V.). then T and L.Y are c-interleaved.
15

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TMctaNet: Topological Meta-Learning Framework for Dynamic Link Prediction
Proof. 1. Natural Correspondence By e-interleaving property, the original tripod R induces mappings:
vt, 3dr Lt + S.L. ve L. S.L.
(14)
2. Structural Preservation For l E LN, using e-net coverage:
d(lz, or o vc(l)) S d(l., S.L," + (triangle inequality)
max + 6 S e (by condition)
3. Temporal Consistency The c-smoothing operator ensures:
t+c S.L. S U
t+c LA and S.L. c U L
(15)
T=1-c
T=t-t
do(G*, )) their Dowker Zigzag Persistence diagrums satisfy: Theorem B.10 (Dowker Zigzag Persistence Stability), For c-interleaved dynamic graphs g*,g ,g with structural metric
dg (Dgmpzp(G*). Dgmpzp(9")) (egmpzer""")DDmmmz(( S K dg(g".97
(16)
where constant K depends only the homological dimension.
Proof. By Theorem B.9, e-nets inherit e-interlcaving with:
dc(Lx x L < 2dg
(17)
where the factor 2 comes from:
(18)
original
smoothing compensation
1. Construct Interleaving Diagrams
Given de(Lx, _ = 8, build commutative diagrams for each timestamp t:
D(L
D(L U
1
(19)
@++1
D(LX)* - D(LY LX,1)* D(L
where vertical maps satisfy 5-interleaving conditions.
2. Verify 8- Interlleaig
For any simplex a E D) find matching o' E D(LL) via:
(x(a) = fr E ((T.)) 3 witness w d(a,, w) $ 8) vi(0') (r e D(L 3 witness w/ d(o', (') 6}
This guarantees;
D(I yO -
(20)
3. Bottleneck Distance Bound. In short, the classical statement in persistent homology is that if two persistence modules are 5-interleaved, then their bottleneck distance is at most 6 (Cohen-Steined et al., 2005). We provide a concise sketch of that reasoning below, then indicate how it generalizes to the zigzag case.
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
Case: Single-parameter Filtration (classical statement). parameter ax.. They called "6-interleaved" if for all o:
Consider single-parameter filtrations Pa and G. (for real
Fa > Ga+s and Ga >> Fa+6-
(21)
From these inclusions, one deduces: Whenever a homology class is "born" in Fa, it must appear in Go+6 the latest (so its birth can shift by most 8). 2. Whenever that class "dies" in F3, it disappears in G3+5 at the latest (so its death can shift by at most 8). Thus, homology class that persists during the interval [a. [8] in F. will persist during an interval at most [a + ò, 3 + 6] in G. Equivalently, the barcodes (interval decompositions) of F. and G. can be matched so that all intervals differ by at most in endpoints-this precisely implies their bottleneck distance at most ô.
Case: Zigzag Filtration. For zigzag modules, we do not have simple monotone filtration, but rather a sequence
E
F2
Fr
(22)
with forward/backward inclusions. While conceptually more involved, the same principle holds: if we have a d ò-interleaving between two zigzag modules (compatible inclusions that shift by at most d), then the birth and death of any homology class can shift by at most Hence the bottleneck distance the two zigzag persistence modules is at most 6.
cimension-dependent constant K". Sometimes, one sees constant factor K' (depending on the dimension or the grading) in front of ò. Intuitively, in higher-dimensional homology, identifying mapping homology classes from one module to the other might require extra factor. However, in many discrete standard cases, absorb these small local expansions into a constant K' independent of the filtration size the number of time steps. Thus write
dB(F., G.) I K'8 S K dolg* )
(23)
where K does not depend on T.
C. Method Details
s-net
In this section, we provide a detailed explanation of the construction method for the e-net of single snapshot, followed by the method to construct the e-net for the entire dynamic graph based on the single snapshot E-net construction method.
Taking the UCT dataset an example, where and 5 represent the parameters for constructing landmarks, the statistics of landmarks under different parameter configurations follows:
1_1
2.2
3.3
4_4
average proportion
43%
21%
16%
12%
average overlap rate
35%
22%
17%
16%
C.2. Zigzag Persistence Image
The zigzag persistence diagram space equipped with distances derived from the Hausdorff metrie. However, the structure of the metric space alone is insufficient as input for machine-learning techniques that require the inner product structure. To address this limitation, we transformed the ZPDs to zigzag persistence image (ZPI) representation, pixel arrays in Euclidean space. For each persistence diagram Dgm,,,4GG.). we computed its ZPI.
exp)exp (- (-12-111 1112
PDgma,
Σ
202
(24)
3c peDgm' (9.)
Dgm (G)) (9i') (2',)))))))))) (x, -x),
(25)
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TMctaNet: Topological Meta-Learning Framework for Dynamic Link Prediction
Algorithm 1 Construct e-net for Snapshots
Input: dist. malrix (The distance matrix representing the shortest path distances between nodes), (A threshold value defining the minimum allowed distance between nodes in the e-net), nodes (A set of nodes to be considered for the e-nct), mapped laappe..lndmarkk seed (A set of predefined seed nodes to be prioritized for inclusion)
Output: s_nel (The constructed e-net containing nodes whose pairwise distances are greater than E)
covered_nodes 0 {Set keep track of selected nodes)
2: node degrees +- (Dictionary store the degree of each node}
3: for each node E nodes do
4: degree +0
5: for each i {l....., |nodes|] do
6: dist_matrizli, node] S E A # node then
degree degree + 8: end if
9: end for
10: mode degresssses/ + degree
11: end for
12: sorted_nodes + Sort(nodes by node node plegrees in descending order)
13: if mapped landmark seed is not empty then
14: sorted_seed +- Sondrmaapeed..nddmark
landmark kecoorksseeee seed by node node_degrer degrees in descending order)
for each start_node sorted_seed do
16: e_net +- e_net U (start.node)
covered.nodes +- covered_node U {start_node] 18: break (Only include the highest-degree seed node first)
17:
19:
end for
20: for each node E sorted_seed do
21:
node & covered_nodes then
if V'selecteed e net, dist matria (node, selected] e then
s_net s_net U {node}
24:
codered.nodes codered.nodes U {node}
25:
end if
26: end if
27: end for
28: end if
for each node E sorted nodes do
30: if node covered_nodes then
31: if Viellccted net, dist matrix (node, selected] > then
32:
e_net + s_net U (node)
33:
covered.nodes t covered.nodes U {node}
34: end if
35: end if
end for returne_ned
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end for node modeedegrees [node] + degree, end for TMctaNet: Topological Meta-Learning Framework for Dynamic Link Prediction
Algorithm Construct e-nets for Discrete-Time Dynamic Graphs
Input: {G+ (The discrete-time dynamic graph), (Threshold for the e-net construction), T (Number of time snapshots in the dynamic graph)
Output: Landmarks and Witnesses for all time snapshots
1: landmarks; - { Set to store landmarks for each snapshot]
2: witnesses; t 0 (Set to store witnesses for each snapshot)
3: napped_landmarkk _seed t (Set store the mapped landmarks from previous snapshot)
4: for 1 to T do
5: nodes; t- Get nodes from G. (Get the set of nodes for the current snapshot} t- 9 (Dictionary to store degrees of nodes in current snapshot}
6: node.degrees
7: for each node E nodes, do
8: degree; +-
9: for each E (1...., |nodes;]) do
if dist_matrix, node] E + node then degree; degree; 11: end if
12:
13:
14:
15:
16:
sorted_nodes + Sort(nodes; by node degreess in descending order)
ift then
18:
landmarks; First few nodes from sorted
19:
witnesses +- Construct witnesses from londmarks, in G1 else
20:
21:
maappad + Landmarks from previous snapshot G landmarks; +- First few nodes from sorted_nodes,
22:
23: wilnesses: +- Construct witnesses using lanppe..andmaaree
G,
end if
25: Output landmarks, and witnesses, for time snapshot
26: end for
27:
returnAll landmarkss ks; and wilnesses, for all time snapshots
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
where D (G.) is the transformed multiset of Dgm,,.(G))) The weighting function g(n) is defined with mean u == (Mar My) E R² and variance 0², which depends on the distance from the diagonal. PDgm., represents the zigzag persistence surface.
