# Limit theorems for eigenvectors of the normalized Laplacian for random graphs

arXiv: https://arxiv.org/abs/1607.08601 · HTML: https://mlsanity.com/paper/1607.08601/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jul 2016.

## Built on by (most cited first, 17 of 17)

- [Spectral clustering under degree heterogeneity: a case for the random walk Laplacian](https://mlsanity.com/paper/2105.00987/) (2021)
- [Matrix factorisation and the interpretation of geodesic distance](https://mlsanity.com/paper/2106.01260/) (2021)
- [Hypothesis Testing for Equality of Latent Positions in Random Graphs](https://mlsanity.com/paper/2105.10838/) (2021)
- [Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks](https://mlsanity.com/paper/2306.06155/) (2023)
- [Exact Recovery of Community Structures Using DeepWalk and Node2vec](https://mlsanity.com/paper/2101.07354/) (2021)
- [Clustered Graph Matching for Label Recovery and Graph Classification](https://mlsanity.com/paper/2205.03486/) (2022)
- [Identifying Peer Influence in Therapeutic Communities Adjusting for Latent Homophily](https://mlsanity.com/paper/2203.14223/) (2022)
- [Implications of sparsity and high triangle density for graph representation learning](https://mlsanity.com/paper/2210.15277/) (2022)
- [SIMPLE-RC: Group Network Inference with Non-Sharp Nulls and Weak Signals](https://mlsanity.com/paper/2211.00128/) (2022)
- [Detection of Model-based Planted Pseudo-cliques in Random Dot Product Graphs by the Adjacency Spectral Embedding and the Graph Encoder Embedding](https://mlsanity.com/paper/2312.11054/) (2023)
- [Maximum a Posteriori Inference of Random Dot Product Graphs via Conic Programming](https://mlsanity.com/paper/2101.02180/) (2021)
- [Asymptotics of ℓ2 Regularized Network Embeddings](https://mlsanity.com/paper/2201.01689/) (2022)
- [Spectral embedding and the latent geometry of multipartite networks](https://mlsanity.com/paper/2202.03945/) (2022)
- [Adversarial contamination of networks in the setting of vertex nomination: a new trimming method](https://mlsanity.com/paper/2208.09710/) (2022)
- [Dynamic Network Sampling for Community Detection](https://mlsanity.com/paper/2208.13921/) (2022)
- [Adjacency Spectral Embeddings of Correlation Networks](https://mlsanity.com/paper/2602.21055/) (2026)
- [The ASE-LSE Disagreement Landscape: An End-to-End Characterisation of Extremes and Structural Drivers](https://mlsanity.com/paper/2605.22346/) (2026)

## Abstract (arXiv)

We prove a central limit theorem for the components of the eigenvectors corresponding to the d largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.