Figure 2: These two graphs cannot be distinguished by 1-WL-test. The COMBINE step takes the result of AGGREGATE and the previous representation of current node asinput. Wereduce theFFN inner-layer dimension of4din [47] tod, which does not appreciably hurt the performance but significantly save the parameters. The embedding dropout ratio is set to 0.1 by default in many previous Transformer works[11,34]. The rest of hyper-parameters remain unchanged. Table 8 summarizes the hyper-parameters used for fine-tuning Graphormer on OGBGMolPCBA.

We thank all the reviewers for their constructive feedback. Below we provide specific responses to each reviewer. We will add more results in the paper. In the following response 2, we further highlight our important improvements ignored by existing work. The Method In Fig.1(e), Tables 4 and 5, S-GWL can be slightly worse than GWL on node correctness.

Abstract--V ehicular ad hoc networks (V ANETs) are a crucial component of intelligent transportation systems; however, routing remains challenging due to dynamic topologies, incomplete observations, and the limited resources of edge devices. Existing reinforcement learning (RL) approaches often assume fixed graph structures and require retraining when network conditions change, making them unsuitable for deployment on constrained hardware. We present TrajA ware, an RL-based framework designed for edge AI deployment in V ANETs. TrajA ware integrates three components: (i) action space pruning, which reduces redundant neighbour options while preserving two-hop reachability, alleviating the curse of dimensionality; (ii) graph cross-attention, which maps pruned neighbours to the global graph context, producing features that generalise across diverse network sizes; and (iii) trajectory-aware prediction, which uses historical routes and junction information to estimate real-time positions under partial observations. We evaluate TrajA ware in the open-source SUMO simulator using real-world city maps with a leave-one-city-out setup. Results show that TrajA ware achieves near-shortest paths and high delivery ratios while maintaining efficiency suitable for constrained edge devices, outperforming state-of-the-art baselines in both full and partial observation scenarios. OMMUNICA TION and routing are challenging in a vehicular ad hoc network (V ANET) [1], as vehicles can observe only part of the network, and the network's structure shifts rapidly; a previously obtained observation may soon become obsolete (as shown by Figure 1). Although compared to classical software algorithms, RL routing algorithms can potentially deal with more complex objectives (e.g., optimising delay while minimising the bandwidth overhead) [2], the problems of partial observation and network dynamics put a strain on the RL routing models. Several studies have shown that graph neural networks (GNNs) generalise better on routing tasks compared to other neural networks like multilayer perceptrons (MLPs) [3]-[7]. This work will be submitted to the IEEE for possible publication. Xiaolu Fu is an AI research engineer at Unicom Data Intelligence, China Unicom, Hangzhou, China (fuxl67@chinaunicom.cn), and a former student of the Computing Department, Imperial College London, London, UK (email: andy.fu23@alumni.imperial.ac.uk). Ziyuan Bao is an independent researcher and a former MSc student of the Computing Department, Imperial College London, London, UK (email: ziyuan.bao23@alumni.imperial.ac.uk).

Link prediction is one of the most fundamental tasks in graph mining, which motivates the recent studies of leveraging contrastive learning to enhance the performance. However, we observe two major weaknesses of these studies: i) the lack of theoretical analysis for contrastive learning on link prediction, and ii) inadequate consideration of node degrees in contrastive learning. To address the above weaknesses, we provide the first formal theoretical analysis for contrastive learning on link prediction, where our analysis results can generalize to the autoencoder-based link prediction models with contrastive learning. Motivated by our analysis results, we propose a new graph augmentation approach, Edge Balancing Augmentation (EBA), which adjusts the node degrees in the graph as the augmentation. We then propose a new approach, named Contrastive Link Prediction with Edge Balancing Augmentation (CoEBA), that integrates the proposed EBA and the proposed new contrastive losses to improve the model performance. We conduct experiments on 8 benchmark datasets. The results demonstrate that our proposed CoEBA significantly outperforms the other state-of-the-art link prediction models.

Graph Neural Networks (GNNs) have shown remarkable success in graph classification tasks by capturing both structural and feature-based representations. However, real-world graphs often exhibit two critical forms of imbalance: class imbalance and graph size imbalance. These imbalances can bias the learning process and degrade model performance. Existing methods typically address only one type of imbalance or incur high computational costs. In this work, we propose SamGoG, a sampling-based Graph-of-Graphs (GoG) learning framework that effectively mitigates both class and graph size imbalance. SamGoG constructs multiple GoGs through an efficient importance-based sampling mechanism and trains on them sequentially. This sampling mechanism incorporates the learnable pairwise similarity and adaptive GoG node degree to enhance edge homophily, thus improving downstream model quality. SamGoG can seamlessly integrate with various downstream GNNs, enabling their efficient adaptation for graph classification tasks. Extensive experiments on benchmark datasets demonstrate that SamGoG achieves state-of-the-art performance with up to a 15.66% accuracy improvement with 6.7$\times$ training acceleration.