Next, discretized subdomain of the zigzag persistence surface ( onto grid. The ZPI, represented matrix of pixel values, is obtained by integrating over each grid cell. Specifically, the value of each pixel E IR? within the ZPI is defined as:
s(H)ezp! { 1e-g12 d2nddy:
Zn, = //. Σ
(26)
WEDgm (9.)
Zn,k represents the k-dimensional ZPI under parameters En and on This results in a matrix where each pixel value encapsulates the density topological features within its corresponding grid cell, weighted by their proximity to the mean μ..
C.3. Toy Example of different task splitting strategies
Figure illustrates a comparative analysis of distinct task splitting methodologies. ROLAND implements a snapshot-wise partitioning strategy, wherein each temporal snapshot G, partitioned into training, validation, and testing subsets. The model undergoes training on the training subset of - validation on the validation subset of G,, and evaluation on the testing subset of G,, thereby leveraging the complete temporal sequence for both training and testing phases. In contrast, WinGNN employs chronological partitioning approach, segregating the temporal sequence into distinct training and testing periods. For instance, given sequence comprising 6 temporal snapshots, WinGNN utilizes the initial 4 snapshots for training purposes and the remaining snapshots for testing.
G₁
G2
G3
G4
G5
G6
Dynamic Graph with 6 snapshots
ANNN N
Train
Test
ROLAND Task Splitting Strategy
Train
Test
WinGNN Task Splitting Strategy
Figure 5. Toy Example of different task splitting strategies.
D. Experimental Details
D.I. Datasets Details
The datasets used in the experiments summarized in Table 3.
D.2. Baseline Details
The baseline models used the experiments are:
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TMctaNet:
Topological Meta-Lcarning Framework for Dynamic Link Prediction
Dataset
# Edges
# Nodes
Range
# Snapshots
ETH- Yocoin (Li al., 2020)
746, 397
15, 682
Jul 2016 Feb 2018
426
Reddit-Title (Kumar al., 8a)
571,927
54,075
Dec 31, 2013
Apr 30, 2017
178
Reddit-Body (Kumar et al., 201 201
286,561
35,776
Dec 31, 2013
Apr 30, 2017
178
UCI-Message (Panzarasa et al., 2009)
59,835
1,899
Apr 15, 2004 Oct 26, 2004
29
Bitcoin-Ore -TC (Kumar et al., 2018b)
35,592
5,881
Nov 8, 2010 Jan 24, 2016
279
Bitcoin-Ar Alpha (Kumar et al., 2016)
24,186
3,783
Nov 7, 2010 JJan 21, 2016
274
Table Summary the datasets used our experiments.
GCN-L and GCN-G: These combine structural encoding GCN modules with temporal encoding LSTM and GRU modules, widely used in dynamic link prediction. (Manessi et al., 2020)
EGC-O and EGC-H: Use RNNs update internal GNN parameters between snapshots, with EGC-O using LSTM encoding and EGC-H using GRU encoding. (Pareja al., 2020)
ROLAND: A meta-learning-basee dynamic graph neural network that uses live update training strategy, stacking GCN module for capturing structural information and GRU module for temporal encoding, (You et al., 2022)
WinGNN: meeta-carning-baae based dynamic graph neural network that replaces explicit temporal encoding with in- window stochastic gradient aggregation. (Zhu et al., 2023b)
DeGNN: A decoupled graph neural network that uses two-stream architecture capture both structural and temporal information. (Zheng et al., 2023)
D.3. Experimental Settings
All experiments in this paper were conducted on a Linux server with 4 NVIDIA A6000 GPUs. For the ROLAND setting, we used the recommended parameter settings from ROLAND for the baselines. For the WinGNN setting, we used the recommended parameter settings from WinGNN for the baselines, except for GCN-L and GCN-G which were not used in WinGNN, so we selected their optimal parameter combinations through grid search, For TMetaNet, the core parameters include the following parts:
Parameters for computing the Dowker Persistence Diagram. We choose and 5 = 1, with the window size set to full.
Parameters for computing the Zigzag Persistence Image. We set the image size be 50.
Parameters for the TMeetNet meta-learning model, mainly including the meta-learning parameter update model's learning rate Ir and dropout rate. We use the grid search to select the optimal parameter combinations for each dataset.
Table 4. Running time comparison between TeettNet and other meta- meta-learning methods under different settings.
LIVE UPDATE SETTINGROLLAND)
Method
Alpha
OTC
Body
Title
UCI
ETH
ROLAND
0.49
0.53
1.21
1.90
0.75
0.43
TMETANET
1.04
1.15
2.58
4.15
1.81
1.45
TIME INCREASE
112%
117%
113%
118%
141%
237%
WINGNN SETTING
WINGNN
16.36
15.37
15.52
74.54
1.93
23.92
TMETANET
18.78
16.26
16.46
75.97
2.19
25.72
TIME INCREASE
14%
6%
6%
2%
13%
7%
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TMctaNet:
Topological Meta-Lcarning Framework for Dynamic Link Prediction
D.4. Running Time
We compare the running time of TMeetNet with other meta-learning methods under different settings. Please refer to Table 4. For ROLAND, the time shown is the average training time per snapshot, while for WinGNN, the time shown is the average training time per epoch. As can be seen, after adding the persistent homology-basse meta-learning parameter adjustment, TMetaNet's s running time increases. In the ROLAND setting, the model needs to run for multiple epochs at each snapshot, so we take the average training time per snapshot. Under the ROLAND setting, the meta-model parameter update needs to consider the parameter results from each epoch, resulting in a larger proportion time increase. In the WinGNN setting, the model trains on the entire training set in each epoch, so we take the average training time per epoch. For meta-learning parameter updates within each epoch's 's snapshots, only one update is needed, resulting in smaller proportion of time increase.
