LaplaceGNN: Laplacian-Enhanced GCN

• LaplaceGNN is a graph neural network architecture that integrates graph Laplacian principles to enforce local invariance in node representations.
• It combines standard GCN propagation with an explicit Laplacian regularization term applied to both features and labels.
• Empirical results on benchmark citation datasets show improved semi-supervised classification, especially in low-label scenarios.

LaplaceGNN refers to a class of graph neural network architectures that integrate the graph Laplacian and Laplacian regularization principles directly into their learning objectives and propagation mechanisms, with the aim of achieving robust, locally consistent representations for semi-supervised node classification on graphs. The foundational model discussed, termed gLGCN (“graph Laplacian GCN”), was introduced to explicitly address the local invariance constraint often neglected by standard GCNs, thus providing improved performance and robustness in semi-supervised learning contexts.

1. Foundations and Motivation

Traditional graph convolutional networks (GCNs) learn node representations using both node features and the structure of the input graph. However, classic GCNs typically lack an explicit mechanism for enforcing the local invariance constraint: if two nodes are close in the intrinsic geometry of the data (either in feature space or graph topology), their learned representations and predicted labels should also be similar. This local invariance principle is vital in manifold learning and semi-supervised learning, and is commonly operationalized through Laplacian regularization—penalizing large differences in embeddings or labels between neighboring nodes.

gLGCN (LaplaceGNN) explicitly incorporates this principle, bridging classical manifold regularization approaches and deep learning-based GCNs.

2. Local Invariance and Laplacian Regularization

The central regularization term in LaplaceGNN is formulated as:

$$\mathcal{L}_{\mathrm{reg}} = \sum_{i,j=1}^n S_{ij} \|f_i - f_j\|^2$$

Here, $S_{ij}$ is a similarity measure between nodes $i$ and $j$, and $f_i$ denotes either the label prediction vector $Z_i$ (for output regularization) or the latent feature vector $X^{(K)}_i$ (for embedding regularization). The core effect of this term is to make representations smooth over the graph structure, such that similar or connected nodes have similar outputs or embeddings. This regularization can be applied:

• In the label prediction space (“gLGCN-L”).
• In the feature representation space (“gLGCN-F”).
• Or jointly in both (“gLGCN-F-L”).

The similarity matrix $S$ can be chosen based on adjacency, feature similarity, or a combination thereof.

3. Model Architecture and Training Objective

The LaplaceGNN extends the multilayer GCN architecture by adding Laplacian-inspired regularization to its loss function. The forward propagation at each layer remains similar to standard GCNs:

$$\begin{align*} & X^{(1)} = \mathrm{ReLU}(\widetilde{A} X W^{(0)}) \ & X^{(k)} = \mathrm{ReLU}(\widetilde{A} X^{(k-1)} W^{(k-1)}), \quad k=2,\ldots,K \ & Z = \mathrm{softmax}(\widetilde{A} X^{(K)} W^{(K)}) \end{align*}$$

where $\widetilde{A}$ is the normalized adjacency matrix.

The training objective combines:

• The cross-entropy loss on labeled nodes.

$$\mathcal{L}_{\mathrm{GCN}} = -\sum_{i \in L} \sum_{j=1}^{d} Y_{ij} \log Z_{ij}$$

• The Laplacian regularization, with a weighted sum over node pairs:

For label prediction:

$$\mathcal{L}_\text{gLGCN-L}(Z) = \mathcal{L}_{\mathrm{GCN}}(Z) + \lambda \sum_{i, j} S_{ij} \|Z_i - Z_j\|^2$$

For feature representations:

$$\mathcal{L}_\text{gLGCN-F}(X^{(K)}) = \mathcal{L}_{\mathrm{GCN}}(Z) + \lambda \sum_{i, j} S_{ij} \|X^{(K)}_i - X^{(K)}_j\|^2$$

Or jointly:

$$\mathcal{L} = \mathcal{L}_{\mathrm{GCN}}(Z) + \lambda\,\mathcal{L}_{\mathrm{reg}}(X^{(K)})$$

This framework can also include a correlation-based regularization using known labels through a matrix $C_{ij}$, encoding agreement or disagreement between labeled node pairs.

4. Semi-supervised Classification with LaplaceGNN

Semi-supervised node classification is targeted by computing the cross-entropy loss only over the labeled subset of nodes ($L$), while using Laplacian regularization to propagate label information and feature smoothness across the entire graph. In practice, this means that unlabeled nodes receive indirect guidance via their similarity to labeled neighbors, mediated by the Laplacian term.

LaplaceGNN is particularly effective in settings where labeled nodes are sparse, as it utilizes local consistency regularization to propagate supervision effectively throughout the graph. This provides an advantage over purely supervised or naive GCN baselines when label scarcity is an issue.

5. Empirical Performance and Benchmarking

LaplaceGNN was evaluated on standard citation network datasets (Citeseer, Cora, Pubmed) for node classification under low label rates:

Method	Citeseer	Cora	Pubmed
ManiReg	60.1	59.5	70.7
SemiEmb	59.6	59.0	71.1
LP	45.3	68.0	63.0
Planetoid	64.7	75.7	77.2
GCN	70.4	81.4	78.6
gLGCN-F	70.8	82.2	79.2
gLGCN-L	71.3	82.7	79.2
gLGCN-F-L	71.4	83.3	79.3

Both label and feature space regularization (gLGCN-F-L) produces the best performance, but improvements are also apparent when only one space is regularized. LaplaceGNN outperforms or matches high-performing baselines, especially as the proportion of labeled nodes decreases.

6. Significance, Limitations, and Future Directions

LaplaceGNN demonstrates the practical benefit of fusing classical graph Laplacian regularization—emphasizing local invariance—with modern graph convolutional neural architectures. The result is a hybrid model that better aligns with manifold learning principles, leading to superior performance in data-scarce regimes and greater robustness to label sparsity.

Potential future research avenues include:

• Extending Laplacian regularization to other tasks such as link prediction or generalizing to inductive and dynamic graph settings.
• Investigating more adaptive or learned similarity/correlation matrices for constructing the Laplacian regularizer.
• Scaling LaplaceGNN to very large or heterogeneous graphs, possibly with further architectural or algorithmic innovation for efficiency.

LaplaceGNN thus serves as a blueprint for integrating local consistency principles directly into the learning dynamics of graph neural networks, combining expressive representation power with robustness and local smoothness properties necessary for effective semi-supervised learning.