The multilayer random dot product graph
This work addresses the problem of modeling multiple interconnected graphs for researchers in network analysis, offering a novel framework with proven asymptotic properties, though it is an incremental extension of existing latent position models.
The authors extended the random dot product graph model to handle multiple graphs with shared nodes, providing a joint embedding method with theoretical guarantees of convergence to latent positions. They demonstrated improved link prediction in a cyber-security example and achieved comparable or better results than existing spectral methods in statistical inference tasks.
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.