Topology-Guided Graph Pre-training and Prompt Learning on Directed Graphs
Authors: Peiyu Liang
Chenguang Yang
Yixuan He
Rong Pan
Yuzhou Chen
Venue: Accepted by TMLR ()

Abstract: In recent years, graph neural networks (GNNs) have been the dominant approach for graph representation learning, leading to new state-of-the-art results on many classification and prediction tasks. However, they are limited by the fact that they cannot effectively learn expressive node representations without the guide of labels, thus suffering from the labeled data scarcity problem. To address the challenges of labeling costs and improve robustness in few-shot scenarios, pre-training on self-supervised tasks has garnered significant attention. Additionally, numerous prompting methods have been proposed as effective ways to bridge the gap between pretext tasks and downstream applications. Although graph pre-training and prompt tuning methods have explored various downstream tasks on undirected graphs, directed graphs have been largely under-explored, and these models suffer limitations in capturing directional and topological information in directed graphs. In this paper, we propose a novel topology-guided directed graph pre-training and prompt tuning model, named TopoDIG, that can effectively capture intrinsic directional structural and local topological features in directed graphs. These features play essential roles in transferring  knowledge from a pre-trained model to downstream tasks. TopoDIG consists of an encoder in the form of a magnetic Laplacian matrix, a topological encoder, and a graph prompt learning function. Experimental results on both real-world and synthetic directed graphs demonstrate the superior performance of TopoDIG compared to prominent baseline methods.

Published in Transactions on Machine Learning Research (03/2026)
Topology-Guided Graph Pre-training and Prompt Learning on Directed Graphs
Peiyu Liang Temple University
peigu.kang
Chenguang Yang
cyong314 &uuer. edu
University of California, Riverside
Yixuan He Arizona State University
Yiruan. He@asu edu
Rong Pan Arizona State Universsit
Dong.Pan@asu. edu
Yuzhou Chen
yuzhou, cachhu..cheBeee Sucr..d edu
University of California, Riverside
Reviewed on OpenReview: Sttps::/openreeee net/ forum? id= KNI dkL dwyTT Tys8
Abstract
In recent years, graph neural networks (GNNs) have been the dominant approach for graph representation learning, leading to new saate-of-thh-arr results on many classification and prediction tasks. However, they are limited by the fact that they cannot effectively learn expressive node representations without the guide of labels, thus suffering from the labeled data scarcity problem. To address the challenges of labeling costs and improve robustness in few-shot scenarios, pre-training self-supervised tasks has garnered significant attention. Additionally, numerous prompting methods have been proposed as effective ways to bridge the gap between pretext tasks and downstream applications. Although graph pre-training and prompt tuning methods have explored various downstream tasks on undirected graphs, directed graphs have been largely under-explored, and these models suffer limitations in capturing directional and topological information in directed graphs. In this paper, we pro- pose novel topology-guided directed graph pre-training and prompt tuning model, named TopoDIG, that effectively capture intrinsic directional struetural and local topological features in directed graphs. These features play essential roles in transferring knowledge from pre-trained model to downstream tasks. TopoDIG consists of an encoder in the form of a magnetic Laplacian matrix, a topological encoder, and a graph prompt lcarning func- lion. Experimental results on both real-world and synthetic directed graphs demonstrate the superior performance of TopoDIG compared to prominent baseline methods.
1 Introduction
With the growing prevalence of applications where data originate from non-Euclides domains and are nat- urally represented as graphs-ase such as social nelworks, citation networks, and biochemical struct ures graph data, rich in relational information, play a crucial role in numerous learning tasks (Zhou et al.) 2020 He ct al. 2024b). These tasks include predicting and modeling social interactions, protein interfaces, classifying discases, learning molecular fingerprints, and modeling physical systems (Zhou al. 2020 He al. 2025) In many cases, these relationships inherently exhibit. sense of direction. For instance, the WebKB dataset

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(Pei et al. 2020 consists of university websites connected by hyperlinks. In this setting, one website may link to another without necessarily receiving a reciprocal link. Such datasets are naturally represented by di- rected graphs staphs/networds nntworks.. Directed networks, with asymmetric sending and receiving patterns, are important with many applications, such as clustering time-series data with lead-lag relationships (He ct al. 2022b), ranking from pairwise comparisons (He el al., 2022a), angular synchronization (He al. 2024a), detect- ing influential groups social networks (He et al. 2022b Zhang et al. 2021b), and -boed cognitive manufacturing (Liu et al.) 2022a). Indeed, He al. (2022b) points out that even in the absence of any edge density differences, directionality (i.e., edge orientation) can reveal latent properties of network flows. Moreover, Zhang et al. (2021b) introduces MagNet, a spectral Graph Neural Network (GNN) designed for directed graphs. MagNet is built on the magnetic Laplacian, complex Hermitian matrix that captures both geometric and dorectional structure: the magnilude of its entries encodes undirected geometric rela- tionships (here we refer to connection patterns), while their phase represents directional information (e.g.. how information flows on a graph, or source and target information for edges). A tunable "charge' parameter is introduced, which asymptotically allows control over the emphasis on directionality for small parameter values. More generally, different charge parameter values induce sensitivity to different directed subgraph motifs. Although these models achieve promising results in the directed graph domain, they cannot simulta- neously capture directional and topological structures which may result in the loss of important higher-order directional information (which is crucial for certain tasks).
Besides, recent trends in graph transfer learning have led to a proliferation of studies that learn useful graph representations that can be applied to various downstream tasks or different domains. Among transfer learning methods, graph prompt. learning is a major area of interest and number of recent efforts have been made on graph pre-training, graph prompt design, and graph fine-tuning (Sun et al. 2023b). For example, GPPT (Sun et al. 2022) employs edge prediction as a pre-training objective and redefines node classification within this framework by introducing labeled tokens into the original graph and aligning the classification task with the pretext objective. GraphPrompt (Liu et al. 2023) unifies pre-training and downstream tasks within a common task template while incorporating a learnable prompt that guides the downstream task in identifying the most relevant knowledge from the pre-trained model in a lask-specific manner. All-in- One (Sun al.) (202aa) introduces a meta-learning technique the design of graph prompt which improve multi-task performance. Nonetheless, most graph prompt learning do not take directed graph structure into account or may discard this salient information, e.g., Zi el al. (2024) lists six different pre-training mel hods and 5 graph prompt learning methods, however, none of these methods has been applied to directed graphs.
To address those two issues, in this work, we first propose a Topology-Guided Directed Graph Pre- training and Prompt Learning (TopoDIG) approach on directed graphs which learn from direc- tional and topological information generated by magnetic Laplacian-based graph convolutional networks and Dowker complex-based topological representation learning module, respectively. In essence, our main contributions can be summarized as follows:
We propose an innovative directed graph pre-training framework, which brings the concepts of mag- netic Laplacian, Dowker complex- complex-based topological features, and topological representation learning to the directed graph domain.
We design topology-eno -eplooeed graph prompt function that improves the transfer and generaliza- tion capabilities of GNN.
We present extensive experimental studies on both real-world and synthetic datasets and find that our proposed TopoDIG outperforms the competitive baselines in various few-shot node classification tasks. These observations demonstrate the effectiveness of our framework in practical applications.
2 Related Work
2.1 Graph Neural Networks for Directed Graphs
Graph Neural Networks (GNNs) are specifically designed to learn node representations by leveraging neu- ral networks that operate directly on graph structures. Among the various deep learning approaches, the
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message-passing framework has emerged the dominant paradigm in recent works, including models such as Graph Convolutional Networks (GCN) (Kipf & Welling. 2017], GraphSAGE (Hamilton et al.) 2017), and Graph Attention Networks (GAT) (Veličković et al. 2018). Through iterative aggregation steps, these meth- ods enable cach nodc's representation to incorporate information from its neighboring nodes across multiple hops which is crucial for enhancing the performance of downstreem lasks. However, in directed graphs, these traditional GNNs struggle to fully capture the directional relationships between nodes. That is, the inherent symmetry of messagg-passing methods often disregards the edge direction, leading to loss of important structural information. This limitation becomes particularly pronounced in tasks where the directionality of the graph plays critical role such as in the analysis of causal relationships or in recommendation systems. To relieve this limitation, recently, several methods focus on handling directional structure during the prop- agation and leverage the power of directed Laplaciann to uncover complex palterns. For instance, DGCN (Tong et al.) 2020b) leverages first- and second-order proximity by constructing three Laplacians, but it can be incfficient terms of space and computation speod. DiGCN (Tong et al., 2020a) simplifics DGCN by introducing a directed Laplacian based on PageRank and aggregating information while considering higher- order proximity. MagNet (Zhang et al.) 20211b) builds complex Hermitian matrix where the magnitude of its entries encodes undirected geometric structure, and their phase captures directional information. Tong ct al. (2021) proposes a directed network data augmentation technique, Laplacian perturbation, and applies it to rastivv learning in directed graphs. DiGCI. (He et al.) 2022b) focuses on directed net work clustering, introducing novel imbalance objectives and evaluation metrics based on flow imbalance measures.