D.5. Supplementary Experimental Results
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TMctaNet: Topological Meta-Lcarning Framework for Dynamic Link Prediction
Table Supplementary dynamic link prediction performance of different models under different settings. The best results highlighted in bold, and the second-best results are underlined. Each experiment is repeated five times, and the mean (omitting %) and standard deviation reported. indicates statistical significance (p value < 0.05 between TMetaNet and the second-best result).
LIVE UPDATE SEOTING(ROLAND)
Dataset
Metric
GCN-L
GCN-G
EGC-O
EGC-H
ROLAND
TMetaNet
Impr.
AUC
87.01 + 1.00
89.72 + 0.82
83.42
0.78
85.64 + 1.01
93.27
1.00
93.93 0.75
0.71 %
@@1
7.23 0.50
7.01 +
6.34 0.89
6.12 1.01
7.76 + 0.80
*8.16 0.49
5.15%
ALPHA
@@3
14.99 0.23
15.23 + 0.47
12.45
16.60 0.85
*17.17 + 0.90
3.43%
R R@10 10
28.73 + 1.23
29.87
+ 0.79
20.98 = 0.99
25.82
32.76
0.84
*35.96 + 0.93
9.77%
AUC
80.12 1.20
74.56 + 1.78
73.45 = 1.34
93.36
= 0.48
*93.52 + 0.49
0.17%
R@I
0.96
6.01
0.89
5.54
0.78
4.53
+.76
*9.01 = 1.05
2.74%
OTC
@@3
1
12.01 + 1.82
1.78
8.53 1.89
1
19.68 1.82
6.67%
@@l
27.54 1.98
26.01
1.82
23.54
1.78
22.53 1 1.89
36.47 1.19
1.74 37.78
3.59%
AUC
88.77 0.42
89.65
0.35
82.77
- 0.69
80.61
0.42
96.81 0.42
96.18 0.25
R
18.00
0.23
15.28 4: 0.15
12.90 = 0.78
10.82
0.67
24.10
0.48
#27.35 0.59
13.49
BODY
@
33.92 + 1.20
30.13 + 0.82
25.92 0.76
20.98 + 0.78
42.42
= 0.42
42.77 + 0.58
0.83%
R @ 10
52.23 0.20
49.23
+ 1.00
39.20 = 0.89
33.90 +: 0.23
62.28 = 0.37
62.17 + 0.28
AUC
94.71 + 0.03
95.02 + 0.03
92.30 + 0.03
90.35
0.02
97.93 + 0.00
97.64 + 0.00
R@I
22.34 0.23
20.98 + 0.72
14.57 + 0.76
12.28
0.46
26.34 1.30
#30.90 + 1.02
17.31 %
TITLE
@
40.28 2.03
41.23 1.28
37.28
1.72 1.72
35.32 1.24
47.23 # 1.19
48.33 0.95 68.30 0.35
2.33%
R @@1
56.88 0.87
57.88
0.66
52.32 2.98
49.23
1.89
0.71 68.32
AUC
86.27 0.74
1..23+0.00
81.29
+ 0.93
80.81
0.61
88.96 0.51
88.98 0.60
0.02%
R@I
3.45 0.93
3.22 0.73
2.99 1.09
2.38 0.78
4.63 1 1.13
*5.30 0.67
14.47%
UCI
R@3
8.00 0.92
7.77 0.81
5.16
9.49 1 1.80
#10.28
1.14
8.32%
R 10
20.82 1 1.01
15.27 11.87
12.23 1 1.05
11.28
0.98
21.51 + 2.35
#22.34 1: 2.05
3.86%
AUC
90.23 + 1.10
88.92
+ 0.72
87.92
1 1.98
85.89
+ 1.34
94.41 + 1.06
94.75
1.13
0.36%
R@I
18.72 0.93
19.92
+ 1.92
17.82
+ 2.03
15.37
+ 1.28
28.94 + 1.62
#32.58 + 1.61
12.58%
ETH
R@3
35.29
3.94
30.29
+ 1.82
24.23
1.96
20.92
+ 0.82
38.49
+ 1.57
40.17
1.67
4.36%
@
38.28
#2.23
40.92
= 0.23
34.82
1.37
34.29
+
48.51 0.79
48.42
1.39
WINGNN SETTING
DATASET
METRIC
GCN-L
GCN-G
EGC-O
EGC-H
WINGNN
TMETANET
IMPR.
