# Spectral embedding for dynamic networks with stability guarantees

arXiv: https://arxiv.org/abs/2106.01282 · HTML: https://mlsanity.com/paper/2106.01282/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 55 references that are in the map.
- First posted on arXiv 2 Jun 2021.

## Builds on (most cited first, 27 of 27)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Spectral redemption: clustering sparse networks](https://mlsanity.com/paper/1306.5550/) (2013)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Dynamic Network Embedding Survey](https://mlsanity.com/paper/2103.15447/) (2021)
- [The impossibility of low rank representations for triangle-rich complex networks](https://mlsanity.com/paper/2003.12635/) (2020)
- [Statistical clustering of temporal networks through a dynamic stochastic block model](https://mlsanity.com/paper/1506.07464/) (2015)
- [Latent Space Models for Dynamic Networks](https://mlsanity.com/paper/2005.08808/) (2020)
- [Perfect Clustering for Stochastic Blockmodel Graphs via Adjacency Spectral Embedding](https://mlsanity.com/paper/1310.0532/) (2013)
- [Multivariate Time Series Classification Using Dynamic Time Warping Template Selection for Human Activity Recognition](https://mlsanity.com/paper/1512.06747/) (2015)
- [Latent Space Model for Road Networks to Predict Time-Varying Traffic](https://mlsanity.com/paper/1602.04301/) (2016)
- [Scalable Link Prediction in Dynamic Networks via Non-Negative Matrix Factorization](https://mlsanity.com/paper/1411.3675/) (2014)
- [Dynamic stochastic blockmodels for time-evolving social networks](https://mlsanity.com/paper/1403.0921/) (2014)
- [The multilayer random dot product graph](https://mlsanity.com/paper/2007.10455/) (2020)
- [A Review of Dynamic Network Models with Latent Variables](https://mlsanity.com/paper/1711.10421/) (2017)
- [Manifold structure in graph embeddings](https://mlsanity.com/paper/2006.05168/) (2020)
- [Spectral clustering in the dynamic stochastic block model](https://mlsanity.com/paper/1705.01204/) (2017)
- [Network Representation Using Graph Root Distributions](https://mlsanity.com/paper/1802.09684/) (2018)
- [A Survey on Dynamic Network Embedding](https://mlsanity.com/paper/2006.08093/) (2020)
- [Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodel](https://mlsanity.com/paper/2011.04558/) (2020)
- [Nonparametric Bayes dynamic modeling of relational data](https://mlsanity.com/paper/1311.4669/) (2013)
- [Spectral Clustering for Multiple Sparse Networks: I](https://mlsanity.com/paper/1805.10594/) (2018)
- [Stochastic Block Transition Models for Dynamic Networks](https://mlsanity.com/paper/1411.5404/) (2014)
- [Sparse and Smooth: improved guarantees for Spectral Clustering in the Dynamic Stochastic Block Model](https://mlsanity.com/paper/2002.02892/) (2020)
- [On Two Distinct Sources of Nonidentifiability in Latent Position Random Graph Models](https://mlsanity.com/paper/2003.14250/) (2020)
- [Bayesian Learning of Dynamic Multilayer Networks](https://mlsanity.com/paper/1608.02209/) (2016)
- [Link prediction in dynamic networks using random dot product graphs](https://mlsanity.com/paper/1912.10419/) (2019)

## Built on by (most cited first, 13 of 13)

- [Synergistic Graph Fusion via Encoder Embedding](https://mlsanity.com/paper/2303.18051/) (2023)
- [Valid Conformal Prediction for Dynamic GNNs](https://mlsanity.com/paper/2405.19230/) (2024)
- [Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks](https://mlsanity.com/paper/2306.06155/) (2023)
- [A Simple and Powerful Framework for Stable Dynamic Network Embedding](https://mlsanity.com/paper/2311.09251/) (2023)
- [Refined Graph Encoder Embedding via Self-Training and Latent Community Recovery](https://mlsanity.com/paper/2405.12797/) (2024)
- [Gradient-Based Spectral Embeddings of Random Dot Product Graphs](https://mlsanity.com/paper/2307.13818/) (2023)
- [Principal Graph Encoder Embedding and Principal Community Detection](https://mlsanity.com/paper/2501.14939/) (2025)
- [Valid Bootstraps for Network Embeddings with Applications to Network Visualisation](https://mlsanity.com/paper/2410.20895/) (2024)
- [Generalized Grade-of-Membership Estimation for High-dimensional Locally Dependent Data](https://mlsanity.com/paper/2412.19796/) (2024)
- [Unsupervised Attributed Dynamic Network Embedding with Stability Guarantees](https://mlsanity.com/paper/2503.02859/) (2025)
- [Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation](https://mlsanity.com/paper/2508.12674/) (2025)
- [Inferring Network Evolutionary History via Structure-State Coupled Learning](https://mlsanity.com/paper/2601.02121/) (2026)
- [Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities](https://mlsanity.com/paper/2603.05703/) (2026)

## Abstract (arXiv)

We consider the problem of embedding a dynamic network, to obtain time-evolving vector representations of each node, which can then be used to describe changes in behaviour of individual nodes, communities, or the entire graph. Given this open-ended remit, we argue that two types of stability in the spatio-temporal positioning of nodes are desirable: to assign the same position, up to noise, to nodes behaving similarly at a given time (cross-sectional stability) and a constant position, up to noise, to a single node behaving similarly across different times (longitudinal stability). Similarity in behaviour is defined formally using notions of exchangeability under a dynamic latent position network model. By showing how this model can be recast as a multilayer random dot product graph, we demonstrate that unfolded adjacency spectral embedding satisfies both stability conditions. We also show how two alternative methods, omnibus and independent spectral embedding, alternately lack one or the other form of stability.