2.2 Graph Pre- Pre-Tranning and Prompt Tuning
Most transfer learning approaches in graph representation learning follow the 'pre-train & fine-tune" paradigm. This framework involves leveraging readily available information through a pretext task (e.g.. node-level, edge-level, or graph-level) to learn meaningful representations which are subsequently fine-tuned on a downstream task using the pre-trained model as initialization. The primary objective of graph pre- training is to capture structural patterns from input graphs in self-supervieed manner. In general, pre- training methods can be categorized into three approaches based on their tailored learning architectures, i.e., node-level, edge-level, and graph-level respectively. Significant. advancements have been made in each category. For the graph-level learning, BRep-BERT (Lou et al.) 2023) integrates GNNs with Transformers and incorporates graph structural information into the Transformer to learn both global and local entity feature representations. At the edge-level, EdgePreGPPTT (Sun et al. 2022) estimates link probabilities between node pairs using dot product calculations, and enhances the similarity of contextual subgraphs of linked pairs and reduces similarity for unlinked pairs by sampling triplets from label-free graphs. At the node-level, MoAMa (Inac et al., 2023) introduces novel node-masking technique that enables the model to capture long-range inter-motif structures for graph pre-training. Additionally, the adaptation of knowledge from an unlabeled graph to a target downstream task is typically achieved through fine-tuning where the pre- trained GNN is refined using a limited amount of labeled data (Lu et al. 2021). Furthermore, graph prompt functions mandate a pre-training task that can be readily emulated and integrate the pre-trained model into downstream tasks smoothly. ProG (Zi et al.] 2024) provides a comprehensive overview of 5 state-of-thheart graph prompt techniques from GPPT ((unn et al. 2022) to GPF-plus (Fang et al., 2023) to All-in-one (Sun) et al.] 2023a The fine-tuning strategy is inherently related to the pre-training method. Specifically, if pre-training is conducted at the graph-level, an appropriate graph-level fine-tuning approach should be em- ployed, c.g., S2PGNN (Lu et al. 2021) provides an adaptive fine-tuning framework for preserving the global information that optimally tailors the fine-tuning process to the characteristics of the pre-trained GNN and the downstream dataset in graph-level tasks. However, all these approaches fall short in their ability to incorporate directional structure and topological information, the essence of directed graphs.
2.3 Deep Learning with Topology
Topological data analysis (TDA) Wasserman 2018 Chazal & Michel 2021 Carlsson et al. 2012 Chen &
Gel 2025 Dixon et al. 2025 Chen et al. 20211). collection of methods derived from algebraic topology, has demonstrated significant utility across various machine learning tasks due to its robustness to noise and adaptability to diverse data modalities, including images, time series, and graphs. Among TDA techniques,
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persistent homology (PH) has gained prominence in image classification, graph learning, and text mining by capturing topological structures across multiple intensity levels. Additionally, approaches such discrete Morse theory, topological interactions, and center-line transforms have further contributed to performance improvements in these domains. In graph learning tasks, TDA has been leveraged in graph neural networks for topology learning by incorporating lopological descriptor embeddings or PH-based loss functions to en- hance GNN's performance (Carrière al. 2020 Chen et al., 2021a Zhao Wang 2019 Arafat et al. 2025; Horn et al. 2022; Chen et al. 2025b 2024). Despite these advancements,, conventional persistent homology pipelines often face substantial scalability challenges when applied to large-scale graphs. Standard construc- tions such as Vietoris-Rij Rips or clique complexes typically incur prohibitive computational and memory costs, as the number of simplices grows exponentially with graph density and size. To address this issue, new approaches based on weaker complexes such as Dowker complex which can effectively reduce computational complexity (Dowker 1952. Liu et al. 2022b; Choi et al. 2024 Li et al. 2025), which constitute a promising research direction. However, prior rescarch primarily concentrate on undirected graphs, and Dowker complex has untapped potential in transfer learning for directed graphs where learning meaningful representations across tasks remains challenging. Our work, TopoDIG, introduces novel framework that integrates di- rected geometric structure learning, topology-level information learning, graph prompting, and fine-tuning to facilitate knowledge transfer in the directed graph domain. More specifically, by leveraging PH to extract topological features, our approach enhances the adaptability of pre-trained models while preserving critical directional structural properties. This represents a significant departure from previous applications of TDA and extends its impact beyond unsupervised topology discovery to informed transfer learning across complex graph domains.
3 Method
Problem Definition. Denote (possibly weighted) directed graph (digraph) with node attributes G = (V,8, w, X), with V the set of nodes, & the set of directed edges/links, and w E [0, oo)lEl the set of edge weights. G may have self-loops, but no multiple edges. The number of nodes is [V], and X E Buxd is a matrix whose rows encode node features. This network can be represented by the node feature matrix X and the adjacency matrix A = (Aij)s,ev. with Aij = if no edge exists from &2 to My; if there is an edge e from iy to Uj, we set As₃ = We, the edge weight.
In this section, we will introduce, in detail, the TopoDIG framework, which leverages directional structural features and topological information of the directed graph to enhance the exprossiveness and generalization capability of GNNs. TopoDIG mainly comprises three modules: magnetic Laplacian-based GNN module, topological signature modeling module, and graph prompt learning module. Figure | provides an intuitive illustration of the design of pre-training and prompting framework (including magnetic Laplacian-based GNN modulc and topological signature modeling module) within TopoDIG.
3.1 Magnetic Laplacian- Based Graph Neural Networks
MagNet (Zhang et al. 2021b) applies the concept of a magnetic Laplacian to directed graph neural networks, building on a parameterized family of magnetic Laplacians (Fanuel et al.) 2017 F. de Resende & F. Costa 2020). Our spectral GNN part is built upon MagNet. We first define a symmetrized adjacency matrix and a corresponding degree matrix as follows:
(Aid Ag,)), 15i, Sn,
D
Σ (Ai + Aja), isis
with D. = 0 for i i j. To incorporate directional information, Zhang et al. (2021b) introduces a phase matrix (()) by [ 2mq(Ai.) Ag.), where g E R is the so-called "charge paramcter', Using elementwise multiplication (denoted by @) and the imaginary unit i, we define a complex Hermitian matrix as H(9)) = À @ exp(i where (x) is applied elementwise by exp(ie(4));, = exp(ie (o)).
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MLGC
MLGC
MLGC
Taratnodepar
i
+
MLP
DC
TPL
Link Prediction
MLGC
MLGC
MLGC
Taken Polre
MLP
DC
TPL
(ii)
Figure 1: Overview the TopoDIG pre-training and prompting framework. Given an input directed graph with node attributes, ToGoDIG learns transferable representations through two complementary pathways (with the gray box). The first branch (i) applies magnetic Laplacian- based graph convolution (MLGC) layers to capture directional and local structural information. In parallel, the second branch (ii) constructs Dowker complex to encode higher-order relational structures, which are transformed into compact topolog- ical embeddings via the topological representation learning (TPL) module. The structural and topological representations are fused to form joint node embeddings. Pre-training is performed using a pairwise pre- diction objective on both node-node and token-node pairs with a shared MLP and pretext loss, enabling topology-aware and prompt-compatible representations for downstream tasks.
Since À is symmetric and (4) is skew-symmetric, H()) is Hermitian. Notably, setting 0 results in H A, effectively symmetrizing the input graph and removing directional information.
Given ((),, we define the unnormalized and normalized magnetic Laplacians as follows
L = D H (0) =D= --Ac cxp((1(()))
and
(g) -I - 1/2 exp(ie
These Laplacian matrices are Hermitian, positive-definite and the eigenvalues L lie in [0,2].
Given L (one of the Laplacian matrices, in our experiments, the normalized one). let u..... u, be an orthonormal basis of eigenvectors satisfying Luç Au Let U be matrix whose k-th column is ug. for < k < n. The Fourier transform of a signal x V - C is defined as 8(k) (x, us) = u x, and equivalently, x U Since U is unitary, we obtain the Fourier inversion formula
x == U = Σ x(k)
(1)
1=1
The convolution of with filter is defined as the elementwise multiplication in the Fourier domain: y* x(k) = {k)) By equation [ this implies x = UDiag(E)2 = (UDiag()) where Diag(z) denotes a diagonal matrix with the vector z on its diagonal. Thus, we define a generalized convolution matriz as
Y = UUU!
