Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding

Abstract

Standard Adjacency Spectral Embedding (ASE) relies on a global low-rank assumption often incompatible with the sparse, transitive structure of real-world networks, causing local geometric features to be 'smeared'. To address this, we introduce Local Adjacency Spectral Embedding (LASE), which uncovers locally low-dimensional structure via weighted spectral decomposition. Under a latent position model with a kernel feature map, we treat the image of latent positions as a locally low-dimensional set in infinite-dimensional feature space. We establish finite-sample bounds quantifying the trade-off between the statistical cost of localisation and the reduced truncation error achieved by targeting a locally low-dimensional region of the embedding. Furthermore, we prove that sufficient localisation induces rapid spectral decay and the emergence of a distinct spectral gap, theoretically justifying low-dimensional local embeddings. Experiments on synthetic and real networks show that LASE improves local reconstruction and visualisation over global and subgraph baselines, and we introduce UMAP-LASE for assembling overlapping local embeddings into high-fidelity global visualisations.

Figures (10)

• Figure 1: Left: the top 100 eigenvalues of the adjacency matrix, $\mathbf{A}$, of the Bristol Road Network (see Section \ref{['sec:road_network']} for details of the dataset), where $n = 3857$. Right: the top 20 eigenvalues of $\mathbf{W}^{1/2}\mathbf{A}\mathbf{W}^{1/2}$, where $\mathbf{W}$ is a diagonal matrix of weights $w_1, \ldots, w_n > 0$ selected to 'localise' LASE (Algorithm \ref{['alg:lase']}) around a node of interest, $z$, based on the graph distance from $z$ (see Section \ref{['weights_computation']} - graph-distance-based weighting).
• Figure 2: First two dimensions of the Mercer feature map $\phi$ (leftmost column) and the locally weighted feature map $\phi_w$ for increasing localisation (subsequent columns), controlled by the concentration parameter $\tau$. We use $\mu = \text{Uniform}[0,10]$ and $f(x,y)= \exp(-\tfrac{1}{2} (x-y)^2)$. Top row: localisation around $z \approx 1.5$. Bottom row: localisation around $z \approx 7.5$. In each row, $z^\star$ is indicated by a red circle and latent positions $z^\star \pm 0.5$ are indicated by red crosses. As $\tau$ increases, $\phi_w$ “zooms in” and better represents inner products for points likely under $\mu_w$. All plots align $\phi_w(z^\star)$ to $\phi(z^\star)$ using Procrustes analysis.
• Figure 3: The top 6 eigenvalues of $\mathbf{P}$ are plotted as the measure $\mu_w$ concentrates about a point. In each plot, the dimension $d$ of the latent space $\mathcal{Z}$ is identified in the plot subtitle. Notice the distinct change in rate of decay between eigenvalues $d+1$ and $d+2$.
• Figure 4: Comparison of reconstruction accuracy under different LASE weighting strategies. Top left: Continuous weighting functions $w(Z_i)$ with varying concentration parameters $\tau \in \{1, \dots, 10\}$, centered at $x = 4$. Top right: RMSE of reconstructed probabilities $\hat{\mathbf{P}}_{ij}$ versus true $\mathbf{P}_{ij}$, computed over nodes with latent positions in $[3.5, 4.5]$, for each $\tau$. Bottom left: Top-hat weighting functions of varying width, also centered at $x = 4$, corresponding to subgraph ASE on different-sized subgraphs. Bottom right: RMSE of reconstructed probabilities for each top-hat width. Shaded regions indicate $\pm 1$ SE over 10 sampled graphs.
• Figure 5: Column 1: Latent positions within radius 1 of a central point, $z^\star$. We call this the neighbourhood of $z^\star$, and represent it by $\mathcal{N}_{z^\star}$. Column 2: ASE is performed on the full graph into 3D, followed by PCA into 2D on embeddings of nodes in $\mathcal{N}_{z^\star}$. Column 3: Subgraph SE is performed into 3D on the subgraph consisting of at least the nodes in $\mathcal{N}_{z^\star}$, followed by PCA into 2D on embeddings of nodes in $\mathcal{N}_{z^\star}$. Column 4: LASE is performed into 3D on the full graph, with weights corresponding to latent position $x \in \mathcal{Z}$ set to $\exp(- 0.4 \times \Vert x-z^\star \Vert^2$), followed by PCA into 2D on embeddings of nodes in $\mathcal{N}_z$. The reconstruction RMSE associated with each embedding is recorded in the titles.

Theorems & Definitions (24)

• Definition 1: Latent position model
• Theorem 1
• Lemma 1
• Theorem 2
• Proposition 1
• Theorem 3
• Theorem 4
• Lemma 2
• proof
• Lemma 3