# Spectral embedding and the latent geometry of multipartite networks

arXiv: https://arxiv.org/abs/2202.03945 · HTML: https://mlsanity.com/paper/2202.03945/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 71 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 25 of 25)

- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [SimplE Embedding for Link Prediction in Knowledge Graphs](https://mlsanity.com/paper/1802.04868/) (2018)
- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [The impossibility of low rank representations for triangle-rich complex networks](https://mlsanity.com/paper/2003.12635/) (2020)
- [Concentration and regularization of random graphs](https://mlsanity.com/paper/1506.00669/) (2015)
- [On a 'Two Truths' Phenomenon in Spectral Graph Clustering](https://mlsanity.com/paper/1808.07801/) (2018)
- [Perfect Clustering for Stochastic Blockmodel Graphs via Adjacency Spectral Embedding](https://mlsanity.com/paper/1310.0532/) (2013)
- [The multilayer random dot product graph](https://mlsanity.com/paper/2007.10455/) (2020)
- [Limit theorems for eigenvectors of the normalized Laplacian for random graphs](https://mlsanity.com/paper/1607.08601/) (2016)
- [Manifold structure in graph embeddings](https://mlsanity.com/paper/2006.05168/) (2020)
- [On the Power of Edge Independent Graph Models](https://mlsanity.com/paper/2111.00048/) (2021)
- [Understanding Regularized Spectral Clustering via Graph Conductance](https://mlsanity.com/paper/1806.01468/) (2018)
- [On estimation and inference in latent structure random graphs](https://mlsanity.com/paper/1806.01401/) (2018)
- [Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities](https://mlsanity.com/paper/1602.02114/) (2016)
- [Spectral clustering under degree heterogeneity: a case for the random walk Laplacian](https://mlsanity.com/paper/2105.00987/) (2021)
- [On Two Distinct Sources of Nonidentifiability in Latent Position Random Graph Models](https://mlsanity.com/paper/2003.14250/) (2020)
- [Learning 1-Dimensional Submanifolds for Subsequent Inference on Random Dot Product Graphs](https://mlsanity.com/paper/2004.07348/) (2020)
- [Matrix factorisation and the interpretation of geodesic distance](https://mlsanity.com/paper/2106.01260/) (2021)
- [Estimating Graph Dimension with Cross-validated Eigenvalues](https://mlsanity.com/paper/2108.03336/) (2021)
- [Root and community inference on the latent growth process of a network](https://mlsanity.com/paper/2107.00153/) (2021)
- [Consistency of adjacency spectral embedding for the mixed membership stochastic blockmodel](https://mlsanity.com/paper/1705.04518/) (2017)

## Abstract (arXiv)

Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of a properly constructed matrix, and has found applications throughout science and technology. Many networks are multipartite, meaning that they contain nodes of fundamentally different types, e.g. drugs, diseases and proteins, and edges are only observed between nodes of different types. When the network is multipartite, this paper demonstrates that the node representations obtained via spectral embedding lie near type-specific low-dimensional subspaces of a higher-dimensional ambient space. For this reason we propose a follow-on step after spectral embedding, to recover node representations in their intrinsic rather than ambient dimension, proving uniform consistency under a low-rank, inhomogeneous random graph model. We demonstrate the performance of our procedure on a large 6-partite biomedical network relevant for drug discovery.