(2)
for diagonal matrix E. This generalizes the spectral convolutions introduced in Bruna et al. 2014
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Following Zhang al. [2021b] we approximate spectral convolutions using polynomials of L. We set Σ Σk= O (A) as a polynomial in L, using Chebyshev polynomials Tx(+), where À - A normalizes eigenvalues to [-1, 1]. For 0 < K, Ta is the Chebyshev polynomials defined by To( 1, Ti (4) I, and Tx(x)) 2xTx-1(2))
Xmpz
2(x) for h 2 Since U is unitary, we have (UAUty UA and thus, letting L = 2 L I, we have
Amax
Yx U L AUUAA = (I)x
(3)
Let Zz be the imput matrix at layer I (with Zg X being G....... &g,P)), and L denotes the number of convolution layers. As in Zhang et al. (2021b), we use a complex activation function defined by a(g) = q. if 2 I arg(q) < T/2, and o(g) = 0 otherwise, where arg(-) is the complex argument of E C. Tet F3 be the number of channels in the f-th layer. For S & L. 1s S Fe, and S j S Fe-1, let Y be convolution matrix defined by Eq. equation 2 or Eq. equation Given the (e 1)-st layer hidden representation matrix z
x Fu input matrix with columns
we define z columnwise by
(0)
(t-1) G + +b,0) }
where b/ is a bias vector with equal real and imaginary parts, Real(b))) Imag b (9). In matrix form, we write H 9 (()) (H)- where Q(1) is hidden layer of the form Eq. equation In experiments, we utilize convolutions of the form equation with L = L and set K = 1, in which case we obtain
z = -
18 self
L t---- W( neigh + B(()).
(5)
where w(( self and w(f) neigh are learned weight matrices corresponding to the filter weights of different channels bl)
and (
3.2 Persistent Homology
PH is a subfield in computational topology, where the main goal is to detect, track, and encode the evolution of shape pattorns in the observed object along various user-selected gcometric dimensions (Edelsbrunner let al. 2000 These shape patterns represent topological properties such as connected components, loops, and, general, n-dimensional "holes', that is, the characteristics of the graph that remaín preserved at different resolutions under continuous transformations. By employing such multi-resolution approach, PH addresses the intrinsic limitations of classical homology and allows for retrieving the latent shape properties of G which may play an essential role in given learning task. The key approach here is to select some suitable scale parameters y and then to study changes in the shape of G that occur as G evolves concerning v. That is, we no longer study G as a single object but as filtration Gu, C C... C Gr., = G, induced by monotonic changes of v. To ensure that the process of pattern selection and counting is objective and efficient, we build an abstract simplicial complex X (Gw,) ) on each Ge,, which results in filtration of complexes combinatorial structure specifying (Edelsbrunner & Harer 2010). For example, for an edge- set Gev, = (V,E, w w e;]))
H (Go,) C... S X (Gv,). Note that, the abstract simplicial complex the adjacency of nodes, edges, triangles, and weighted graph (V, E, w), with the edge-weight function : - R, for cach Vj, j = 1,..., yiclding the induced sublevel edig-weighted filtration. Similarly, we can consider a function on a node set V, for example, node degree, which results in a sequence of induced subgraphs of G with a maximal degree of vy for each j = 1,..., n and the associated degree sublevel set filtration. We can then record scales b, (birth) and de (death) at which each topological feature first and last appear in the sublevel filtration gr, S Iv, S Iyy... S Gr.. Figure 2 shows examples of degree-based filtration and power filtration.
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Dagree-based filtration
Although the inherent nature of PH appears as perfect fit to capture the topological characteristics of the graph, computa- tional complexity remains the major roadblock on the way of wider adoption of PH in practice. For example, for 0-dimensional PH, the currently best available algorithm to compule PH has the complexity of O = (ma(m)); where m denotes the number of simplices and ar-)) denotes inverse of the Ackermann function. One intuitive idea to address this fundamental problem is to use somehow only subset of the available nodes when computing PH. However, can we do so, without sacrificing the topological information! The answer to this question is posilive we invoke the notion a Dowker complex on graphs. Dowker complex belongs to the family of weaker complexes which also includes, for example, witness complex (Ghrist 2014
f1 Power filtration
52
93
F2
91
Degree-based filtration (upper) and power filtration (bottom).
Figure 2:
Chazal al. 2014 Chowdhury & Mémoli 2018), and also may be viewed the witness complex counterpart similarities and differences of witness and Dowker complexes see Aksoy et al. 2023)) ). The ultimate idea is to assess the shape of the graph based only substantially smaller sub- set of nodes, called landmarks, while using all other remaining nodes as witnesses which dictate appearances of simplices in the Dowker complex.
graphs (for more discussion
Definition 1. Dowker Complex. For graph G (V,E), the Dowker Compler D(I, W) is simplicial complex construcled from the lundmark set L and the wilness set W as follows:
D(G) = for S Ew E W such that WeO d(l. w) sc,
where represents the marimumm allowable distance between landmark nodes and wilness nodes. Each simplex in D(L, W) corresponds to subset of landmark nodes. A k--simplzz is included in D(L, W) if there erists at least one witness node w E W that is within distance from every landmark node in o.
To select the suitable set L of landmark nodes for the Dowker complex, we leverage the s-nets algo- rithm (De Silva & Carlsson 2004 Arafat et al. 2020 2025), which ensures computational efficiency without loss of topological information. Given the Dowker complex of the graph D(G), we compute the Dowker complex-based persistence diagram (i.e., DC-PD) as follows DCDg = E(D(G) where E\t denotes the function that computes Dowker persistence.
In this paper, to encode the above topological information presented in a DC-PD DCDg into the embedding function, we use its vectorized representation, i.e., persistence image (PI) AAamns el al. 2017). The PI is a finite-dimensional vector representation obtained through a weighted kernel density function and can be computed in the following two steps (sce more details in Definition 2 First, we map the DC-PD DCD, to integrable function @DCp, 2 + R2, which is referred to as persistence surface. The persistence surface @DCp, is constructed by summing weighted Gaussian kernels centered at each point in DCDy- Second, we integrate the persistence surface @DCp, over each grid box to obtain the value of the Dowker complex- PI (i.e., DCp1) (see its definition in Appendix A
For a given DC-PD vectorization (e.g., the differentiable distribution-bant transformation described above). stability is a fundamental property from a statistical perspective. That is, stability requires that small perturbations in the input DC-PD lead to only small changes in its vectorized representation. To formalize this notion, distances between DC-PDs are measured using the Wasserstein distance, which provides a principled metric for comparing persistence diagrams. A vectorization function is said to be stable if the distance between vectorized representations is bounded by a constant multiple of the Wasserstein distance between the corresponding DC-PDs. This guarantees robustness of the learned topological features to small perturbations in the underlying graph structure. Formal definitions of the Wasscrstein distance and the associated stability results are provided in Appendix A Figure 3 displays different network structures and their corresponding Dowker complex-based persistence images.
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(a)
(b)
(c)
Figure 3: Illustration of network structures and their corresponding Dowker complex- persistence
images for (a) Texas, (b) Cornell, and (c) Wisconsin respectivoly.
3.3 Topological Representation Learning
To capture the underlying topological features of the graph 9, we employ K. filtration functions: Si I for = (K). Each filtration function fi gradually reveals specific topological structure different levels of connectivity, c.g., degree centrality score, betweenness centrality score, closeness centrality score, and other node centrality measurements. With each filtration function fi, we construct set of two persistence images of resolution P x P using tools in PH analysis (since we focus on both 0- and 1- dimensional topological features). Combining two persistence images of resolution P P from K. different filtration functions, we construct multi view topological representation, i.e., the of Dowker complex- based PIs [DC DCP), DCPL DCPI.......... DCP DC with the dimension K Xx22 P x P. We design a topological convolutional layer ((-) to (i) jointly extract and learn the latent lopological features and (ii) leverage and preserve the mulii-dimensional graph structural information. Firstly, hidden representations of the set of Pls are achieved through a combination of CNN-based model and global pooling, which can be defined
DC Z7 ([X, DC DCH, DCM, DC/. .......... -.. PCr DCP 1));
(6)
where ((-) embeds Dowker complex-basse PIs to high-dimensional space, and [....... denotes the con- calenation operation. This step is crucial for graph representation learning as the resulting topological embedding contains rich topological information encoded in the graph and can be integrated into any graph prompting functions. Based on Zg and Zy, we utilize the aggrogation and combination operation to obtain the node representations which is defined as follows
AGG-COMB(ZZ, Z7),
where AGG-COMB operation collects the node and topology embeddings, and combincs them using sum, mean, or max functions to generate the joint embedding for nodes. In this work, we employ the link prediction task for pre-training, ie., deciding if nodes are connected which is generated as ming, Σ (pre(fe(ee z,): g(u, u)), where fc is the projection head with trainable parameters Ç (we adopt the Multi-Layer Perceptrons (MLP)) which computes the similarity between node embeddings, and g(u, v) of node pair (i.e., if any pair of nodes connected or not)) based on the adjacency matrix A.
