Laplacian Position Encoding (LPE)

• Laplacian Position Encoding is a spectral approach that uses graph Laplacian eigenvectors to capture spatial and temporal structure in node representations.
• It generalizes classical Laplacian eigenmaps by incorporating p-norm penalties, learnable spectral filters, and supra-Laplacian frameworks for dynamic graphs.
• Practical implementations employ efficient eigendecomposition techniques to significantly enhance performance in tasks like link prediction and graph analysis.

Laplacian Position Encoding (LPE) is a spectral approach for encoding spatial or spatio-temporal position information in graph learning architectures. It leverages eigenvectors of (normalized or unnormalized) graph Laplacians to provide node representations that capture graph structure beyond what is easily accessible via message-passing alone. As a general framework, LPE encompasses classical Laplacian eigenmaps, its generalizations to p-norm penalties, dynamic-graph extensions via supra-Laplacian constructions, and recent learnable parameterizations designed to adapt to homophilous and heterophilous settings.

1. Classical Laplacian Position Encoding for Static Graphs

Given an undirected graph $G = (V, E)$ with $n = |V|$ nodes, adjacency matrix $A \in \mathbb{R}^{n \times n}$, and degree matrix $D = \operatorname{diag}(A\cdot 1)$, the normalized Laplacian is defined as $L = I - D^{-1/2} A D^{-1/2}$. The standard Laplacian positional encoding seeks an embedding $X \in \mathbb{R}^{n\times k}$, where the columns of $X$ are the first $k$ nontrivial eigenvectors of $L$. This can be formulated as

$$\min_{X \in \mathbb{R}^{n \times k}} \operatorname{Tr}(X^T L X) \quad \text{s.t.} \quad X^T D X = I_k.$$

The resulting node embeddings $P_\mathrm{LPE} = U_k W$, where $U_k$ are the selected eigenvectors and $W$ is an optional learned projection, encode coarse-to-fine geometric structure: lower-frequency modes correspond to smooth, global variations; higher frequencies capture localized structure. These encodings strictly increase the expressive power of message-passing neural networks (MPNNs) beyond the 1-WL test when included as node features (Maskey et al., 2022).

2. Extensions to Generalized and Learnable Encodings

The classical 2-norm-based Laplacian embedding can be generalized by replacing the squared Euclidean metric with arbitrary $p$-norm or other dissimilarity functions:

$$\min_{X \in \mathbb{R}^{n \times k}} \sum_{i<j} a_{ij} \|X_i - X_j\|_p^p \quad \text{s.t.} \quad X^T X = I_k,$$

yielding p-Laplacian encodings. For $p \approx 2$, the representations are smooth and global; for $p \to 1$, embeddings become more piecewise-constant, accentuating partition-like features; $p \to \infty$ emphasizes maximal separations for encoding shortest-path information. Each regime targets different structural graph properties, enabling expressivity to be tuned for specific downstream tasks. Practical computation uses Riemannian optimization on the Stiefel manifold with a continuation strategy from $p=2$ downward (Maskey et al., 2022).

Recently, learnable LPEs (LLPEs) have been introduced to address the limitations of fixed-basis encodings, especially in heterophilous graphs where high-frequency Laplacian modes are often more informative. LLPEs parameterize the encoding using the full spectrum:

$$P_\mathrm{LLPE} = U W_\mathrm{LLPE}, \qquad [W_\mathrm{LLPE}]_{i,j} = h(\lambda_i; \theta_j),$$

where $h$ is a spectral filter expanded as a Chebyshev polynomial with coefficients $\theta_j$. This parameterization allows the model to adaptively attend to the most relevant spectral components, and can uniformly approximate a broad class of graph distances and community structures (Ito et al., 29 Apr 2025).

3. Supra-Laplacian Position Encoding for Dynamic Graphs

The “supra-Laplacian” framework generalizes LPE to discrete-time dynamic graphs $G = \{G_1,\ldots,G_T\}$, where each snapshot $G_t$ shares the same node set $V$ but has possibly time-varying edges. For a window of $w$ snapshots, a connected multi-layer graph is constructed via:

• Removing isolated nodes in each $G_t$
• Adding a single virtual node in each layer for inter-component connectivity
• Inserting temporal self-edges linking node $u$ in layer $t$ to its instance in $t{-}1$

The supra-adjacency matrix $\bar{A}$ has block structure with diagonals $A_{t-w+1},\ldots,A_t$ and off-diagonals as identity matrices encoding the temporal links. The normalized supra-Laplacian is

$$\bar{L} = I - \bar{D}^{-1/2} \bar{A} \bar{D}^{-1/2} \in \mathbb{R}^{N'\times N'},$$

where $N'$ is the node count after pruning and virtual node addition. By eigendecomposition,

$\bar{L} = \Phi \Lambda \Phi^\top,$

the bottom $k$ nontrivial eigenvectors $\{\phi_1, \ldots, \phi_k\}$ are used for positional encoding across space and time. Node $u$ at time $t$ is assigned coordinates $[\phi_1^{(u,t)},\ldots,\phi_k^{(u,t)}]$, jointly capturing spatio-temporal structure; the lowest mode ($\phi_1$) aligns with global temporal shifts, while higher modes localize to more specific patterns (Karmim et al., 26 Sep 2024).