AUC
72.32 +: 0.65
70.82 + 1.24
62.78 + 1.18
69.83 + 1.57
81.57 : 1.18
*94.50 + 1.45
15.85%
R@I
1.32 0.05
1.82
1.23
1.33
23.26
1.56
18.73 1.49
ALPHA
R@3
4.32 0.15
0.12
3.57
0.47
43.21 1.59
39.40
1.61
R 10
8.23 0.32
0.49
2.72
6.12
65.82
1.64
*75.31 1.90
14.42%
AUC
52.83
1.34
56.23 1.23
59.83
1.57
52.78
87.38 + 1.21
#89.14 2.22
2.01%
R@1
3.16
0.17
0.112
5.71
0.44
20.76
1.47
*22.51 #.93
8.43%
OТC
R@ R@3
9.16 0.12 4.992
10.45
13.01
14.01
1.05 49.96
49.15 4.90 *80.33 -..33+2221 2.21
R @ 10
16.90
19.67
20.67
73.95 1.10
8.63%
AUC
+..20
76.34
3.45
99.34 2.67
99.54 0.27
0.20%
R @
1.01 0.01
0.99
1.99 0.12
2.24
5.39
#11.57
114.66%
BODY
R@3
0.12
0.12
0.112
1.221 4.24
24.05 2.688
52.12%
R
0.12
0.45
0.11
8.24
43.78 + 3.01
46.29 + 2.52
5.73%
AUC
88.45
87.01
9.00
0.12
99.49
99.90 0.01
0.41%
R@1 1
0.000
0.000
0.0140.00
0.0110.00
16.65 + 1.45
21.87 1 0.47
31.35%
TITLE
R@3
0.00
0.03
1,01+0.00
0.54+0.00
34.56 + 3.88
38.48 - 0.89
11.34%
R @ 10
0.05
0.02
2.42
5.08 1.661.69 1.68
60.62
62.94 2.92
3.83%
AUC
2.334
54.01 + 1.32
65.08
3.91
71,54 2.89
2.89
14.37+0.22
91.19 + 1.83
R@1
3.45
7.01 0.18
0.06
12.56
14.95 = 1.31
19.03%
UCI
R@3
0.12
11.02
0.23
10.54
0.48
22.20 1.41
27.72 1.56
24.86%
R 10
15.76
18.81 0.76
0.9220099
15.08 + 1.99
40.62 1.51
*45.13 1.46
11.10
AUC
89.76
87.98 + 1.82
82.98
2.34
85.67 1.01
95.08 0.01
*99.60 0.01
4.75%
R@1
13.56 2,09
12.34
15.92 0.89
14.02 + 1.23
54.87 1.21
*63.81 1.87
16.30%
ETH
R 3
24.34 2.97
25.98
+ 3.58
28.974 2.23
25.78
+ 3.23
67.08 + 1.83
*92.04 + 1.27
37.21%
R @ 10
42.38 + 1.84
42.77
1.70
45.98 + 2.34
40.98 + 3.45
80.08 + 0.78
*98.97 + 0.45
23.59%
23


Reviews:
[{"review_id": "ay41LLLgZ5", "reviewer": "Reviewer_K9Wc", "summary": "This paper proposes TMetaNet, a topological meta-learning framework for dynamic link prediction that integrates DZP to capture high-order topological features in dynamic graphs. The authors claim that DZP provides a computationally efficient and stable representation of dynamic graph evolution, which is then used to guide meta-learning parameter updates. Theoretical stability guarantees for DZP are provided, and ablation studies validate the necessity of topological features.", "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": "**Q1**: Lemma B.3.\n\n**A**: To clarify briefly: given two tripods \n$R_1:\\mathcal{G}^X \\leftarrow W_1 \\rightarrow \\mathcal{G}^Y$ and $R_2:\\mathcal{G}^Y \\leftarrow W_2 \\rightarrow \\mathcal{G}^Z$ , each satisfying temporal consistency, their composite tripod is defined via fiber product: $W=\\{(w_1,w_2)\\in W_1\\times W_2\\mid\\pi_2^{(1)}(w_1)=\\pi_1^{(2)}(w_2)\\}.$ Temporal consistency holds because nodes align through intermediate set $V_t^Y$: $(\\pi_1^R)^{-1}(V_t^X)\\leftrightarrow(\\pi_2^{(1)})^{-1}(V_t^Y)=(\\pi_1^{(2)})^{-1}(V_t^Y)\\leftrightarrow(\\pi_2^R)^{-1}(V_t^Z)$\n\n**Q2**: Task-splitting of Fig.5.\n\n**A**:  ROLAND splits each snapshot $G_t$ into train/val/test sets. The model trains on $G_{t-1}$'s training set, validates on $G_t$'s validation set, and tests on $G_t$'s test set, utilizing all snapshots for both training and testing. WinGNN uses chronological splitting, dividing the sequence into training and testing periods. E.g., in a 6-snapshot sequence, WinGNN uses the first 4 snapshots for training and the last 2 for testing.\n\n**Q3**:  Recent SOTA baselines.\n\n**A**:  We have run DeGNN [1] as a baseline (∗ indicates statistical significance). We found that TMetaNet outperforms DeGNN in both settings. We are currently implementing [2] and will post the results later. The topology [3] refers to the specific connection states between nodes in a dynamic graph and their changes over time. In our paper, we use term topology in terms of algebraic and computational topology on graphs, i.e., shape characteristics of various orders which provide important information about higher-order structural organization of the graph.\n\nROLAND setting\n||ALPHA|OTC|BODY|TITLE|UCI|ETH|\n|-|-|-|-|-|-|-|\n|DeGNN|ACC|76.7±1.33|76.9±1.33|89.7±3.6|92.6±3.5|76.4±1.12|62.6±1.73|\n||MRR|12.5±1.03|15.0±1.21|26.7±3.5|40.1±4.8|9.2±4.3|33.0±2.62|\n|TMetaNet|ACC|86.84±1.02*|85.89±1.22*|89.59±1.17|93.96±0.02*|80.88±0.08*|85.10±1.46*|\n||MRR|17.68±0.55*|18.06±1.22*|34.93±1.07*|42.72±1.01*|10.99±0.92*|38.08±1.57*|\n\nWinGNN setting\n||ALPHA|OTC|BODY|TITLE|UCI|ETH|\n|-|-|-|-|-|-|-|\n|DeGNN|ACC|81.48±2.87|81.87±0.08|OOM|OOM|75.11±1.02|OOM|\n||MRR|32.36±0.90|29.85±0.59|OOM|OOM|20.15±1.13|OOM|\n|TMetaNet|ACC|89.92±1.84*|90.43±1.17*|98.26±1.29*|99.63±0.07*|86.37±5.63*|97.83±1.53*|\n||MRR|38.93±3.06*|39.98±2.16*|28.93±2.06*|34.96±2.06*|25.31±1.02*|78.07±1.09*|\n\n**Q4**: Scalability of TMetaNet.\n\n**A**: On ALPHA data, compared to VR complexes Zigzag Persistence, our method reduces the computational overhead by 46% on average per snapshot during complex construction. When dealing with extremely large-scale graphs, we can sample from snapshots or remove nodes according to degree centrality from low to high to obtain subgraphs that preserve global higher-order features, and then calculate the learning rates.\n\n**Q5**: Insufficient explanations.\n\n**A**: For Reddit-Body data, the differences between adjacent snapshots are relatively small, leading to suboptimal performance under the ROLAND setting, as the complementary information gains yielded by our method are less profound. However, under the WinGNN settings where we aggregate training within a certain window length, extracted topological signals are more prominent; TMetaNet more efficiently leverages the underlying higher-order graph topology, and outperforms the baseline. These findings indicate that extracted time-evolving topological information and the associated induced learning rates have higher value under more heterogeneous dynamics and less value for more homogeneous cases.\n\n**Q6**: The starting point.\n\n**A**: Topological information has been indeed incorporated into graph learning tasks before, largely for static scenarios and most recently for dynamic scenarios. However, there are no studies yet on explicit integration of time-evolving topological information for dynamic link prediction. This is also the first work that specifically focuses on using time-evolving topological representation to guide parameter updates in meta-learning models. Finally, models for dynamic link prediction and graph meta-learning using even conventional topological information are yet in their nascency. Our method makes a step forward toward this important direction by explicitly incorporating the essential time-evolving topological information in a mathematically rigorous and computationally more efficient manner.\n\n**Q7**: CTDGs.\n\n**A**: Indeed, CTDGs provide a more fine-grained node evolution process through event streams, by using representations via ODEs. However, CTDGs primarily focus on pairwise node interactions and do not account for simultaneous multi-node structures. TMetaNet and CTDGs can effectively complement each other. One approach is to use PDE rather than ODE in CTDG, where PDE describes dependencies across all nodes in the neighborhood and TMetaNet is used to parametrize the resulting coupled dynamics. Another direction is to use CTDG for identification and reconstruction of the dynamic landmarks in TMetaNet.", "claims_and_evidence": "Claim 1: we propose the Dowker Zigzag Persistence (DZP), a computationally efficient and stable dynamic graph persistent homology representation method...