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3.4 Prompting for Downstream Tasks
Most existing graph prompling methods construct prompts purely from node-level altribules, where a prompting function fprompa(+) maps a node's standalone feature vector into structured token. While effective in undirected or feature-rich settings, such approaches are often insufficient for directed graphs where label semantics are governed by asymmetric flows and higher-order structural patterns that cannot be fully captured by node features alone. To address this limitation, in this paper, we introduce a topology- empowered prompting strategy that injects the joint directional and topological representations learned during pre-training directly into the downstream prompl lokens. Specifically, after fusing the magnetic Laplacian- based structural embedding Zg and the Dowker complex- based topological embedding ZT into joint node representation Z (see Eq. 7 we treat the resulting node embedding Zu as a topology-assisted structure token for each node u E Vrrompt C CV, denoted by
</ Zu
(8)
That is, the obtained representation encodes both local directional connectivity and higher-order topolog- ical signatures which enables the prompt to reflect intrinsic graph organization rather than isolated node attributes. Following the GPPTTbased prompting paradigm (Sun et al.) 2022), we reformulate downstream node classification as a compatibility prediction task between a node and a class-specific task token. Tet i./...., } denote the set of class labels (total C' classes), and associate each class ye with a learnable task token Trssk The final prompt loken for node u under class e is constructed as
TGG Stask, u.C = Sprompp (u, c) = [Uea, Su cTG].
(9)
where [ denotes concatenation. Intuitively, the task token Tiask acts as a class-specific query, while the topology-assisted structure token ETG provides a topology-guided representation of node u, allowing the model to retrieve task-relevant structural knowledge encoded during pre-training. Given a pre-trained projection head f ()), we compute a compatibility score between node u and class as
s(u, c) f (Stask,u,e)
(10)
The downstream objective is then formulated as a cross-entropy loss over the prompted nodes
Care (s(u, e); T(Teask (wc). s.t. ginit = @ree sinit
min 8 EM
(11)
(m.
where I(-) denotes the ground-truth label associated with a node pair, E E',..., EM denote M sets of learnable prompt embedding matrices (i.e., following Sun et al. (2022), we first adopt METIS (Karypis & Kumar 1998)
to pre-process and split nodes into M non-overlapped clusters, as varying task tokens in different clusters may provide enhanced lask embeddings), where each ET [eTe ,e.....,e"" [,]] RCxd stores the class-specific prompt tokens em associated with label ye (these embeddings are optimized during downstream prompt tuning), and (::);;) denotes the cross-entropy loss function. This optimization strategy presorves the pre-trained encoder and projection parameters while enabling task adaptation through prompt tuning, i.c., reducing task mismatch and improving generalization in few-shot settings. In contrast to conventional graph prompting approaches (which rely solely on node attributes), our proposed method prompts on topology- assisted representations that intograte directional structural cues and highor-order connectivity patterns. This design allows downstream tasks to directly leverage causally relevant graph structure learned during pre-training, leading to more robust and transferable performance directed graph classification tasks.
4 Experiments
4.1 Datasets
In this study, we evaluate TopoDIG on 10 real-word and synthetic datasets. For 5 real-world directed graphs, we use (i) Texas, Wisconsin, and Cornell which belong to the WebKB collection (Pei et al. 2020) and these
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datasets capture hyperlink connections between webpages from computer science departments at different universities. In each network, nodes represent webpages, edges indicate hyperlinks, and features are based on bag-of-words model. The webpages are manually classified into 5 categories such as student, project, course, staff, and faculty; and (ii) ogbn-arxiv and ogbn-Papers100M from the Open Graph Benchmark (OGB) (Hu et al. 2020) which are large-scale citation networks representing paper paper citation relationships. In these datasets, nodes denote scientific papers, directed edges denote citation links, and node features are derived from paper abstracts. The node labels indicate subject areas, with 40 classes in ogbn-arxiv and 172 classes in For synthetic data, ct al. (2022b), which generate directed graphs with clusters defined by network flows between groups groups/blocks. blocks. Each block represents a cluster in the directed graph, where clustering is partition of the set of nodes into K disjoint sets (clusters) V = Co U CI U .... U Cx-1 (ideally, K 2 2). In Appendix A we provide detailed descriptions of synthetic graph generation, and Tables and show dataset statistics of real-world and synthetic graph datasets.
SgbnnPapeers.ee
Directed Stochastic Block Models (DSBMs) introduced in He
Table I: Overview of real-world datasets.
Dataset
Cornell
Texas
Wisconsin
ogbn-arxiv
Sgbn-Paperss 100M
# Nodes
183
183
251
169,343
111,059,956
# Edges
295
309
499
1,166,243
1,615,685,872
# Features
1,703
1,703
1,703
128
128
Classes
5
5
5
40
172
Table 2: Overview of the synthetic datasets.
Dataset
Cycle (4, 0.1) Cycle (4, 0.01) Complete (5, 0.01) Star (5, 0.02) Path (5, 0.02)
# Nodes
1,000
1,000
1,000
1,000
1,000
# Edges
50,052
5,021
4,900
10,309
9,985
# Features
8
8
8
# Classes
5
5
5
Table 3: Evaluation results for node classification on Texas, Cornell, and Wisconsin, reporting the mean plus/minus one standard deviation over 5 runs. Best results are in bold.
Method
Texas
Cornell
Wisconsin
GCN
57.81 + 0.78
53.91 = 1.98
47.54 + 3.10
GAT
62.50 + 0.78
56.56 + 4.09
49.49 + 3.82
GIN
60.16 2.34
51.41 = 0.58
41.14
GranhTrrnsformer
62.50 3.12
56.87 = 3.18
58.17 1.93
DIGCN
48.44 9.94
44.53 4.50
47.66 + 3.50
MagNet
58.98 + 2.73
60.00 = 1.67
55.54 + 3.85
GCN+CNA
59.73 6.01
60.12 = 1.55
56.79 + 3.61
AdapterGNN
57.47 +
59.61 = 1.79
57.80 + 1.52
ToGoDIG (Ours)
64.84 2.34
61.56 + 2.49
57.94 + 2.86
4.2 Baselines and Experiment Setups
In our experiments, compare our approach with representative graph neural networks, i.c., Graph Con-
volutional Networks (GCN) Kipp & Welling 2017), Graph Isomorphism Network (GIN) [Xu et al. 2019 -
Graph Attention Networks (GAT) (Veličković et al. 2018). and
(CNA)
modules (Skryagin et al.) 2024). a transfer learning method, i.c., AdapterGNN (Li al. 2024). and a
graph transformer model (i.e., GraphTTransformer) (Shi al. 2021]. For directed graphs specifically, we
compare TopoDDG with DiGCN (Tong et. al.] 2020ah and MagNet. (Zhang et. al.) 2021b). We train all
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Table 4: Evaluation results for 3-shot node classification (in %) on ogbn-arxiv.
Method
ogbn-arxiv
GPPT (Sun ct al. 2022
23.35 + 0.70
All-in-one (unn et al. 2023a
16.79 = 6.81
GraphPrompi (Liu et al 2023
82.07 3.19
GPF (Fang et al. 2023
77.50 7.23
TopoDIG (Ours)
86.78 + 2.20
Table 5: Results (in %) of fine-tuning the pretrained GNN with 1% and 5% labeled training data on ogbn- Papers100M.
Method
Label ratio 1%
Label ratio 5%
CCA-SSG (Zhang et al. 2021a)
55.68 + 0.24
59.78 + 0.08
GRACE Zhu et al. 2020
55.45 + 0.23
59.38 + 0.15
BGRL Thakoor et al. 2022
55.12 + 0.23
60.40 0.54
GraphMAE (Hou et al. 2022
58.29 + 0.15
62.00 0.12
GraphMAF2 (Hou et al.
2023)
58.69 0.38
62.87 0.64
TIGoDIG (Ours)
59.42 = 0.41
64.05 + 0.59
Table 6: Evaluation results for node classification on synthetic datasets, reporting the mean plus/ monssmmmusua one standard deviation over 5 runs. Best. results are bold.
Method
Cycle (4, 0.1) Cycle (4, 0.01)
Complete (5, 0.01)
Star (5, 0.02)
Path (5, 0.02)
GCN
26.97 + 1.48
31.94 1.25
36.91 + 0.98
37.20 1.87
39.26 + 1.74
GAT
24.11 = 1.07
31.40 1.47
35.29 + 1.59
30.66 + 1.20
31.46 1.61
GIN
27.66 5.73
38.00 1.12
43.29 + 0.52
25.71 1.91
23.66 1.15
Graph Transforeee
38.86 2.16
41.31 1.56
45.57 +: 2.08
37.03 1.51
43.71 + 2.28
DIGCN
24.60 1.57
27.11
27.49 + 1.66
20.94
22.51 + 1.62
MagNet
26.69 = 8.35
39.26 + 1.48
46.89 + 0.735
35.97 4.02
43.26 + 2.06
GCN+CNA
38.74 4.98
46.65 2.81
44.01 + 2.30
36.25 1.25:...5
41.65 + 1.53
AdapterGNN
38.00 + 4.01
35.15
2.57 2.57
40.74 + 3.06
39.10 1.02
TopoDIG (Ours)
50.26 + 4.52
71.43
12.39
47.31 + 1.16
43.11
43.97 3.91
Table 7: Ablation experiment results, reporting the mean hlus/minus one standard deviation over 5 runs. Best results are in bold.
