The multilayer random dot product graph

Abstract

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.

arXiv comment45 pages, 15 figures

References in corpus (6)

• Assortative mixing in networks
• Joint Embedding of Graphs
• Inference for multiple heterogeneous networks with a common invariant subspace
• A central limit theorem for an omnibus embedding of multiple random graphs and implications for multiscale network inference
• The Multiple Random Dot Product Graph Model
• Link prediction in dynamic networks using random dot product graphs

Cited by in corpus (6)

• Link prediction in dynamic networks using random dot product graphs
• Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodel
• Spectral embedding for dynamic networks with stability guarantees
• Nonparametric two-sample hypothesis testing for low-rank random graphs of differing sizes
• Latent structure blockmodels for Bayesian spectral graph clustering
• Matrix factorisation and the interpretation of geodesic distance