  Network data are ubiquitous in modern machine learning, with tasks of
interest including node classification, node clustering and link prediction. A
frequent approach begins by learning an Euclidean embedding of the network, to
which algorithms developed for vector-valued data are applied. For large
networks, embeddings are learned using stochastic gradient methods where the
sub-sampling scheme can be freely chosen. Despite the strong empirical
performance of such methods, they are not well understood theoretically. Our
work encapsulates representation methods using a subsampling approach, such as
node2vec, into a single unifying framework. We prove, under the assumption that
the graph is exchangeable, that the distribution of the learned embedding
vectors asymptotically decouples. Moreover, we characterize the asymptotic
distribution and provided rates of convergence, in terms of the latent
parameters, which includes the choice of loss function and the embedding
dimension. This provides a theoretical foundation to understand what the
embedding vectors represent and how well these methods perform on downstream
tasks. Notably, we observe that typically used loss functions may lead to
shortcomings, such as a lack of Fisher consistency.