Method w/o MIGC
Texas
Cornell
Wisconsin
60.09 5.74
59.56 = 1.36
56.14 4 3.80
w/o Topo
58.32 + 4.18
60.25 2.02
57.83 + 1.37
TopoDIG (Ours)
64.84 2.34
61.56 + 2.49
57.94 2.86
"pre-train and prompt" models on NVIDIA RTX A5000 GPU cards with 24GB memory, and report the mean accuracy (in %) and standard deviation over 5 runs with different random seeds. In the pre- training phase,
cross-entropy loss positive (i.e., adjacency matrix) and negative (via random sampling) connected edges for link prediction. We save the best-performed pre-trained model parameter within 1000 epochs using an early stop stralegy for downstream prompting tasks. Since this paper mainly focuses on the pretraining strategy, we leverage a prompting strategy proposed in Sun et al. (2022) for downstream few-shot node classification tasks. To provide fairness comparisons, we the seed to 42 and tune each model with learning rate in (5c-3 5. 5e-5.- 5e of GNN layers in (2,3). Additionally, 'charge parameter' q used in MagNet and TopoDIG tune within
7,5e"), "}, hidden dimension in (128, 256, 512), number
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rrizann
moan
60
40
0.2
03
04
03
CD
07
0.5
02
0.5
0.4
0.3
0o
07
0.50
0.05
0.50
0.15
320
0.5
045
0.05
0.50
0.15
0.0
0.25
p xel size
pixei size
(a)
(b)
(c)
(c)
Figure 4: (a) and (b): Sensitivity study of the pixel size of the persistence image on Cornell and Cycle (5, 0.02). (c) and (d): Sensitivity study of the parameter on Cornell and Cycle (4, 0.1). All plots show the mean (dot) and variance (error bar) on 5 runs.
Table 8: Graph prompt function performance comparison (Accuracyistd),
Method
Texas
Cornell
Wisconsin
TopoDIG with GPPT (ours)
64.84 = 2.34
61.56 + 2.49
$7.94 2.86
TopoDIG with GPrompt
63.12 + 2.58
59.91 + 2.41
55.47 + 2.77
TopoDIG with GPF
65.27 + 2.41
60.38 + 2.57
58.42 2.73
{0.05, 0.15, 0.25), and pixel size for generating the topological features tuned within {0.2, 0.5, 0.8). Code can be found at http://///ggggaoo/////ooaaaaaaaa hhom//hhhagggaggggggggi///rooooo
4.3 Results
We provide few-shot node classification results within the pre-train and prompt paradigm in Tables 3 and 6
while seeing that TopoDIG performs well across all datasets. As shown in Table 3 our proposed TopoDIG outperforms all baselines Texas and Cornell except for Wisconsin (which may be due to Graph- Transformer utilizes global attention to enhance information aggregation from sparse data and TopoDIG
still achieves the runner-up result on Wisconsin). Based on the results in Tables 4 and 5 our method ToGoDIG consistently and substantially out perporromss all competing baselines across both few-shot and low- label regimes. On the 3-shot node classification task on ogbn-arxiv (see Table 4 TopoDIG achieves the highest accuracy, surpassing the strongest bascline Gprompt by a clear margin, while dramatically outper- forming earlier prompling-based and fine-tuning approaches. This demonst rates the effectiveness of TopoDIG in extremely label-scarce settings large-scale directed citation graphs. For fine-tuning with limited la- beled data (see Table 5 TopoDIG again delivers the best performance under both 1% and 5% label ratios, respectively. Compared to the strongest self-supervise baseline, GraphMAE2, TopoDIG yields consistent gains while maintaining comparable variance, highlighting its superior representation quality and robust- ness. Overall, these results confirm that incorporating topology-awar representations enables TopoDIG more effectively leverage structural information, leading to superior performance on large-scalc directod graphs. In Table we use 5 synthetic graphs with different structures. While setting the total number of nodes per graph invariant to 1000, and observe that TopoDIG outperforms all baselines across various graph densities and connectivity patterns with relative average gain of 25.84% overall 5 synthetic datasets. Furthermore, we provide detailed computational complexity analysis in Appendix B
4.4 Ablation Studies
To evaluate the effectiveness of cach module in TopoDIG, we compare it with 2 model variants. Specifically, "w/o SGNN" and "w/o Topo" represent methods without using MLGC module and topological represen- tation learning module (Topo). Table shows results on Texas, Cornell, and Wisconsin. is shown that TopoDIG consistently outperforms 2 variants on all datasets, thereby demonstrating the offectiveness of each module in TopoDIG. For the ablation scope, we have conducted additional experiments, i.c., except GPPT, we utilize GraphPrompt (Liu al. 2023) and GPF (Fang et al. 2023 for directed graph learning on
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Published in Transactions on Machine Learning Research (03/2026)
Texas, Cornell, and Wisconsin data. As shown in Table we observe that (1) TopoDIG with GPPT always outperform TopoDIG with Gprompt; (2) TopoDIG with GPF achieves the highest accuracy Texas and Wisconsin data, however, it achieves a slightly lower accuracy compared to TopoDIG with GPPT on Cor- nell; (3) compared to other baselines (shown in Table 3), TopoDIG with different prompt functions achieves state-of-the-art performances.
4.5 Sensitivity Analysis
We conduct sensitivity analysis on pixel size in generating topological features, where increasing the pixel count results in smaller persistence image sizes. As shown in Figure 4 (a & b), different datascts achieve optimal performance with different pixel sizes. For downstream node classification tasks, it is important lo ensure that the connectivity signal does not dominate the node features themselves. Therefore, the size of the input node feature plays a crucial role when selecting the pixel size. In the cases of Cornell and Cyclo(5, 0.02), which have input feature sizes of 1,703 and 8, respectively, the best performance is observed at pixel sizes of 0.5 and 0.8. We also conduct sensitivity analysis of the parameter q. Figure I (c & d) shows results of sensitivity analysis on Cornell and Cycle (4, 0.1) datasets. We find that, Cornell data, the performance remains relative stable across different values of which suggests that our model is not highly sensitive to the on this data which is a sparse graph. However, on Cycle (4, 0.1), the performance fluctuates significantly with different values of q (especially, we can observe a sharp decline at = 0.05 and q = 0.15) which indicates that the parameter selection crucial for dense graph analysis.
5 Conclusion
In conclusion, this work introduces the Topology-Guided Directed Graph Pre-training and Prompt Learn- ing (TopoDIG),, pioneering framework designed to the graph prompting study on directed graphs with applying magnetic Laplacian and persistent homology. Through a novel integration of magnetic Laplacian- based graph convolutional works module and Dowker complex-based topological representation learning module, TIG effectively captures directional and local topological information of directed graphs. Our experiments demonstrate our TopoDIG's superior performance on various real-world and synthetic datasets. In the future, will extend our approach to dynamic directed graph learning tasks.
Acknowledgments
Y.H. supported by an Nvidia Academic Grant award and a Jetstream2 NAIRR A1 Fellowship. Y.C. was supported in part by the NSF DMS-2523488 DMS-2420959, and UCR SoCal OASIS Funding Award. Any opinions, findings, and conclusions or recommendations expressed in this document. are those of the author(s) and do not necessarily reflect the views of the National Science Foundation other funding parties.
References
Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: stable vector representation of persistent homology. JMLR, 2017.
Sinan G Aksoy, Ryan Bennink, Yuzhou Chen, José Frías, Yulia R Gel, Bill Kay, Uwe Naumann, Carlos Ortiz Marrero, Anthony V Petyuk, Sandip Roy, et al. Seven open problems in applied combinatorics. arXiv preprinl ar Xiv:2333. 1 1464, 2023.
Naheed Anjum Arafat, Debabrota Basu, and Stéphane Bressan. epsilon-net induced lazy witness complexes on graphs. arXiv preprint 20Xiv:2009.1300 2020.
Nahcod Anjum Arafat, Debabrota Basu, Yulia Gol, and Yuzhou Chen. When witnesses defend: A witness graph topological layer for adversarial graph learning. In Proceedings of the AAAI Conference on Artificial Intelligence, 2025.
13

---

Published in Transactions on Machine Learning Research (03/2026)
Joan Bruna, Wojeiech Zaremba, Arthur Szlam, and Yann LeCun. Spectral Networks and Deep Locally Connected Networks on Graphs. In TCLR, 2014.
Gunnar Carlsson, Rick Jardine, Dimitry Feichtner-Kozlo,, Dmitriy Morozov, Frédéric Chazal, Vin de Silva, Brittany Fasy, Jesse Johnson, Matt Kahle, Gilad Lerman, et al. Topological data analysis and machine lcarning theory. In BIRS Workshop, pp. 1-11, 2012.
Mathieu Carrière, Frédéric Chazal, Yuichi lke, Théo Lacombe, Martin Royer, and Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topological signatures. In AISTATS, pp. 2786 2796. PMLR, 2020.
Frédéric Chazal and Bertrand Michel. An introduction topological data analysis: fundamental and practical aspects for data scientists. Frontiers in artificial intelligence, 4:667963, 2021.
Frédéric Chazal, Vin De Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicota, 173(1):193 214, 2014.
