  Graph vertex embeddings based on random walks have become increasingly
influential in recent years, showing good performance in several tasks as they
efficiently transform a graph into a more computationally digestible format
while preserving relevant information. However, the theoretical properties of
such algorithms, in particular the influence of hyperparameters and of the
graph structure on their convergence behaviour, have so far not been
well-understood. In this work, we provide a theoretical analysis for
random-walks based embeddings techniques. Firstly, we prove that, under some
weak assumptions, vertex embeddings derived from random walks do indeed
converge both in the single limit of the number of random walks $N \to \infty$
and in the double limit of both $N$ and the length of each random walk
$L\to\infty$. Secondly, we derive concentration bounds quantifying the converge
rate of the corpora for the single and double limits. Thirdly, we use these
results to derive a heuristic for choosing the hyperparameters $N$ and $L$. We
validate and illustrate the practical importance of our findings with a range
of numerical and visual experiments on several graphs drawn from real-world
applications.
