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

TL;DR

This paper introduces ULSE, a new spectral embedding method for dynamic networks that guarantees stability over time and provides structural insights, validated through experiments.

Contribution

The paper extends spectral embedding techniques to normalized Laplacians with stability guarantees for dynamic networks, a previously unresolved challenge.

Findings

Proves stability of ULSE under a dynamic stochastic block model.

Establishes a dynamic Cheeger inequality linking spectrum to conductance.

Empirical validation on synthetic and real networks supports theoretical results.

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.