Yuzhou Chen and Yulia Gel. Topological zigzag spaghetti for diffusion-based generation and prediction on graphs. In Proceedings of the International Conference on Leurning Represenlations, 2025.
Yuzhou Chen, Baris Coskunuzer, and Yulia Gel. Topological relational learning on graphs. Advances in Newral Information Processing Systems, 34:27029 -427022-27704,, 2021a.
Yuzhou Chen, Ignacio Segovia, and Yulia. R Gel. Z-GCNETs: Time zigzags at graph convolutional networks
for time series forecasting. In Proceedings of the International conference on machine learning, pp. 1684- 1694. PMLR, 2021b.
Yuzhou Chen, Jose Frias, and Yulia R Gel. Topogel: Topological graph contrastive learning. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pp. 1445--1161,, 2024.
Yuzhon Chen, Xiao Guo, and Shujie Ma. Stochastic block model-aware topological neural networks for graph link prediction. Transactions on Machine Learning Research, 2025a.
Yuzhou Chen, Yuanyuan Zhang, Stephen Chan, Jeffrey Chu, and Nicholas Lord. Multilayer topology-aware graph contrastive learning for fraud detection in the ethereum transaction network. Journal of the Royal Statistical Society Series A: Statistics in Society, pp. quaf135, 2025b.
Jae Won Choi, Yuzhou Chen, José Frias, Joel Castillo, and Yulia Gel. Revisiting link prediction with the dowker complex. In Pacifie-Asia Conference on Knowledge Discovery and Data Mining, pp. 418-430. Springer, 2024.
Samir Chowdhury and Facundo Mémoli. A functorial dowker theorem and persistent homology of asymmetric networks. Journal of Applied and Computational Topology, 2:115 175, 2018.
Vin De Silva and Gunnar E Carlsson. Topological estimation using witness complexes. In PBG, pp. 157 157-166, 2004.
Matthew Dixon, Yuzhon Chen, and Yulia R Gel. Topological k-means clustering in reproducing kernel hilbert spaces. Electronic Journal of Statistics, 19(1):204 239, 2025.
Clifford H Dowker. Topology of metric complexes. American journal of mathematics, 74(3):555-577, 1952.
H. Edelsbrumner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. In FOCS, pp. 454 463, 2000.
Herbert Edelsbrunner and John Harer. Computational topology: An introduction. American Mathematical Soc., 2010.
Bruno Messias F. de Resende and Luciano da F. Costa. Charactcrization and comparison large dirccted networks through the spectra of the magnetic laplacian. Chaos: An Interdisciplinary Journal of Nonlinear Science, 3077)3073 2020.
14

---

Published in Transactions on Machine Learning Research (03/2026)
Taoran Fang, Yunchao Zhang, Yang Yang, Chumping Wang, and Lei Chen. Universal prompt tuning for graph neural networks. Advances in Newral Information Processing Systems, 2023.
Michaël Fanuel, Carlos M Alaiz, and Johan AK Suykens. Magnetic eigenmaps for community detection in directed networks. Physical Review E, (222):0 2017.
Robert W Ghrist. Elementary applied topology, volume 1. Createspace Seattle, 2014.
Will Hamilton, Zhitao Ying. and Jure Leskovec. Inductive representation learning on large graphs. Advances in Neural Information Processing Systems, 30, 2017.
Yixuan He, Quan Gan, David Wipf, Gesine D Reinert, Junchi Yan, and Mihai Cucuringu. GNNRank: Learning Global Rankings from Pairwise Comparisons via Directed Graph Neural Networks. In Proceedings of the International Conference Machine Learning, 2022a.
Yixuan He, Gesine Reinert, and Mihai Cucuringu. DIGRAC: Digraph Clustering Based on Flow Imbalance. In LOG, pp. 21-1. PMLR, 2022b.
Yixuan He, Gesine Reinert, David Wipf, and Mihai Cucuringu. Robust angular synchronization via directed graph neural networks. In ICLR, 2024a.
Yixuan He, Xitong Zhang, Junjie Huang, Benedek Rozemberezki, Mihai Cucuringu, and Gesine Reinert. Pytorch geometric signed directed: A software package on graph neural networks for signed and directed graphs. In Learning on Graphs Conference, pp. 12-1. PMLR, 2024b.
Yixuan He, Aaron Sandel, David Wipf, Mihai Cucuringu, John Mitani, and Gesine Reinert. Learning fuse temporal proximity networks: A case study in chimpanzee social interactions. arXiv preprint arXiv:2502.0088 2025.
Max Horn, Edward De Brouwer, Michael Moor, Yves Moreau, Bastian Rieck, and Karsten Borgwardt. Topo- logical graph neural networks. In Proceedings of the International Conference on Learning Representations, 2022.
Zhenyu Hou, Xiao Liu, Yukuo Cen, Yuxiao Dong, Hongxia Yang, Chunjie Wang, and Jie Tang. Graphmae: Self-supervised masked graph autoencoders. In Proceedings of the 28th ACM SIGKDD conference on knowledge discovery and data mining, pp. 594-604, 2022.
Zhenyu Hou, Yufei He, Yukuo Cen, Xiao Liu, Yuxiao Dong, Evgeny Kharlamov, and Jie Tang. Graphmac2: A decoding--nhanced masked self-supervised graph learner. In Proceedings of the ACM web conference, 2023.
Weihua Hu, Matthias Fcy, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. Advances in Newral Information Processing Systems, 33:22118 22133, 2020.
Eric Inae, Gang Liu, and Meng Jiang, Motif-aware attribute masking for molecular graph pre-training, anXiv preprint ar Xrx::2200..0 04589, 2023.
George Karypis and Vipin Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on scientific Computing, 20(1):359-315 1998.
Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. Tn Proceedings of the International Conference on Learning Representations, 2017.
Hao Li, Hao Wan, Yuzhou Chen, Dongsheng Ye, Yulia Gel, and Hao Jiang. TMetaNet: Topological meta-
learning framework for dynamic link prediction. In Forty-second International Conference on Machine Learning, 2025.
Shengrui Li, Xueting Han, and Jing Bai. AdapterGNN: Parameter-cfficient fino-tuning improves generaliza- tion in gnns. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pp. 13600-13608, 2024.
15

---

Published in Transactions on Machine Learning Research (03/2026)
Mingfei Liu, Xinyu Li, Jie Li, Yahui Liu, Bin Zhou, and Jinsong Bao. A knowledge graph-based data ion approach for iiot-enabled cognitive manufacturing. Advanced Engineering Informatics, 51: 101515, 2022a.
Xiang Liu, Huitao Feng, Jie Wu, and Kelin Xia. Dowker complex based machine learning (deml) models for protein-ligand binding affinity prediction. PLoS computational biology, 18(4):c1009943, 2022b.
Zemin Liu, Xingtong Yu, Yuan Fang, and Xinming Zhang. GraphPrompt:: Unifying pre-training and down- stream tasks for graph neural networks. In Web Conference, pp. 417-428, 2023.
Yunzhong Lou, Xucyang Li, Haotian Chen, and Xiangdong Zhou. Brep-bert: Pre-training boundary rep- resentation bert with sub-graph node contrastive learning. In Proceedings of the ACM International Conference on Information and Knowledge Management, pp. 1657-166661 -6666, 2023.
Yao Lu,, Xinyang Jiang, Yiqi Fang, and Chuan Shi. Learning to pre-train graph neural networks. In Proceedings of the AAAI Conference Artificial Intelligence, 2021.
Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. Geom-gen: Geometric graph convolutional networks. arXiv preprint ar Xiv::0002 20:22000.00887, 2020.
Yunsheng Shi, Zhengjie Huang, Shikun Feng, Hui Zhong, Wenjing Wang, and Yu Sun. Masked label predic- tion: Unified message passing model for semi-supervised classification. In Proceedings of the International Joint Conference on Artificial Intelligence, 2021.
Arseny Skryagin, Felix Divo, Mohammad Amin Ali, Devendra S Dhami, and Kristian Kersting, Graph neural networks need fusterrnorrmaliz--cctiva modules. Advances in Neural Information Processing Systems, 37:79819 79842, 2024.
Mingehen Sun, Kaixiong Zhou, Xin He, Ying Wang, and Xin Wang. GPPT: Graph pre-training and prompt tuning to generalize graph neural networks. In ACM SIGKDD, 2022.
Xiangguo Sun, Hong Cheng, Jia Li, Bo Liu, and Jihong Guan. All in One: Multi-task prompting for graph neural networks. In ACM SIGKDD, 2023a.
Xiangguo Sun, Jiawen Zhang, Xixi Wu, Hong Cheng, Yun Xiong, and Jia Li. Graph prompt learning: A comprehensive survey and beyond. arXio preprint urXin:2311. 166344,, 2023b.
Shantamu Thakoor, Corentin Tallec, Mohammad Gheshlaghi Azar, Mchdi Azabou, Eva L Dyer, Remi Munos, Petar Veličkovié, and Michal Valko. Large-scale representation learning on graphs via bootstrapping In International Conference on Learning Representations, 2022.
