An individual vertex is fully identified through its feature, which works as the vertex attribute. 's neighborhood, including both the internal connectivity information and contained vertex features Here we show more details about the graph datasets used in our experiments of both graph classification and vertex classification. Note that, three social network datasets, IMDB-B, IMDB-M and COLLAB do not provide specific vertex features, where the vertex dimension is denoted as 1 and the maximum vertex degrees are shown in addition. The detailed information of graph datasets used in the experiments of vertex classificationDataset Cora Citeseer Pubmed # V ertices 2708 3327 19717 # Edges 5429 4732 44338 # Classes 7 6 3 V ertex Dimension 1433 3703 500 # Train V ertices (full-sup.) The classification results are illustrated in Figure 1.

In the quest of accurate vertex classification, we introduce GRAVITY (Graph-based Representation leArning via Vertices Interaction TopologY), a framework inspired by physical systems where objects self-organize under attractive forces. GRAVITY models each vertex as exerting influence through learned interactions shaped by structural proximity and attribute similarity. These interactions induce a latent potential field in which vertices move toward energy efficient positions, coalescing around class-consistent attractors and distancing themselves from unrelated groups. Unlike traditional message-passing schemes with static neighborhoods, GRAVITY adaptively modulates the receptive field of each vertex based on a learned force function, enabling dynamic aggregation driven by context. This field-driven organization sharpens class boundaries and promotes semantic coherence within latent clusters. Experiments on real-world benchmarks show that GRAVITY yields competitive embeddings, excelling in both transductive and inductive vertex classification tasks.

Additional Feedback: This paper proposes GXN for modeling graph data. Overall, the proposed approach is quite intuitive, and extensive experiment on graph classification and vertex classification proves the effectiveness of the approach. I have the following concerns regarding the paper: 1. The relation to existing graph pooling techniques is not clear. One key component of GXN is vertex informax pooling, which is used to create multi-scale graphs and is thus related to existing graph pooling techniques.

This paper introduces a refined graph encoder embedding method, enhancing the original graph encoder embedding using linear transformation, self-training, and hidden community recovery within observed communities. We provide the theoretical rationale for the refinement procedure, demonstrating how and why our proposed method can effectively identify useful hidden communities via stochastic block models, and how the refinement method leads to improved vertex embedding and better decision boundaries for subsequent vertex classification. The efficacy of our approach is validated through a collection of simulated and real-world graph data.

This information can be captured by an n n adjacency matrix A, where A(i, j) = 1 indicates the existence of an edge between vertices i and j, and A(i, j) = 0 indicates the absence of an edge. Traditionally, the adjacency matrix is binary and primarily used for representing unweighted graphs. Its capacity extends to handling weighted graphs, where the elements of A are assigned the corresponding edge weights. Moreover, graph representation can be used for any data via proper transformations. For instance, when vertex attributes are available, they can be transformed into a pairwise distance or kernel matrix, thus creating a general graph structure. This adaptability enables the incorporation of additional information from vertex attributes, making graph representation a powerful tool for data analysis that goes beyond traditional binary or weighted graphs. Graph embedding is a fundamental and versatile approach for analyzing and exploring graph data, encompassing a wide range of techniques such as spectral embedding [7, 8], graph convolutional neural networks [9, 10], node2vec [11, 12], among others. By projecting the vertices into a low-dimensional space while preserving the structural information of the graph, graph embedding yields vertex representations in Euclidean space that facilitate various downstream inference tasks, such as community detection [13, 14], vertex classification [15, 9], outlier detection [16, 17], etc.

In this paper we propose a lightning fast graph embedding method called graph encoder embedding. The proposed method has a linear computational complexity and the capacity to process billions of edges within minutes on standard PC -- an unattainable feat for any existing graph embedding method. The speedup is achieved without sacrificing embedding performance: the encoder embedding performs as good as, and can be viewed as a transformation of the more costly spectral embedding. The encoder embedding is applicable to either adjacency matrix or graph Laplacian, and is theoretically sound, i.e., under stochastic block model or random dot product graph, the graph encoder embedding asymptotically converges to the block probability or latent positions, and is approximately normally distributed. We showcase three important applications: vertex classification, vertex clustering, and graph bootstrap; and the embedding performance is evaluated via a comprehensive set of synthetic and real data. In every case, the graph encoder embedding exhibits unrivalled computational advantages while delivering excellent numerical performance.