\nEvidence: The complexity analysis of DZP are mentioned in Section 4, however, it lacks direct comparisons with traditional Zigzag Persistence. Table 4 shows significant runtime increases for TMetaNet under the ROLAND and WINGNN settings, although the authors analyzed the reasons, which may limit practicality.", "methods_and_evaluation_criteria": "Metrics (ACC, MRR) are appropriate for the dynamic link prediction task.", "theoretical_claims": "Theorem 4.5 (DZP stability) relies on discrete ϵ-smoothing and tripod constructions in Appendix B. However, critical lemmas (e.g., Lemma B.3) lack rigor, for instance, the composite tripod construction and its temporal consistency are not fully justified.", "experimental_designs_or_analyses": "Cross-dataset experiments are comprehensive. The improvements on link prediction performance are visible under most datasets. In addition, the ablation study and noise analysis experiment are relatively complete.", "relation_to_broader_scientific_literature": "This work is related to the research of topology.", "essential_references_not_discussed": "This work overlooks recent SOTA baselines on DTGB, such as:\n\n[1] Yanping Zheng, Zhewei Wei, and Jiajun Liu. 2023. Decoupled Graph Neural Networks for Large Dynamic Graphs. Proc. VLDB Endow. 16, 9 (May 2023), 2239–2247. https://doi.org/10.14778/3598581.3598595\n\n[2] Fu, J., Guo, X., Hou, J. et al. SEGODE: a structure-enhanced graph neural ordinary differential equation network model for temporal link prediction. Knowl Inf Syst 67, 1713–1740 (2025)\n\nIn addition, some past work combining topology and dynamic graphs can be included in related work, such as:\n[3] Zhou, Zhengyang et al. “GReTo: Remedying dynamic graph topology-task discordance via target homophily.” International Conference on Learning Representations (2023).", "other_strengths_and_weaknesses": "Strengths: \n1.\tThe paper is well-written with comprehensive experiments covering LP tasks, Ab study, noise robustness and hyperparameter sensitivity analysis.\n2.\tThe integration of Dowker complexes with Zigzag Persistence is relatively novel.\n\nWeaknesses: \n1.\tThe model has high computational overhead (as shown in Table 4), which limits the scalability.\n2.\tThere is insufficient explanations for certain results (e.g., Reddit-Body performance drop in Table 1). The authors are expected to further explain the performance drop.\n3.\tThe idea of introducing graph structure topological information to graph learning to improve model performance is not a new idea for both static and dynamic graphs. Hence, the starting point of this work is a relatively incremental motivation.", "other_comments_or_suggestions": "1.\tDiscuss the scalability of TMetaNet on large-scale graphs.\n2.\tDiscuss the possible solution of reconstructing TMetaNet to deal with continuous-time dynamic graphs (CTDGs) since CTDGs are more consistent with real dynamic graph scenes and more informative.", "questions_for_authors": "See above weaknesses and comments.", "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": "WqOXvlC0cA", "reviewer": "Reviewer_GZjY", "summary": "The paper introduces TMetaNet, a meta-learning framework leveraging topological information for dynamic link prediction. The authors integrate Dowker Zigzag Persistence with graph neural networks to capture evolving topological structures. The work demonstrates competitive performance across six datasets compared to state-of-the-art methods.\n\n## update after rebuttal: \nThank the authors for their detailed responses. Most of my initial concerns have been addressed. As a result, I kept my original score.", "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": "**Q1**: This approach, while innovative, may not fully address the complexity of evolving graph structures.\n\n**A1**: We agree that our approach may not fully address the full complexity of evolving graph structures. However, we argue that given the currently existing methods, we achieve almost as much as possible. In particular, by using zigzag persistence, our proposed model inherently captures higher-order topological structures across different timestamps in both spatial and temporal dimensions. Additionally, Dowker complex can efficiently handle large-scale graphs by focusing on more compact and expressive components of the graph topology, resulting in higher scalability, comparing to more conventional persistence homology methods for graphs (and scalability nowadays is arguably one of the primary roadblocks on adopting PH tools, especially, for dynamic scenarios).  Considering the uncertainty and complexities of evolving graph structures, zigzag persistence and Dowker complex provide stable representations of the graph’s topology even in presence of noise or incomplete data. Certainly, our approach can be extended further, for example, by using zigzag persistence along multiple geometric dimensions, i.e. zigzag multi-persistence. However, such techniques are barely explored even in pure mathematics.\n\n**Q2**: The motivation for employing adaptive learning rates derived from topological features requires more explanation or stronger theoretical justification. \n\t\n**A2**: Thank you for your suggestion. Qualitatively speaking, persistent homology captures essential \"shape\" features across scales, directly reflecting graph structure. By shape here, we understand properties invariant under continuous transformations, e.g., bending, stretching, and twisting. When structural changes between snapshots are minimal, persistent homology features remain stable, allowing smaller learning rates as parameter updates stabilize. Conversely, significant structural changes produce substantial shifts in persistent homology features, necessitating larger learning rates to adapt.\n\nWe designed an experiment to illustrate this point. Starting from a random BA network A with 500 nodes, we randomly changed m% of the edges between each snapshot 50 times. Experiments are run under both TMetaNet and fixed learning rate methods, where m takes values of 5, 20, 50, 90. The experimental results of MRR are:\n\n||5%|20%|50%|90%|\n|-|-|-|-|-|\n|TMetaNet|4.53±0.24*|4.15±0.11|3.56±0.15*|2.64±0.10*|\n|fixed|3.90±0.37|3.82±0.16|2.51±0.22|1.74±0.06|\n\nThe performance of the fixed learning rate method drops faster with increasing m, while TMetaNet's performance drops relatively less, indicating that TMetaNet can better adapt to the changes in the graph structure and maintain good performance when facing random perturbations of the graph structure.