  This paper shows that graph spectral embedding using the random walk
Laplacian produces vector representations which are completely corrected for
node degree. Under a generalised random dot product graph, the embedding
provides uniformly consistent estimates of degree-corrected latent positions,
with asymptotically Gaussian error. In the special case of a degree-corrected
stochastic block model, the embedding concentrates about K distinct points,
representing communities. These can be recovered perfectly, asymptotically,
through a subsequent clustering step, without spherical projection, as commonly
required by algorithms based on the adjacency or normalised, symmetric
Laplacian matrices. While the estimand does not depend on degree, the
asymptotic variance of its estimate does -- higher degree nodes are embedded
more accurately than lower degree nodes. Our central limit theorem therefore
suggests fitting a weighted Gaussian mixture model as the subsequent clustering
step, for which we provide an expectation-maximisation algorithm.
