Bitcoin transaction networks are large scale socio- technical systems in which activities are represented through multi-hop interaction patterns. Graph Neural Networks(GNNs) have become a widely adopted tool for analyzing such systems, supporting tasks such as entity detection and transaction classification. Large-scale datasets like Elliptic have allowed for a rise in the analysis of these systems and in tasks such as fraud detection. In these settings, the amount of transactional context available to each node is determined by the neighborhood aggregation and sampling strategies, yet the interaction between these receptive fields and embedding geometry has received limited attention. In this work, we conduct a controlled comparison of Euclidean and tangent-space hyperbolic GNNs for node classification on a large Bitcoin transaction graph. By explicitly varying the neighborhood while keeping the model architecture and dimensionality fixed, we analyze the differences in two embedding spaces. We further examine optimization behavior and observe that joint selection of learning rate and curvature plays a critical role in stabilizing high-dimensional hyperbolic embeddings. Overall, our findings provide practical insights into the role of embedding geometry and neighborhood depth when modeling large-scale transaction networks, informing the deployment of hyperbolic GNNs for computational social systems.
Figures & tables
Dataset
Models
Metrics
Geometry
Paper
New BTC Graph
Elliptic
GCN
GraphSAGE
GAT
HyperGNN
Precision
Recall
F1
Euclidean
Hyperbolic
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TABLE I: Comparison of prior work by dataset, models, metrics, and embedding geometry.
Fig. 1: Workflow for Bitcoin Address Classification.
Fig. 2: HyperbolicSAGE architecture showing the computational flow across layers.
Layers
Metric
GCN
HGCN
GraphSAGE
HGraphSAGE
GAT
HGAT
2-layer
Precision
0.68
0.67
0.81
0.82
0.79
0.82
Recall
0.49
0.55
0.67
0.71
0.65
0.71
Macro-F1
0.55
0.59
0.72
0.76
0.71
0.75
3-layer
Precision
0.67
0.75
0.82
0.87
0.79
0.86
Recall
0.54
0.57
0.67
0.75
0.69
0.72
Macro-F1
0.59
0.64
0.73
0.80
0.73
0.78
TABLE II: Node classification performance on depth-two subgraphs (macro-averaged). Columns correspond to different GNN architectures; rows indicate model depth and evaluation metric. The highest value in each row is highlighted in bold.
Layers
Metric
GCN
HGCN
GraphSAGE
HGraphSAGE
GAT
HGAT
2-layer
Precision
0.68
0.74
0.84
0.83
0.77
0.85
Recall
0.52
0.58
0.72
0.71
0.70
0.71
Macro-F1
0.58
0.64
0.77
0.76
0.73
0.77
3-layer
Precision
0.71
0.74
0.84
0.88
0.85
0.86
Recall
0.60
0.63
0.73
0.76
0.73
0.77
Macro-F1
0.64
0.68
0.78
0.81
0.78
0.81
TABLE III: Node classification performance on depth-three subgraphs (macro-averaged). Columns correspond to different GNN architectures; rows indicate model depth and evaluation metric. The highest value in each row is highlighted in bold.
Class
Metric
GCN
HGCN
GraphSAGE
HGraphSAGE
GAT
HGAT
BET
Precision
0.96
0.96
0.97
0.98
0.97
0.98
Recall
0.95
0.96
0.99
0.99
0.99
0.99
F1
0.95
0.96
0.98
0.99
0.98
0.99
EXCHANGE
Precision
0.73
0.67
0.83
0.85
0.84
0.82
Recall
0.40
0.53
0.62
0.67
0.63
0.70
F1
0.51
0.59
0.71
0.75
0.72
0.76
TABLE IV: Per-class node classification performance (macro-averaged metrics) for selected Euclidean and hyperbolic GNNs. Bold indicates the best value per class; ties are all bolded.
Fig. 3: Macro-F1 versus learning rate for different curvature ( c ) in tangent-space Hyperbolic GraphSAGE. Low curvature values at low to moderate learning rates produce large regions of convergent but uninformative solutions. Moderate to high curvature values achieve effective performance across a wider learning-rate range, highlighting the importance of joint learning-rate and curvature selection.
The consensus that GCN, GraphSAGE, GAT, and EvolveGCN outperform feature-only baselines on the Elliptic Bitcoin Dataset is widely cited but has not been rigorously stress-tested under a leakage-free evaluation protocol. We perform a seed-matched inductive-versus-transductive comparison and find that this consensus does not hold. Under a strictly inductive protocol, Random Forest on raw features achieves F1 = 0.821 and outperforms all evaluated GNNs, while GraphSAGE reaches F1 = 0.689 +/- 0.017. A paired controlled experiment reveals a 39.5-point F1 gap attributable to training-time exposure to test-period adjacency. Additionally, edge-shuffle ablations show that randomly wired graphs outperform the real transaction graph, indicating that the dataset's topology can be misleading under temporal distribution shift. Hybrid models combining GNN embeddings with raw features provide only marginal gains and remain substantially below feature-only baselines. We release code, checkpoints, and a strict-inductive protocol to enable reproducible, leakage-free evaluation.
Hyperbolic spaces provide a natural geometry for representing hierarchical and tree-structured data due to their exponential volume growth. To leverage these benefits, neural networks require intrinsic and efficient components that operate directly in hyperbolic space. In this work, we lift two core components of neural networks, Multinomial Logistic Regression (MLR) and Fully Connected (FC) layers, into hyperbolic space via Busemann functions, resulting in Busemann MLR (BMLR) and Busemann FC (BFC) layers with a unified mathematical interpretation. BMLR provides compact parameters, a point-to-horosphere distance interpretation, batch-efficient computation, and a Euclidean limit, while BFC generalizes FC and activation layers with comparable complexity. Experiments on image classification, genome sequence learning, node classification, and link prediction demonstrate improvements in effectiveness and efficiency over prior hyperbolic layers. The code is available at https://github.com/GitZH-Chen/HBNN.
Hyperbolic geometry has emerged as an effective latent space for representing complex networks, owing to its ability to capture hierarchical organization and heterogeneous connectivity patterns using low-dimensional embeddings. As a result, numerous hyperbolic graph representation learning methods have been proposed in recent years. However, their practical adoption and systematic comparison remain challenging, as implementations are fragmented and shared tools for reproducible and fair evaluation are lacking. In this work, we introduce a unified open-source framework for hyperbolic graph representation learning that integrates several widely used embedding methods under a common optimization interface. The novel framework enables consistent training, visualization, and evaluation of hyperbolic embeddings, and interfaces seamlessly with standard network analysis tools. Leveraging this unified setup, we conduct an experimental study of hyperbolic embedding methods on real-world networks, focusing on two canonical downstream tasks: link prediction and node classification. Beyond predictive accuracy, the study offers practical insights into the strengths and limitations of existing approaches, thereby facilitating informed method selection and fostering reproducible research in hyperbolic graph representation learning.
Facultad de Ingenier´ıa, Universidad de la Rep´ublica, Uruguay · Centro Interdisciplinario en Ciencia de Datos y Aprendizaje Autom´atico (CICADA), Universidad de la Rep´ublica, Uruguay