Statistical inference on random dot product graphs: a survey

TL;DR

This survey reviews statistical inference methods for random dot product graphs, emphasizing spectral embeddings for hypothesis testing, community detection, and network analysis, with applications to social networks and connectome data.

Contribution

It provides a comprehensive overview of spectral inference techniques for RDPGs, including theoretical results and practical applications, highlighting open problems in the field.

Findings

Spectral embeddings are consistent and asymptotically normal for RDPGs.

Spectral methods can effectively detect communities and classify nodes.

Applications include social network analysis and connectome exploration.

Abstract

The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describe a comprehensive paradigm for statistical inference on random dot product graphs, a paradigm centered on spectral embeddings of adjacency and Laplacian matrices. We examine the analogues, in graph inference, of several canonical tenets of classical Euclidean inference: in particular, we summarize a body of existing results on the consistency and asymptotic normality of the adjacency and Laplacian spectral embeddings, and the role these spectral embeddings can play in the construction of single- and multi-sample hypothesis tests for graph data. We investigate several real-world applications, including community detection and classification in large social networks and the determination of functional and biologically relevant network properties from an exploratory data analysis of the Drosophila connectome. We outline requisite background and current open problems in spectral graph inference.