Lorentzian Graph Convolutional Networks

• GCNs use feature transformation and neighborhood aggregation to learn higher-level node features from local neighborhoods.Source: §1 · p. 1
• Hyperbolic geometry is pursued because it can embed scale-free or hierarchical graphs more powerfully than Euclidean geometry.Source: §1 · p. 1
• Existing hyperbolic GCN operations may leave node features off the hyperboloid and aggregate in tangent spaces rather than according to hyperbolic-space meanings.Source: §1 · pp. 1–2
• LGCN builds a unified hyperboloid-model framework with Lorentzian feature transformation, nonlinear activation, and centroid-based neighborhood aggregation.Source: §1 · p. 2
• Experiments report LGCN superiority in link prediction and node classification, lower tree-like-graph distortion, and gains from replacing operations for molecular-property prediction.Source: §1 · p. 2

2.1 Graph neural networks

• Graph neural networks include spectral convolution methods and spatial approaches such as gated propagation, attention, sampling, and dynamic-edge modeling.Source: §2.1 · p. 2

2.2 Hyperbolic graph representation learning

• Hyperbolic representation work addresses hierarchical structures and extends neural models, attention, and graph convolution to hyperbolic spaces, including HGCN and HAT [60].Source: §2.2 · p. 2

3.1 Hyperbolic geometry

• The hyperboloid model represents hyperbolic geometry as a manifold with constant negative curvature and a Lorentzian metric on tangent spaces.Source: §3.1 · pp. 2–3
• Exponential and logarithmic maps connect points on the hyperboloid with tangent-space vectors, while H(·, ·) denotes intrinsic distance.Source: §3.1 · p. 3

3.2 Hyperbolic graph convolutional networks

• HGCN message passing includes feature transformation and neighborhood aggregation, with features mapped from hyperbolic space into tangent spaces.Source: §3.2 · p. 3
• The described transformation violates the tangent-space Lorentzian constraint, so projected node features can fall outside hyperbolic space.Source: §3.2 · p. 3
• Neighborhood aggregation is framed as a weighted mean or geometric centroid, but HGCN performs it in tangent spaces, potentially distorting features.Source: §3.2 · p. 3

4 LGCN: OUR PROPOSED MODEL

• LGCN combines feature transformation, centroid-based aggregation with self-attention weights, and Lorentzian pointwise activation to produce output node features.The curvature is a trainable parameter; the paper notes that adjusting it matters in practice given machine precision and normalization.Source: §4 · p. 3

4.1 Mapping feature with different curvature

• Euclidean input features are mapped into hyperbolic space, while hyperbolic inputs with a different curvature are transformed to the current model’s curvature.The paper also describes mapping Euclidean features from the tangent space at the origin using the exponential map.Source: §4.1 · p. 3

4.2 Lorentzian feature transformation

• Using logarithmic and exponential maps, Lorentzian transformation applies Euclidean maps only to the last n tangent coordinates and fixes the first at 0.Lorentzian matrix-vector multiplication is equivalent to Möbius multiplication in the Poincaré ball.Source: §4.2 · p. 4
• Lorentzian matrix-vector multiplication uses an m×n matrix for an n-to-m feature transformation, unlike the (m+1)×(n+1) matrices used by other methods.The paper says those other matrix sizes can violate the tangent-space constraint.Source: §4.2 · p. 4

4.3 Lorentzian neighborhood aggregation

• LGCN aggregates neighbors using a centroid of squared Lorentzian distance with a closed-form solution, while attention weights are softmax-normalized.Unlike tangent-space aggregation, this centroid is defined in the hyperboloid model; the weights include the node itself.Source: §4.3 · pp. 4–5

4.4 Lorentzian pointwise non-linear activation

• Lorentzian pointwise non-linearity keeps transformed features in hyperbolic space and is equivalent to Möbius pointwise non-linearity in the Poincaré ball.The equivalence is stated for specific activations such as ReLU and leaky ReLU.Source: §4.4 · p. 5

4.5.1 Hyperbolic graph operations.

• LGCN is the only compared hyperbolic GCN satisfying all three operation properties, including aggregation as an expectation in hyperbolic space.The properties are feature transformation and non-linearity staying inside hyperbolic space, and expectation-based aggregation there.Source: §4.5.1 · p. 6

4.5.2 Hyperbolic centroids.

• Only the Lorentzian centroid both minimizes summed squared distances and has a closed-form solution; the paper is first to use it in a hyperbolic GCN.Fréchet means lack a closed form, while Einstein and Möbius gyromidpoints do not minimize summed squared distances.Source: §4.5.2 · p. 6

5.2 Link prediction

• LGCN performs best in all reported link-prediction cases, with a more significant advantage at lower latent dimensions.The comparison uses AUC across the evaluated datasets.Source: Table 4 · p. 7, §5.2 · p. 7