These embeddings can be leveraged in various downstream tasks, including node classification, link prediction, clustering, exploratory data analysis, and network visualization. The method has found broad application across diverse domains, such as fraud detection in financial networks (van Belle et al. 2023), friendship recommendation and bot detection in social networks (Saxena et al. 2022; Dehghan et al. 2023), knowledge discovery in knowledge graphs (Egami et al. 2023; Liu et al. 2023), analysis of biological networks (Jiang et al. 2021; Pasquier et al. 2023), and fake review detection on online platforms (Zaki et al. 2024). A key challenge in Node Embedding is developing a scalable method for preserving the structural properties of nodes suitable for the required structural patterns of a downstream application task. The type of structural patterns in which a node is embedded within the graph can vary depending on the role or function of the node in a specific application task. For instance, fraudulent activities such as money laundering can be embedded in particular money flow patterns among illicit entities, resulting in characteristic structural patterns within the financial transaction network, such as suspicious cyclic transaction chains (Granados Vargas 2022). These structural patterns differ significantly from those observed in social networks, where specific roles such as bridge and core nodes define the network's connectivity and influence (Huang et al. 2014). As Node Embedding methods cannot preserve all types of structural patterns simultaneously, they must align with the requirements of a specific application task when defining types of structural identities.

Large data applications rely on storing data in massive, sparse graphs with millions to trillions of nodes. Graph-based methods, such as node prediction, aim for computational efficiency regardless of graph size. Techniques like localized approximate personalized page rank (APPR) solve sparse linear systems with complexity independent of graph size, but is in terms of the maximum node degree, which can be much larger in practice than the average node degree for real-world large graphs. In this paper, we consider an \emph{online subsampled APPR method}, where messages are intentionally dropped at random. We use tools from graph sparsifiers and matrix linear algebra to give approximation bounds on the graph's spectral properties ($O(1/\epsilon^2)$ edges), and node classification performance (added $O(n\epsilon)$ overhead).

Deep generative models have recently achieved significant success in modeling graph data, including dynamic graphs, where topology and features evolve over time. However, unlike in vision and natural language domains, evaluating generative models for dynamic graphs is challenging due to the difficulty of visualizing their output, making quantitative metrics essential. In this work, we develop a new quality metric for evaluating generative models of dynamic graphs. Current metrics for dynamic graphs typically involve discretizing the continuous-evolution of graphs into static snapshots and then applying conventional graph similarity measures. This approach has several limitations: (a) it models temporally related events as i.i.d. samples, failing to capture the non-uniform evolution of dynamic graphs; (b) it lacks a unified measure that is sensitive to both features and topology; (c) it fails to provide a scalar metric, requiring multiple metrics without clear superiority; and (d) it requires explicitly instantiating each static snapshot, leading to impractical runtime demands that hinder evaluation at scale. We propose a novel metric based on the \textit{Johnson-Lindenstrauss} lemma, applying random projections directly to dynamic graph data. This results in an expressive, scalar, and application-agnostic measure of dynamic graph similarity that overcomes the limitations of traditional methods. We also provide a comprehensive empirical evaluation of metrics for continuous-time dynamic graphs, demonstrating the effectiveness of our approach compared to existing methods. Our implementation is available at https://github.com/ryienh/jl-metric.

Link Prediction (LP) is a crucial problem in graph-structured data. Graph Neural Networks (GNNs) have gained prominence in LP, with Graph AutoEncoders (GAEs) being a notable representation. However, our empirical findings reveal that GAEs' LP performance suffers heavily from the long-tailed node degree distribution, i.e., low-degree nodes tend to exhibit inferior LP performance compared to high-degree nodes. \emph{What causes this degree-related bias, and how can it be mitigated?} In this study, we demonstrate that the norm of node embeddings learned by GAEs exhibits variation among nodes with different degrees, underscoring its central significance in influencing the final performance of LP. Specifically, embeddings with larger norms tend to guide the decoder towards predicting higher scores for positive links and lower scores for negative links, thereby contributing to superior performance. This observation motivates us to improve GAEs' LP performance on low-degree nodes by increasing their embedding norms, which can be implemented simply yet effectively by introducing additional self-loops into the training objective for low-degree nodes. This norm augmentation strategy can be seamlessly integrated into existing GAE methods with light computational cost. Extensive experiments on various datasets and GAE methods show the superior performance of norm-augmented GAEs.