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Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation

Haruka Ezoe, Hiroki Matsumoto, Ryohei Hisano

TL;DR

This work proves that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model, and yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings.

Abstract

Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying node embeddings, whose usefulness depends on well defined stability properties across nodes and across time. We introduce Unfolded Laplacian Spectral Embedding (ULSE), a principled extension of unfolded adjacency spectral embedding to normalized Laplacian operators, a setting where stability guarantees have remained out of reach. We prove that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model. Moreover, the Laplacian formulation yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings. Empirical results on synthetic and real world dynamic networks validate the theory.

Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation

TL;DR

This work proves that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model, and yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings.

Abstract

Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying node embeddings, whose usefulness depends on well defined stability properties across nodes and across time. We introduce Unfolded Laplacian Spectral Embedding (ULSE), a principled extension of unfolded adjacency spectral embedding to normalized Laplacian operators, a setting where stability guarantees have remained out of reach. We prove that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model. Moreover, the Laplacian formulation yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings. Empirical results on synthetic and real world dynamic networks validate the theory.

Paper Structure

This paper contains 34 sections, 22 theorems, 157 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Setup and Stability Requirements
  3. Dynamic Network Embedding Setup
  4. Stability Conditions
  5. Dynamic Stochastic Block Model (DSBM)
  6. Unfolded Adjacency Spectral Embedding (UASE)
  7. Unfolded Laplacian Spectral Embedding
  8. Degree Quantities and Notation
  9. Unfolding Normalized Laplacian Operators
  10. ULSE-n1: Per-Snapshot Normalization
  11. ULSE-n2: Partially Aggregated Normalization
  12. Theoretical Guarantees
  13. Stability Guarantees for ULSE-n1
  14. Stability Guarantees for ULSE-n2
  15. Structural Interpretation via a Dynamic Cheeger Inequality
  16. ...and 19 more sections

Key Result

Theorem 1

Suppose that the community level singular values satisfy Set the embedding dimension $d = K-1$. Then there exist matrices $\mathbf{Y}^{(t)} \in \mathbb{R}^{n\times d}$, $t=1,\ldots,T$, such that and the following stability properties hold:

Figures (7)

  • Figure 1: Comparison of ULSE-n1, ULSE-n2, and TempCut-N on synthetic data. Colors labeled as Community 1–3 correspond to the three latent community trajectories.
  • Figure 2: Counterexample demonstrating the failure of Theorem 4.1 in zhao2021context. Despite structural differences between $G^{(1)}$ and $G^{(2)}$, their spectral embeddings under the context-aware perturbation model are identical.
  • Figure 3: Clustering performance (NMI and ARI) for Experiments 1 and 2 as the intra-community probability $p$ varies.
  • Figure 4: t-SNE visualizations of dynamic embeddings for the second synthetic dataset with $p=0.4$. Colors labeled as Community 1–3 correspond to the three latent community trajectories. Top: ULSE-n1. Bottom: UASE.
  • Figure 5: Three-dimensional dynamic embeddings on the synthetic dataset. Colors labeled as Community 1–3 correspond to the three latent community trajectories. Top: ULSE-n1. Middle: ULSE-n2. Bottom: UASE.
  • ...and 2 more figures

Theorems & Definitions (38)

  • Theorem 1: Stability of ULSE-n1
  • Theorem 2: Convergence of ULSE-n1
  • Theorem 3: Stability of noise-free ULSE-n1
  • Lemma 1: Deviation bound
  • Lemma 2: Spectral structure of the population operator
  • Corollary 1
  • Lemma 3: Projection deviation bound
  • Theorem 4: Stability of ULSE-n2
  • Proposition 1: Dynamic Cheeger bound
  • Lemma 4: Lower bound for the minimum degree
  • ...and 28 more