Zekun Tong, Yuxuan Liang, Changsheng Sun, Xinke Li, David Rosenblum, and Andrew Lim. Digraph inception convolutional networks. Advanees in Neural Information Processing Systems, 33:17907-17918, 2020a.
Zekun Tong, Yuxuan Liang, Changsheng Sun, David S Rosenblum, and Andrew Lim. Directed graph convolutional network. arXiv preprint arXim:200,,1397 2020b.
Zekun Tong, Yuxuan Liang, Henghui Ding, Yongxing Dai, Xinke Li, and Changhu Wang. Directed graph contrastive learning. Advances Neural Information Processing Systems, 34, 2021.
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. In Proceedings of the International Conference on Learning Representations, 2018.
Larry Wasserman. Topological data analysis. Arnual review of statistics and its application, 5(2018):501 532, 2018.
Keyulu Xu, Weihua Hu, Jure Leskovee, and Stefanie Jegelka. How powerful are graph neural networks? In Proceedings of the International Conference Learning Representations, 2019.
16

---

Published in Transactions on Machine Learning Research (03/2026)
Hengrui Zhang, Qitian Wu, Junchi Yan, David Wipf, and Philip S Yu. From canonical correlation analysis to self-supervieed graph neural networks. Advances in Newral Information Processing Systems, 34:76 89, 2021a.
Xitong Zhang, Yixuan He, Nathan Brugnone, Michael Perlmutter, and Matthew Hirn. MagNet: A Neural Network for Directed Graphs. In Advances in Neural Information Processing Systems, 2021b.
Qi Zhao and Yusu Wang. Learning metrics for persistence-based summaries and applications for graph classification. Advances in Neural Information Processing Systems, 32, 2019.
Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph Neural Networks: A Review of Methods and Applications. Al Open, 1: 57 57-81, 2020.
Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep graph contrastive represen- tation learming. ar preprint arXiv:2006.00 04131, 2020.
Chenyi Zi, Haihong Zhao, Xiangguo Sun, Yiqing Lin, Hong Cheng, and Jia Li. Prog: A graph prompt learning benchmark. Advances in Neural Information Processing Systems, 37:95406-954371 2024.
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A Datasets and Visualization
Here we provide more details of the DSBM model. A DSBM model is defined by the number of clusters, K, and edge probabilities, with edges being independently assigned given cluster memberships. In our experiments, the DSBM is characterized by a meta-graph adjacency matrix = version F = O,..,K and noise level 1/ S 0.5. The matrix F is derived from a meta-graph structure M. Here we do not have an ambient background as He et al. 2022b). We follow He et al. (2022b) to define four structures of F without any ambient nodes, where Il denotes the indicator function. In this study, we choose the number of clusters, K, the meta-graph structure, and the edge probability p. We set the number of nodes n to be 1000 and the (approximate) ratio, P, between the largest and the smallest cluster size, to be 1, and the direction flip parameter 1 = 0.05.
K-1, its filled
Cyccle" Fx.1- (1- [1 ((k: + 1) mod K)) 11 (1 ((k: -16-- 1) mod K)) + f1(( = k).
" Patth": Fx, = (1 1j) [(1 = k + 1) + n (1 = k - 1) + 2 ll (1 = k))
" "Complete": assign diagonal entries 1/4 For each pair (k; with ko < 1, let Fx,, be ry and 1 r with FR.
equal probability, then assign FI,e-
select the center node as w = K-1 and set Fx,1 = (1 n) = w.l odd) + 1 (k = w,l even)= + (1 - (1 w, l odd) + 7 = w.l even).
"Star":
For synthetic graph generation, in our experiments, we choose the number of clusters, K, the mela-graph structure, and the edge probability p. We set the number of nodes to be 1000 and the (approximate) ratio, P. between the largest and the smallest cluster size, to be 1, and the direction flip parameter 17 0.05. Our DSBM, which we denote by M(K,p,P) p) is built according to He et al. 2022bb. Due to p is set to 1, we simplify the notation to M(K,p) p) and M represents the structures, such as, " "Cycle", " "Path", " "Complete", and " "ttar". For each node v;- E Ck, and each node v, E C,, independently sample an edge from node ir to node " with probability p- Fx.I- Based on He et al. (2022b), we use the Ovol vol_ sum term with the naive pair selection method to compute an imbalance score lor the ground-truth labels for our datasets, in order to explain the level of flow imbalance related to directionality. Roughly speaking, the larger this imbalance score, the more directed this network is, with respect to its class labels.
Wasserstein distance (or matching distance) is defined as follows. Let DCD,(G+) and DC be Dowker complex- basod persistence diagrams of two graphs g+ and g (We omit the dimonsions in DC-PDs). Let DC = {q, JUA+ and DC = {पr JUA- where A= represents the diagonal (representing trivial cycles) with infinite multiplicity. Here, 9j (b)+ d E DCDg(G+) represents the birth and death times of a k-hole oy. Let Φ (Ct) + DCb, (9-) represent bijection (matching). With the existence of the diagonal A on both sides, we make sure of the existence of these bijections even if the cardinalities K(a; H and Ha. H are different. Then, th Wasserstein distance fw, defined (D, (G+), DCDs (G-))
min l1 - =o(at) )IBc) where p Zt. Here, the bottleneck distance is fw (DCDg(9 DCDg (gT) max, llat (Na))) Then, function yp is called stable d(pp*.p=) S C. C.w, (D,CDDD DD) DCDpyGT)) where s- is a vectorization of (CD,(G*) and d(...) is a suitable metric in the space of vectorizations. Here, the constant C > is independent of g+. This stability inequality interprets that as the changes in the vectorizations are bounded by the changes in DC-PDs.
Definition 2 (Dower Complex-Based Persistence Image). Let g R² + I be non-negative weight function for the persistence plane R. The value of each pizel E R2 is defined as DCpi(z)
(k-pe)2 (u-ny)) dydx, where T(DCDg) is the transformation of the DCDg (i.e., for each (z.y), T(z, y) (z.g- (e.y-))), 4 (Ma)Hl) E R22 and 8e and 8y are the standard deviotions of differentiable probahility distribution in the x and y directions, respecticely-
SI Eper(DC))))
s(H)
282
25020y
18

---

Published in Transactions on Machine Learning Research (03/2026)
B Computational Complexity
The computational complexity of persistent homology (PH) for persistence diagram is O((n + m))) where w 2.3728596. Dowker complex can reduce the complexity of computing 0- and I-dimensional features to O((n' + m²')") where n' denotes the number of landmark nodes, and m' is the number of edges in the subgraph induced by those landmarks. For the magnetic Laplacian module, the computational complexity is O( Ft-1 + |V| Ft) where Fi-1 denotes the embedding dimension of each node at the input of layer and Fg denotes the embedding dimension of each node at. the output of layer l. The complexity of CNN + pooling topological encoder is O(K p2 CCNN) where A. denotes the number of filtration functions, P denotes the resolution size of the persistence image, and CCNN denotes the interaction between input and output channels in standard convolution operations. We also compare the running time (training time per epoch; along with the accuracy (%)) between our TopoDIG model and runner-ups, i.e., DGCN, GCN+CNA, AdapterGNN, GPF (Fang et al., 2023), and GraphMAE2 (Hou et al. 2023). From Table 9 our TopoDIG consistently achieves competitive or faster training efficiency compared all baselines on small- scale benchmarks (i.e., Texas, Cornell, Wisconsin), the large-scale graph (i.e., ogbn-arxiv), and synthetic scttings (i.c., Cycle (4, 0.01) and Complete (5, 0.01)). Importantly, despite its favorable computational efficiency, TopoDIG also delivers consistently superior accuracy across all datasets (see Tables which indicates more effective trade-off between representational expressiveness and computational cost. These results demonstrate that the proposed TopoDIG enables improved direct link prediction performance without incurring additional training overhead, and in many cases, and offers both accuracy and efficiency gains over existing methods. Furthermore, for the ogbn-arxiv data, as shown in Table T0 the topological preprocessing requires 5.0 minutes for Dowker complex construction and 1.5 minutes for persistence image generation, and is performed once per graph and reused across runs. Training takes 25.30 seconds per epoch and full-graps inference requires 2.6 seconds. During topological preprocessing (Dowker complex construction and persistence image computation). the peak CPU memory usage is around 35 GB. During model training, the GPU memory peak is around 9.0 GB. The resulting topological features are compact and reusable, which requires around 0.5 GB.
Table 9: Running time analysis (seconds per epoch).
Model
Texas
Cornell
Wisconsin
ogbn-arxiv
Cycle (4, 0.01)
Complete (5, 0.01)
DGCN
1.38
0.50
0.32
34.43
0.69
0.58
GCN+CNA
0.82
0.29 s
0.18
21.60 s
0.40
0.35
AdapterGNN
0.96
0.34
0.22
24.80
0.48
0.42
GPF
1.10
0.40 s
0.26
28.90
0.56 s
0.49
GraphMAE2
1.25
0.29
31.70
0.62 s
0.54
TopoDIG (Ours)
1.04
0.35
0.21
25.30
0.45
0.38
Table 10: Running time analysis different modules within the TopoDIG on ogbn-arxiv.
