# A statistical interpretation of spectral embedding: the generalised random dot product graph

arXiv: https://arxiv.org/abs/1709.05506 · HTML: https://mlsanity.com/paper/1709.05506/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2017.

## Built on by (most cited first, 30 of 55)

- [Conformal Prediction for Network-Assisted Regression](https://mlsanity.com/paper/2302.10095/) (2023)
- [Spectral embedding for dynamic networks with stability guarantees](https://mlsanity.com/paper/2106.01282/) (2021)
- [Asymptotics of Network Embeddings Learned via Subsampling](https://mlsanity.com/paper/2107.02363/) (2021)
- [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)
- [Online Change Point Detection for Weighted and Directed Random Dot Product Graphs](https://mlsanity.com/paper/2201.11222/) (2022)
- [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)
- [Hypothesis Testing for Equality of Latent Positions in Random Graphs](https://mlsanity.com/paper/2105.10838/) (2021)
- [Discovering Communication Pattern Shifts in Large-Scale Labeled Networks using Encoder Embedding and Vertex Dynamics](https://mlsanity.com/paper/2305.02381/) (2023)
- [Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks](https://mlsanity.com/paper/2306.06155/) (2023)
- [Refined Graph Encoder Embedding via Self-Training and Latent Community Recovery](https://mlsanity.com/paper/2405.12797/) (2024)
- [Popularity Adjusted Block Models are Generalized Random Dot Product Graphs](https://mlsanity.com/paper/2109.04010/) (2021)
- [Exact Representation of Sparse Networks with Symmetric Nonnegative Embeddings](https://mlsanity.com/paper/2111.03030/) (2021)
- [Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels](https://mlsanity.com/paper/2208.08638/) (2022)
- [On the Validity of Conformal Prediction for Network Data Under Non-Uniform Sampling](https://mlsanity.com/paper/2306.07252/) (2023)
- [Improved Algorithm and Bounds for Successive Projection](https://mlsanity.com/paper/2403.11013/) (2024)
- [Transfer Learning for Latent Variable Network Models](https://mlsanity.com/paper/2406.03437/) (2024)
- [Exact Recovery of Community Structures Using DeepWalk and Node2vec](https://mlsanity.com/paper/2101.07354/) (2021)
- [Semi-supervised Community Detection via Structural Similarity Metrics](https://mlsanity.com/paper/2306.01089/) (2023)
- [Learning Joint and Individual Structure in Network Data with Covariates](https://mlsanity.com/paper/2406.08776/) (2024)
- [Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2506.10101/) (2025)
- [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)
- [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)
- [Signed Diverse Multiplex Networks: Clustering and Inference](https://mlsanity.com/paper/2402.10242/) (2024)
- [Bipartite Graph Variational Auto-Encoder with Fair Latent Representation to Account for Sampling Bias in Ecological Networks](https://mlsanity.com/paper/2403.02011/) (2024)
- [Interpretability of Graph Neural Networks to Assess Effects of Global Change Drivers on Ecological Networks](https://mlsanity.com/paper/2503.15107/) (2025)
- [Weighted Random Dot Product Graphs](https://mlsanity.com/paper/2505.03649/) (2025)
- [Multiresolution Analysis and Statistical Thresholding on Dynamic Networks](https://mlsanity.com/paper/2506.01208/) (2025)

## Abstract (arXiv)

Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow interpretation of those vector representations as latent position estimates. The generalisation is needed to model heterophilic connectivity (e.g., `opposites attract') and to cope with negative eigenvalues more generally. We show that, whether the adjacency or normalised Laplacian matrix is used, spectral embedding produces uniformly consistent latent position estimates with asymptotically Gaussian error (up to identifiability). The standard and mixed membership stochastic block models are special cases in which the latent positions take only K distinct vector values, representing communities, or live in the (K-1)-simplex with those vertices, respectively. Under the stochastic block model, our theory suggests spectral clustering using a Gaussian mixture model (rather than K-means) and, under mixed membership, fitting the minimum volume enclosing simplex, existing recommendations previously only supported under non-negative-definite assumptions. Empirical improvements in link prediction (over the random dot product graph), and the potential to uncover richer latent structure (than posited under the standard or mixed membership stochastic block models) are demonstrated in a cyber-security example.