4. Integration into Transformer Architectures

In the SLATE model, static node features $x_u$ are projected via a learned linear map $g_{\theta_E}$, then concatenated with the spatio-temporal positional encoding (SLE) generated by transforming the raw spectral coordinates with a small adapter $g_{\theta_{ST}}$. For all $N$ nodes across a $w$-snapshot window, these token representations $$z_{u,t} = g_{\theta_E}(x_u) \oplus \mathrm{SLE}_{u,t}$$ are collected into a sequence and input to a full-attention Transformer encoder. No further biasing of attention is required, as the spatial and temporal position information is directly embedded in the tokens. For dynamic link prediction, cross-attention modules use the output sequences of node pairs to produce edge embeddings, ensuring that global space-time context is available for scoring candidate links (Karmim et al., 26 Sep 2024).

5. Theoretical and Empirical Analysis

Supra-Laplacian encoding provides algorithmic and theoretical benefits over per-snapshot static LPE and simple temporal concatenation:

• It eliminates the need for separate spatial and temporal encodings, as the spectrum jointly describes the spatio-temporal geometry.
• By construction (removing isolated nodes, virtual nodes, temporal linking), the supra-graph is connected, avoiding degeneracies in the spectrum.
• The bottom $k$ eigenvectors of $\bar{L}$ minimize a regularized loss combining per-slice smoothness and inter-slice temporal smoothness:

$$\sum_{t=1}^T \operatorname{Tr}(X^{(t)T} L_t X^{(t)}) + \mu \sum_{t=2}^T \|X^{(t)} - X^{(t-1)}\|_F^2,$$

for appropriate choice of block decomposition (Galron et al., 2 Jun 2025). This formalizes the intuition that SLPEs align spatially smooth encodings over time while implementing temporal consistency.

Empirically, supra-Laplacian encodings (SLPE) have demonstrated:

• Significant performance gains in temporal link prediction, particularly with uninformative node features
• Robust improvement in both node property and link prediction tasks compared to static LPE
• Efficient computation with iterative solvers (e.g., LOBPCG), achieving up to $56\times$ speedups for large-scale graphs (up to $50,000$ active nodes) versus direct dense eigendecomposition
• Positive results predominantly for temporal graph architectures with weak raw feature informativeness; limited gains when temporal information is less predictive or node IDs alone are highly informative (Galron et al., 2 Jun 2025, Karmim et al., 26 Sep 2024)

6. Algorithmic and Practical Considerations

Computation of LPE and its supra-Laplacian variant involves eigendecomposition, which imposes nontrivial cost:

• For the static case, classical LPE requires $O(|E|\cdot k + k^3)$ for the $k$ bottom eigenpairs (typically via Lanczos or ARPACK).
• For the supra-Laplacian, the matrix size grows to $N = w|V|$, but approximate eigenvectors can be obtained efficiently using LOBPCG or trajectory concatenation.
• In practice, $k = 8$--$16$ is sufficient for encoding; larger $k$ yields diminishing returns (Maskey et al., 2022, Galron et al., 2 Jun 2025).
• Virtual node addition and explicit temporal edge construction guarantee spectral regularity and computational stability.
• Sign and basis ambiguities in eigenvectors necessitate sign-fixing protocols or sign/basis-invariant downstream architectures (e.g., via SignNet or stable PE processing).
• Window size $w$ should balance temporal context with computational tractability (e.g., $w = 3$--$5$) (Galron et al., 2 Jun 2025, Karmim et al., 26 Sep 2024).

7. Comparative Perspective and Limitations

The supra-Laplacian encoding unifies spatial and temporal information in a dynamic-graph context. Unlike static LPE—which must independently encode each snapshot and subsequently concatenate or otherwise combine spatial and temporal signals—SLPE operates in a unified spectral space, naturally capturing dynamic diffusion patterns and global modes. This results in improved performance across a diverse set of graph transformer and temporal-GNN architectures.

However, the method is not universally superior. Limitations include:

• High computational requirement for very large or dense graphs, although iterative algorithms mitigate this
• Potential inefficacy when temporal correlations are weak or node features are sufficiently informative on their own
• Need for careful architectural integration to handle sign/basis invariance and to select effective window size and embedding dimension

Despite these caveats, Laplacian Position Encoding—in both static and dynamic (supra-Laplacian) forms—constitutes a central tool for encoding structural and temporal position in modern graph learning systems (Maskey et al., 2022, Ito et al., 29 Apr 2025, Karmim et al., 26 Sep 2024, Galron et al., 2 Jun 2025).