  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.