\n\n**Q3**: The authors adopted two settings to train the model and the results are clearly provided. According to Table 1, TMetaNet's performance under \"WinGNN Setting\" is significantly better than that under \"Live Update Setting\" consistently while the other methods' performance are not. \n\n**A3**: Our baseline performance is close to what WinGNN and ROLAND reported, which aligns with the observation that performance under live update setting is better than WinGNN. We believe this difference mainly stems from  variations in how our method is implemented under different settings. In the WinGNN setting, we configure TMetaNet's learning rate updates based on WinGNN's model without temporal encoders. Since WinGNN itself uses meta-learning to update node embeddings, TMetaNet's meta-learning rate updates can more significantly enhance WinGNN's performance, as shown in the table. This explains why TMetaNet performs better under the WinGNN setting compared to the live update setting.\n\n**Q4**:Are there any existing works that use PH for dynamic graphs?\n\n**A4**:Yes, there are several works that apply PH to dynamic graphs. For example, [1] designs a stable distance between dynamic graphs based on persistent homology; [2] uses neural networks to approximate Dowker persistent homology for dynamic graphs, and [3] applies PH in diffusion models for dynamic graphs. These papers, along with other related approaches, primarily focus on using PH to represent dynamic graphs for downstream tasks, while our work specifically focuses on using time-evolving PH to guide parameter updates in meta-learning models.  Furthermore, there are yet no studies on explicit integration of {\\bf time-evolving topological information} for link prediction, either in a form of zigzag persistence or any other alternative approach.\n\n[1] Stable distance of persistent homology for dynamic graph comparison\n[2] Dynamic Neural Dowker Network: Approximating Persistent Homology in Dynamic Directed Graphs\n[3] Topological Zigzag Spaghetti for Diffusion-based Generation and Prediction on Graphs", "claims_and_evidence": "No issue here.", "methods_and_evaluation_criteria": "The authors identify the challenge of accounting for structural evolution in parameter updates but address this primarily through topology-informed learning rates. This approach, while innovative, may not fully address the complexity of evolving graph structures.\n\nThe motivation for employing adaptive learning rates derived from topological features requires more explanation or stronger theoretical justification. The connection among learning rate adaptation, capturing structural evolution patterns, and downstream tasks could be more rigorously established.", "theoretical_claims": "No", "experimental_designs_or_analyses": "The experimental design is thorough, including baseline comparisons, ablation studies, and detailed analyses that effectively demonstrate the method's capabilities.", "relation_to_broader_scientific_literature": "The integration of topological analysis with deep learning through traditional persistent homology presents a novel approach to incorporating structural information into graph neural networks.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is well-structured with a clear exposition of background, methodology, theoretical foundations, and empirical validation.", "other_comments_or_suggestions": "The authors adopted two settings to train the model and the results are clearly provided. According to Table 1, TMetaNet's performance under \"WinGNN Setting\" is significantly better than that under \"Live Update Setting\" consistently while the other methods' performance are not. This phenomenon requires deeper analysis.", "questions_for_authors": "Are there any existing works that use PH for dynamic graphs?", "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": "aQwsbtaoAO", "reviewer": "Reviewer_AgPX", "summary": "The paper proposes TMetaNet, a topological meta-learning framework for dynamic link prediction. Key contributions include:\n(1) Dowker Zigzag Persistence (DZP): A method combining Dowker complexes and zigzag persistence to efficiently capture high-order topological features in dynamic graphs.\n(2) TMetaNet Architecture: Integrates DZP into a meta-learning framework where a CNN-based adaptor adjusts learning rates using topological differences between graph snapshots.\nThe empirical validation results show that the proposed method outperforms baselines (e.g., ROLAND, WinGNN) on six datasets (e.g., Bitcoin-OTC, Reddit) in accuracy, MRR, and robustness to noise, with up to 74.7% improvement in MRR.\n\n## update after rebuttal\nI am not an expertise in computational topology, thus I am not quite familiar with the utilization of Lipschitz continuity assumption and various settings. Therefore I will just keep my score.", "questions": "", "limitations": "", "rating": 2, "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": "**Q1**: High-order graph information.\n\n**A**: For higher-order information, we mean various types of graph (sub)structures formed by interactions of multiple nodes simultaneously. Why is this information important and when? Suppose, we design a certain fraudulent scheme for money laundering. To conceal these illegal activities, it is unlikely that only two nodes would be involved (it’ll be easier to identify fraud!). Indeed, to hide criminal traces, money laundering schemes include many parties (nodes). How to identify particular multi-node patterns (i.e., higher-order information) that may be important? We can leverage the tools of persistent homology (PH). PH looks at the graph at multiple resolutions and tracks when specific multi-node patterns (described by simplices of various orders) appear or disappear as we monotonically change resolution scales. Those topological patterns staying with us longer are likelier to be important. (Note the appearance of weird patterns may serve, e.g., as a signal of money laundering.) Generally, wherever the problem at hand can be characterized by the inherent importance of such intrinsic multi-node interactions, e.g., link prediction due to social or protein-protein interactions, extracting higher-order structural information and integrating it into ML model will likely help. Other tasks, such as graph classification may benefit less. What we are doing in this paper is that extract the most essential time-evolving higher-order information for link prediction in dynamic networks and also use it to guide parameter updates in meta-learning models, which  to the best of our knowledge has never been done before.\n\n**Q2**: The stability proof \\& assumptions.\n\n**A**: We follow the standard practice in pure mathematics, where each statement in the derivation chain is either justified by previously published results or if it is newly derived, it is properly placed in a context of the previously obtained and published results. Specifically, we have explicitly restated the Lipschitz continuity assumption within our discrete-time DZP framework, and the ε-interleaving and bottleneck distance bound are also adapted to our discrete DZP setting. The theoretical results Theorem B.9, B.10 are new and have not been appeared before.   \n\n**Q3**: Significance.