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

arXiv: https://arxiv.org/abs/2508.12674 · HTML: https://mlsanity.com/paper/2508.12674/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 18 Aug 2025.

## Builds on (most cited first, 19 of 19)

- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Temporal Graph Networks for Deep Learning on Dynamic Graphs](https://mlsanity.com/paper/2006.10637/) (2020)
- [Inductive Representation Learning on Temporal Graphs](https://mlsanity.com/paper/2002.07962/) (2020)
- [Predicting Dynamic Embedding Trajectory in Temporal Interaction Networks](https://mlsanity.com/paper/1908.01207/) (2019)
- [Towards Better Dynamic Graph Learning: New Architecture and Unified Library](https://mlsanity.com/paper/2303.13047/) (2023)
- [Towards Foundation Models for Knowledge Graph Reasoning](https://mlsanity.com/paper/2310.04562/) (2023)
- [Spectral redemption: clustering sparse networks](https://mlsanity.com/paper/1306.5550/) (2013)
- [TGB 2.0: A Benchmark for Learning on Temporal Knowledge Graphs and Heterogeneous Graphs](https://mlsanity.com/paper/2406.09639/) (2024)
- [Latent Space Models for Dynamic Networks](https://mlsanity.com/paper/2005.08808/) (2020)
- [Deep Temporal Graph Clustering](https://mlsanity.com/paper/2305.10738/) (2023)
- [Strong Consistency, Graph Laplacians, and the Stochastic Block Model](https://mlsanity.com/paper/2004.09780/) (2020)
- [The multilayer random dot product graph](https://mlsanity.com/paper/2007.10455/) (2020)
- [Understanding Regularized Spectral Clustering via Graph Conductance](https://mlsanity.com/paper/1806.01468/) (2018)
- [A Comprehensive Review of Recommender Systems: Transitioning from Theory to Practice](https://mlsanity.com/paper/2407.13699/) (2024)
- [Spectral embedding for dynamic networks with stability guarantees](https://mlsanity.com/paper/2106.01282/) (2021)
- [A Simple and Powerful Framework for Stable Dynamic Network Embedding](https://mlsanity.com/paper/2311.09251/) (2023)
- [Spectral Algorithms for Temporal Graph Cuts](https://mlsanity.com/paper/1702.04746/) (2017)
- [Learning Persistent Community Structures in Dynamic Networks via Topological Data Analysis](https://mlsanity.com/paper/2401.03194/) (2024)
- [A Generative Model for Dynamic Networks with Applications](https://mlsanity.com/paper/1802.03725/) (2018)

## Abstract (arXiv)

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.