# The multilayer random dot product graph

arXiv: https://arxiv.org/abs/2007.10455 · HTML: https://mlsanity.com/paper/2007.10455/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2020.

## Built on by (most cited first, 19 of 19)

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- [Gradient-Based Spectral Embeddings of Random Dot Product Graphs](https://mlsanity.com/paper/2307.13818/) (2023)
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- [Latent structure blockmodels for Bayesian spectral graph clustering](https://mlsanity.com/paper/2107.01734/) (2021)
- [Spectral embedding and the latent geometry of multipartite networks](https://mlsanity.com/paper/2202.03945/) (2022)
- [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)
- [Tensor State Space-based Dynamic Multilayer Network Modeling](https://mlsanity.com/paper/2506.02413/) (2025)
- [Perfect Clustering in Very Sparse Diverse Multiplex Networks](https://mlsanity.com/paper/2507.19423/) (2025)
- [Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation](https://mlsanity.com/paper/2508.12674/) (2025)

## Abstract (arXiv)

We present a comprehensive extension of the latent position network model known as the random dot product graph to accommodate multiple graphs -- both undirected and directed -- which share a common subset of nodes, and propose a method for jointly embedding the associated adjacency matrices, or submatrices thereof, into a suitable latent space. Theoretical results concerning the asymptotic behaviour of the node representations thus obtained are established, showing that after the application of a linear transformation these converge uniformly in the Euclidean norm to the latent positions with Gaussian error. Within this framework, we present a generalisation of the stochastic block model to a number of different multiple graph settings, and demonstrate the effectiveness of our joint embedding method through several statistical inference tasks in which we achieve comparable or better results than rival spectral methods. Empirical improvements in link prediction over single graph embeddings are exhibited in a cyber-security example.