5.3 Node classification

• LGCN outperforms baselines in most node-classification cases; hyperbolic GCNs show no obvious advantage over Euclidean GCNs on Citeseer.Citeseer has the largest δ_avg among the datasets considered.Source: Table 5 · p. 8, §5.3 · p. 8

5.4.1 Ablations study.

• LGCN variants with self-attention and trainable curvature perform best, while centroid-based aggregation improves HGCN and usually benefits LGCN without attention.The ablations compare link prediction and node classification on Disease, USA, and Amazon.Source: §5.4.1 · p. 8, Table 6 · p. 9

5.4.2 Extension to graph-level task: molecular property prediction.

• On ZINC molecular-property prediction, HGNN_L achieves the best MAE among baselines and improves on HGNN after replacing its graph operations with Lorentzian versions.The reported targets are logP, QED, and SAS; the metrics are scaled by 100.Source: §5.4.2 · p. 8, Table 7 · p. 9

5.4.4 Attention analysis.

• LGCN has running time comparable to several hyperbolic GCNs and GAT, while LGCN_F is slowest; its slight performance improvement is not cost-effective.LGCN uses a closed-form aggregation, whereas LGCN_F's Fréchet-mean aggregation is computed with gradient descent.Source: Figure 4 · p. 9, §5.4.5 · p. 9

5.4.3 Distortion analysis.

• LGCN has the lowest average distortion among the evaluated GCNs on Disease, while all evaluated hyperbolic GCNs have lower distortion than Euclidean GCNs.Lower average distortion indicates better preservation of graph structure.Source: Figure 2 · p. 9, §5.4.3 · p. 9

5.4.4 Attention analysis.

• In the Disease example, node 3 assigns higher attention to same-class nodes 0, 3, 14, 15, and 16.Node labels are shown by color and attention values by edge width.Source: Figure 3 · p. 9, §5.4.4 · p. 9

5.4.5 Efficiency comparison.

• Link-prediction performance rises and then declines slowly as attention-matrix dimension grows, remaining stable for dimensions from 32 to 128.The tested horizontal dimension ranges from 4 to 256.Source: Figure 5 · p. 9, §5.4.6 · p. 9

6 CONCLUSION

• LGCN uses rigorous hyperbolic graph operations and neighborhood aggregation, and extensive experiments demonstrate superiority over state-of-the-art methods.The conclusion identifies feature transformation and non-linear activation as examples of the operations it rebuilds.Source: §6 · p. 10

A HYPERBOLIC GEOMETRY

• Hyperbolic geometry has constant negative curvature and can be described using equivalent models, including the hyperboloid and Poincaré ball.The passage also names the Klein ball model.Source: A HYPERBOLIC GEOMETRY · p. 11
• The Poincaré ball and hyperboloid models are isomorphic, with a diffeomorphism mapping one space onto the other.The passage points to for this mapping.Source: A HYPERBOLIC GEOMETRY · p. 11

B.1 Proof of Theorem 3.1

• The proof maps the feature through the logarithmic map at the origin before applying Lorentzian matrix-vector multiplication.It then considers the resulting feature in the Poincaré ball representation.Source: §B.1 · p. 11
• The equivalence uses matching curvature parameters for the Poincaré ball and hyperboloid models.The passage states α = 1/β for the two models to have the same curvature.Source: §B.1 · p. 11
• Lorentzian matrix-vector multiplication is equivalent to Möbius matrix-vector multiplication.The proof considers a shared m×n matrix and maps between the hyperboloid and Poincaré ball models.Source: §B.1 · p. 11

B.2 Proof of Theorem 3.2

• The proof combines Eqs. (15) and (17) with the hyperboloid condition that a point's Lorentzian self-product equals −β.The passage presents this as the constraint for the point under consideration.Source: §B.2 · p. 12

B.3 Proof of Theorem 3.3

• The proof demonstrates the activation equivalence using LeakyReLU as its example.It specifies the function as σ(v_i) = max(kv_i, v_i), with k in (0, 1).Source: §B.3 · p. 12
• For Möbius activation, features are mapped to the tangent space at the origin before applying the activation function.The proof uses the scaling relation for LeakyReLU to rewrite this operation.Source: §B.3 · p. 12
• The proof establishes that Lorentzian pointwise nonlinear activation is equivalent to Möbius pointwise nonlinear activation.The Lorentzian activated feature is projected into the Poincaré ball representation.Source: §B.3 · p. 12

C.1 𝛿-hyperbolicity

• For four distinct nodes, δ+ is half the difference between the largest and middle sums of paired graph distances.Worst-case hyperbolicity maximizes δ+ over quadruples; average hyperbolicity averages it across all four-node combinations.Source: §C.1 · p. 12
• Both worst-case and average hyperbolicity have O(|V|^4) time complexity, while worst-case hyperbolicity focuses on a local quadruple and does not reflect the whole graph.Source: §C.1 · p. 12
• The authors use average hyperbolicity because it is robust to adding or removing an edge and can be approximated by sampling, unlike worst-case hyperbolicity.Source: §C.1 · p. 12