\n\n**A**: TMetaNet learns learning rates based on corresponding models under different settings (ROLAND & WinGNN). Compared to ROLAND, TMetaNet is more suitable for WinGNN which learns from all snapshots within a certain window length which is also reflected in significance, we have more significant results under WinGNN.\n\n**Q4**: Essential References.\n\n**A**: Neural ODEs and Graph Neural ODEs describe dynamical processes on graphs from the perspective of dynamical systems, while SNPE approaches graph probability distributions from a probabilistic inference perspective on high-dimensional data. Our work starts from a known graph structure and leverages higher-order graph topology in a form of time-varying Dowker zigzag persistence, to enhance link prediction performance on dynamic graphs. Given the recent premise of simplicial models for dynamic systems on networks, we believe that combining persistent homology with methods like Graph Neural ODEs would be a highly promising future direction, we will add all suggested papers to Related Work section.\n\n**Q5**: Limited exploration of real-world noise.\n\n**A**: Our robustness experiments consider the widely used structural Evasion and Poisoning attacks, and we have added the new Reddit-title experiment under the ROLAND setting. The below table shows that TMetaNet still maintains competitive robustness under the ROLAND setting.\n\nRobustness MRR\n||e_5|e_10|e_20|e_30|p_5|p_10|p_20|p_30|\n|-|-|-|-|-|-|-|-|-|\nROLAND|33.76|34.49|32.03|33.20|32.99|32.87|30.01|30.23|\nTMetaNet|31.25|32.21|32.23|32.01|30.21|29.80|28.81|29.71|\n\n**Q6**: Citations and eqs.\n\n**A**: We have made the changes in the paper.\n\n**Q7**: Higher-order graphs \\& dynamic link prediction.\n\n**A**: As noted above, we focus on dynamic higher-order graph information, induced by simultaneous multi-node interactions/interdependencies. Correlation methods focus inherently on linear dependencies and are not feasible for assessment of such joint multi-node interdependencies. The suggested papers relate to dynamic systems, assessment of Granger causality in time series using graphs, relational inference on time series through diffusion generative modeling, and structural inference. These papers neither consider higher-order multi-node graph information, nor dynamic link prediction. (However, we believe that our ZDP can potentially be integrated with structural inference). The closest approach to our ZDP is to use network motifs. However, in contrast to ZDP, network motifs are essentially ad-hoc, typically limited to 4-nodes only, are not easily generalizable for weighted networks, and do not enjoy important mathematical properties such as stability.", "claims_and_evidence": "Some claims are supported by the experiments and ablation studies. Evidence includes:\n\n1. Tables showing TMetaNet’s superior performance (e.g., 93.96% accuracy vs. 93.58% for ROLAND).\n\n2. Ablation studies confirming the utility of topological features over random/noise-injected variants.\n\n3. Noise robustness tests (evasion/poisoning attacks) showing stable performance.\nLimitations: Theoretical proofs rely on external work (Ye et al., 2023); real-world noise validation is limited to synthetic perturbations.\n\nYet following important claim need to be clarified: \n- What is high-order graph information / graph structures? How do they help in this work.", "methods_and_evaluation_criteria": "Methods: DZP reduces computational complexity via landmark sampling (ε-nets); TMetaNet uses topological signatures to guide meta-learning updates.\n\nEvaluation Criteria: Standard metrics (AUC, MRR) on benchmark datasets (Bitcoin, Reddit). Baselines include EvolveGCN, ROLAND, and WinGNN.", "theoretical_claims": "The stability proof (Theorem 4.5) cites external work but adapts it to DZP. The appendix provides a detailed proof sketch using ε-interleaving and bottleneck distance. Assumptions (e.g., Lipschitz continuity) are reasonable but not fully self-contained.", "experimental_designs_or_analyses": "Yes, I checked the soundness/validity. Experiments cover multiple settings (live update, WinGNN), noise scenarios, and hyperparameter sensitivity. Variance in results (e.g., ETH dataset MRR: ±1.57) is reported but significance is not deeply analyzed.", "relation_to_broader_scientific_literature": "This work builds on (1) meta-learning: extends ROLAND/WinGNN by incorporating topology, (2) persistent homology: as it uses zigzag persistence for dynamic graphs, improving scalability via Dowker complexes, and (3) dynamic GNNs: as it compares with EvolveGCN and GCN variants.", "essential_references_not_discussed": "1. Neural ODEs (Chen et al., 2018) for continuous-time dynamics.\n2. SNPE (Greenberg et al., 2019) in simulation-based inference.", "other_strengths_and_weaknesses": "Strengths:\n1. Novel integration of topology and meta-learning; \n2. Rigorous benchmarking; \n3. Noise robustness.\n\nWeakness:\n1. Theoretical reliance on external proofs;\n2. Limited exploration of real-world noise", "other_comments_or_suggestions": "1. It would be a better practice to have the citations in the same bracket in chronological order.\n2. Please check the equations, as some of them run over the margins.", "questions_for_authors": "1. In the abstract it is unclear that why we ought to take care of the intrinsic complex high-order topological information of evolving graphs. What is it connection with the dynamic link prediction.\n2. What is high-order structural information? Why do we have to use it in this work? It remains unknown in the introduction and created a huge challenge of getting pace with the motivation of this work.\n3. Since we target at using higher-order graph to help dynamic link prediction, any other method would work for finding higher-order graphs? Such ranging from simple correlation methods to complicated like relational inference / structural inference (e.g., Kipf et al. Neural Relational Inference for Interacting System, 2019; Löwe et al. Amortized Causal Discovery: Learning to Infer Causal Graphs from Time-Series Data, 2022; Zheng et al. Diffusion model for relational inference, 2024; Wang & Pang, Structural Inference with Dynamics Encoding and Partial Correlation Coefficients, 2024)?", "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": {"ethical_review_concerns": "N.A."}}, {"review_id": "Cwhc1PFVer", "reviewer": "Reviewer_rS6w", "summary": "The authors propose a meta-learning framework that leverages the topological information to guide parameter updates of GNN for dynamic graphs. Specifically, it uses the epsilon-net algorithm to select a set of landmark nodes from the complete graph and construct the Dowker Complex. Then it can use the DZP to capture the key topological properties of the dynamic graph and guide the parameter updates. Experimental results show the proposed method outperforms the SOTA baseline up to 74.70%.", "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": "**Q1**: A possible weakness is some of the baselines are not very new. Ablation, sensitivity, and robustness studies under the ROLAND setting ? \n\n**A1**:We have added DeGNN [1] (ICLR23') as a baseline, with experimental results shown below. ∗ indicates statistical significance (p-value < 0.05).