Module
Dowker complex
Persistence image
Training
Inference
ogbn-arxiv
5.0 mins
1.5 mins
25.30 seconds
2.6 seconds
19


Reviews:
[{"review_id": "Z2nLE4q0Lq", "reviewer": "Reviewer_dTdv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose TopoDIG, a method for pretraining and prompting on directed graphs. The approach combines Magnetic Laplacian convolution (encoding edge directionality via complex phase shift) with topology-aware layers operating on persistence images from Dowker complex filtrations. The motivation—bringing persistent homology-style structure into directed graph representation learning—is interesting and timely. The paper evaluates TopoDIG on 3 WebKB datasets and synthetic Directed Stochastic Block Models (DSBMs).\n\nKey strengths:\n\n- Promising combination of directed GNNs (Magnetic Laplacian) with topological features (Dowker/persistent homology)\n- Timely motivation for incorporating topology into directed graph representation learning\n\nKey weaknesses:\n\n- Empirical validation is limited (few real datasets) relative to the breadth of the claims\n- Runtime/memory/scaling analysis is missing despite substantial computational components\n- Space allocation is imbalanced: the main paper spends too much space on non-novel background relative to the core contributions and experiments", "claims_and_evidence_yn": "No", "claims_explanation": "While the paper combines appealing ingredients (Magnetic Laplacian GNNs and Dowker complex-based persistent homology), the current evidence and presentation do not yet support the strongest claims:\n\n**Insufficient Experimental Validation**: Experiments are limited to WebKB (3 small citation networks) and synthetic DSBMs. For a paper positioning itself as a general advance in directed graph learning, broader evaluation on diverse directed graph domains (e.g., molecular, social, biological, traffic) is needed.\n\n**Missing Computational Cost Analysis**: The method includes computationally heavy components (Dowker complexes, persistence images, CNN processing, Magnetic Laplacian operations). Without runtime/memory comparisons and scaling results, it is hard to assess whether the reported accuracy gains justify the added complexity.\n\n**Methodological Clarity Issues**: Several technical details are unclear:\n\n- Integration of node features with the topological CNN is ambiguous (despite $X$ appearing in equation 6)\n- Please devote more space to clearly delineating pretraining vs. prompting: tasks/objectives used in each stage, what supervision is used (if any), and which parameters are updated or frozen\n- Equation 8 and the surrounding text do not make it clear whether the prompt function is novel or a straightforward application of existing prompting templates to the proposed encoder", "audience_yn": "Yes", "audience_explanation": "The intersection of topological data analysis, directed graphs, and transfer learning is relevant to TMLR's audience. Applying Dowker complexes to directed graph learning is novel and addresses an underexplored area. However, presentation and experimental validation need substantial improvement to realize the potential impact.", "requested_changes": "### Critical Changes (Required for Future Acceptance)\n\n1. **Expand Experimental Validation** (Critical): Current experiments are insufficient for claiming state-of-the-art performance:\n\n   - Add new datasets beyond WebKB (molecular, social, biological, transportation networks)\n\n2. **Provide Computational Cost Analysis** (Critical): Include runtime and memory comparisons:\n\n   - Wall-clock time for: (a) Dowker complexes, (b) persistence images, (c) training, (d) inference\n   - Memory requirements vs. baselines\n   - Scalability: performance degradation with graph size\n   - Justify when added complexity is worthwhile\n\n### Additional Changes\n\n3. **Restructure the Methods Section**: The Methods section extensively repeats existing work without highlighting novel contributions:\n\n   - Move detailed Magnetic Laplacian definitions to the appendix (unless diverging from MagNet)\n   - Condense the persistent homology stability discussion (well-known theory) or clearly state any paper-specific theoretical contribution; avoid spending substantial space on stability framing without a new result\n   - Focus on: (a) combining components, (b) novelty, (c) motivation\n   - Use saved space to expand experiments and clarify motivation\n\n4. **Strengthen the Related Work Section**: Several issues need addressing:\n\n   - Quantify DGCN's efficiency concerns and justify choosing MagNet as the directed baseline/starting point\n   - Clarify that MoAMa is tailored to molecular graphs, which limits how broadly its conclusions transfer\n   - Expand the graph prompting discussion: distinguish paradigms and explain which one you adopt and why\n   - Support the claim that prior approaches \"fall short\" with concrete examples (what directional/topological signals are missing, and why they matter)\n   - Add discussion of prior uses of Dowker complexes in machine learning\n   - Cover existing TDA work on directed graphs (even if limited)\n\n5. **Provide Use Case Analysis**: Given computational complexity, clarify:\n\n   - Which graph classes benefit most (small molecular vs. large social networks)?\n   - When to use TopoDIG versus simpler alternatives\n   - Which directed graph properties make topological features valuable", "broader_impact_concerns": "Broader Impact statement not included. No concerns", "additional_comments": "It is reasonable for this paper to focus primarily on methodology and empirical results rather than new theory. That said, the draft would benefit from a stronger experimental story and clearer positioning. In particular, if you claim to fill a significant gap in the literature, please motivate _why_ jointly modeling directionality and topology during pre-training should improve downstream performance in meaningful settings (and when it should not). The idea is promising, but the current experimental evidence is too limited to support definitive, broad claims.", "extra_scores": {}, "extra_text": {}}, {"review_id": "RkQnrdzhrC", "reviewer": "Reviewer_xW9S", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper aims to fuse topological deep learning with direct graph GNNs + a prompt engineering pre-train and then fine tune approach. Input graph is first subjected to two separate feed-forward graphs which are then concatenated and fed into an MLP. The first track consists of convolutions similar to MagNet (Zhang et al.) The second track relies on constructing a Dowker complex and feeding it through a topological learning representation module. The authors validate their method on Texas/Wisconsins/Cornell as well as on a synthetic Directed SBM model.\n\nWhile this paper shows promise in utilizing many different types of important information in the input graph, significant weaknesses remain. These include (i) computational complexity is not discussed; (ii) Section 3.4 provides insufficient detail; and it's not clear how the prompt engineering fits into the rest of the architecture. (Should it be in Figure 1 somewhere). This last issue is particularly concerning since that is one of the primary innovations of this work.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Model performance is validated on standard benchmarks. Model components are (mostly) well explained.", "audience_yn": "Yes", "audience_explanation": "TDL and pretrain/finetune are active and exciting areas of research", "requested_changes": "The claim the charge parameter controls the emphases on directionality is not quite true (except asymptotically as q->0). Instead, different q emphasize different digraph motifs. Please rephrase.\n\nThe first sentence of 3.1 implies that Zhang et al. introduced the magnetic Laplacian. This is false, they merely integrated it into the GNN framework. Please rephrase.\n\nIf definition 1, the delta should be an epsilon.\n\nSomething is wrong with the definitions of the Wasserstein distances, there is no minus on the RHS.\n\nMore details on prompting for downstream tasks should be given in Section 3.4 in order for the paper to be self contained.\n\nQuestions:\n\nIs there a way to get oriented simplices out of the original graph? This might eliminate the need for multiple tracks in the model?\n\nIs there a way to tell if a graph arises from some underlying space with interesting topology e.g., a manifold or degenerate manifold.\n\nIn section 3.2, how are the Gaussian kernels defined if the graph doesn't lie in a Euclidean space.\n\nNone of the experiments report the running time of the model. This is needed in order to determine whether or not the moderate performance increase is enough to justify increases in computational cost.\n\nMinor: \n\nAbstract: capture -> capturing\nPage 1: predicting modeling -> predicting and modeling\nBroken reference on page 2 and another one on page 10. Please check throughout.\nPage 11 \"shwon\"", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "g3xD0RrZak", "reviewer": "Reviewer_c9rd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the graph transfer learning on directed graphs. Specifically, the authors develop a new framework called TopoDIG which consists of several techniques, such as magnetic Laplacian, Dowker complex-based features. Extensive results showcase the effectiveness of TopoDIG for various downstream tasks.\nHowever, the datasets adopted in this paper are not mainstream graph benchmarks, which significantly weakens the technique contributions of this paper.", "claims_and_evidence_yn": "No", "claims_explanation": "The datasets are unsuitable for evaluating the model performance.", "audience_yn": "Yes", "audience_explanation": "Graph Representation learning is an interesting topic in the machine learning community.", "requested_changes": "1.There are some typos, for instance “, e.g., ? lists six different pre-training methods”. Please carefully proofread.\n\n2.More details of the framework should be highlighted in Figure 1.\n\n3.The complexity analysis of the proposed method is required.\n\n4.The real-world datasets are too small. Please select mainstream graph benchmarks, such as Computers, Photo, CS, Physics or datasets in OGB for performance evaluation [1].\n\n[1] DUALFormer: Dual Graph Transformer. ICLR 2025.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)