\n\n*ROLAND setting*\n|||ALPHA|OTC|BODY|TITLE|UCI|ETH|\n|-|-|-|-|-|-|-|-|\n|DeGNN|ACC|76.7±1.33|76.9±1.33|89.7±3.6|92.6±3.5|76.4±1.12|62.6±1.73|\n||MRR|12.5±1.03|15.0±1.21|26.7±3.5|40.1±4.8|9.2±4.3|33.0±2.62|\n|TMetaNet|ACC|86.84±1.02*|85.89±1.22*|89.59±1.17|93.96±0.02*|80.88±0.08*|85.10±1.46*|\n||MRR|17.68±0.55*|18.06±1.22*|34.93±1.07*|42.72±1.01*|10.99±0.92*|38.08±1.57*|\n\n*WinGNN setting*\n|||ALPHA|OTC|BODY|TITLE|UCI|ETH|\n|-|-|-|-|-|-|-|-|\n|DeGNN|ACC|81.48±2.87|81.87±0.08|OOM|OOM|75.11±1.02|OOM|\n||MRR|32.36±0.90|29.85±0.59|OOM|OOM|20.15±1.13|OOM|\n|TMetaNet|ACC|89.92±1.84*|90.43±1.17*|98.26±1.29*|99.63±0.07*|86.37±5.63*|97.83±1.53*|\n||MRR|38.93±3.06*|39.98±2.16*|28.93±2.06*|34.96±2.06*|25.31±1.02*|78.07±1.09*|\n\nWe are currently implementing [2] and will post the results when they are ready. The results demonstrate that TMetaNet outperforms DeGNN under both ROLAND and WinGNN settings, validating the TMetaNet effectiveness of TMetaNet for dynamic graph learning. \n\n[1] Decoupled Graph Neural Networks for Large Dynamic Graphs\n[2] SEGODE: a structure-enhanced graph neural ordinary differential equation network model for temporal link prediction\n\nAdditionally, we have supplemented the Ablation, sensitivity, and robustness experiments under the Roland setting, with results shown below.\n\n*Ablation*\n|SETTINGS|METRIC|ALPHA|UCI|\n|-|-|-|-|\n|TOPO|ACC|86.84±1.02|80.88±0.08|\n||MRR|17.68±0.55|10.99±0.92|\n|RANDOM|ACC|83.07±2.44|80.04±1.26|\n||MRR|15.85±063|10.32±0.36|\n|DIST|ACC|84.24±1.42|79.96±0.94|\n||MRR|15.55±0.72|10.88±0.87|\n\n*Sensitivity MRR*\n||5|10|20|full|\n|-|-|-|-|-|\n|1_1|12.12|10.06|9.61|9.60|\n|2_2|12.90|12.40|9.61|9.94|\n|3_3|10.50|9.75|9.46|9.92|\n|4_4|11.05|9.44|10.39|9.34|\n\n*Robustness MRR*\n||e_5|e_10|e_20|e_30|p_5|p_10|p_20|p_30|\n|-|-|-|-|-|-|-|-|-|\nROLAND|33.76|34.49|32.03|33.20|32.99|32.87|30.01|30.23|\nTMetaNet|31.25|32.21|32.23|32.01|30.21|29.80|28.81|29.71|\n\nThese findings are consistent with the results under the WinGNN setting, proving that TMetaNet is also effective under the Roland setting.\n\n**Q2**: As a submission to a conference related to computer science, the paper could give more space to address how the algorithm is implemented. For example, move the content in the Appendix to the main sections. This will make the paper more coherent and help readers understand how the method is implemented. \n\n**A2**:Thank you for your suggestion. We will adjust the paper layout to accommodate more algorithmic details.\n\n**Q3**: The upper part of Figure 2 (G1, D1, G1∪G2) or similar illustration can be additionally presented near the background section to help understand the important concepts. \n\n**A3**: In Fig 2 (in the main body), $D_1$ should actually be $D(G_1)$, representing the Dowker complex corresponding to graph $G_1$, and we have updated the figure accordingly. Additionally, we have added the definition of $D(G_t)$ in Definition 4.1. In this paper, $D(G_t)$ is equivalent to $D(L_t,W_t)$. The definition of $G_1 \\cup G_2$ has been added to the background section, representing the intersection of two adjacent graphs.\n\n**Q4**:What proportion of nodes are selected as landmarks and witnesses in some exemplary cases? Can the authors present some examples? \n\n**A4**:Section 6.3 of the paper discusses DZP's parameter sensitivity on the UCI dataset, where ε and δ represent the parameters for constructing landmarks. The statistics of landmarks under different parameters are as follows:\n\n||1_1|2_2|3_3|4_4|\n|-|-|-|-|-|\n|average proportion|43%|21%|16%|12%|\n|average overlap rate|35%|22%|17%|16%|\n\nAs can be seen from Figure 3 in the main body, the experimental results corresponding to 1_1 are generally better, indicating that too few landmarks will reduce the ability to capture higher-order structures.", "claims_and_evidence": "The claims are found to be supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "Yes, they make sense. \n- Topology is an important aspect when analyzing graphs, and focusing on topology meta-learning should be effective and beneficial. \n- Evaluation datasets cover both web community and financial domain and vary in size and snapshot amount.", "theoretical_claims": "Mathematical formulations in sections 3-5 are checked but not very closely. They seem to be conherent and could support the proposed method but their detailed correctness is not rigorously verified by the reviewer.", "experimental_designs_or_analyses": "The experiments are extensive and rigorous. The main experiments compare the proposed with 6 baseline methods, on 6 datasets, under two settings (Rol. and Win.), and in terms of 6 metrics. Standard deviation is presented and statistical significance is presented. A possible weakness is some of the baselines are not very new, and the results can be more convincing if more methods are in 2021-2024 (unless no such methods are remarkable). \nAblation, sensitivity, and robustness studies are presented clearly. The question is, why they are only conducted under the WinGNN setting but not the ROLAND?", "relation_to_broader_scientific_literature": "TMetaNet proposes to use the topology features of graphs for meta-learning and updating the parameters of dynamic GNN. It is a proper and significant extension based on existing literature.", "essential_references_not_discussed": "No essential references are found not discussed.", "other_strengths_and_weaknesses": "Strengths:\ns1: the time complexity is better than existing methods.\ns2: topology is an important aspect of graph learning and should be considered in graph meta-learning\ns3: experiments are extensive (see Experimental Designs Or Analyses)\n\nWeaknesses: \nw1: some of the baselines are not very new (see Experimental Designs Or Analyses)\nw2: some writing is not very clear (see Other Comments Or Suggestions)", "other_comments_or_suggestions": "1. As a submission to a conference related to computer science, the paper could give more space to address how the algorithm is implemented. For example, move the content in the Appendix to the main sections. This will make the paper more coherent and help readers understand how the method is implemented\n2. The upper part of Figure 2 (G1, D1, G1∪G2) or similar illustration can be additionally presented near the background section to help understand the important concepts.", "questions_for_authors": "What proportion of nodes are selected as landmarks and witnesses in some exemplary cases? Can the authors present some examples?", "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: The paper presents a topological meta-learning framework for dynamic link prediction. All reviewers agree that the approach is novel and claims are largely backed up by experiments. From my perspective, the (extensive) author rebuttal also addresses the main concerns, which is even appreciated in (some) reviewer's response(s); I am recommending (weak) acceptance at this point, but encourage the authors (1) to include any additional experiments in the final version and (2) to include the rebuttal comments as best as possible.
Decision: Accept (poster)