  Node embedding is the task of extracting informative and descriptive features
over the nodes of a graph. The importance of node embeddings for graph
analytics, as well as learning tasks such as node classification, link
prediction and community detection, has led to increased interest on the
problem leading to a number of recent advances. Much like PCA in the feature
domain, node embedding is an inherently \emph{unsupervised} task; in lack of
metadata used for validation, practical methods may require standardization and
limiting the use of tunable hyperparameters. Finally, node embedding methods
are faced with maintaining scalability in the face of large-scale real-world
graphs of ever-increasing sizes. In the present work, we propose an adaptive
node embedding framework that adjusts the embedding process to a given
underlying graph, in a fully unsupervised manner. To achieve this, we adopt the
notion of a tunable node similarity matrix that assigns weights on paths of
different length. The design of the multilength similarities ensures that the
resulting embeddings also inherit interpretable spectral properties. The
proposed model is carefully studied, interpreted, and numerically evaluated
using stochastic block models. Moreover, an algorithmic scheme is proposed for
training the model parameters effieciently and in an unsupervised manner. We
perform extensive node classification, link prediction, and clustering
experiments on many real world graphs from various domains, and compare with
state-of-the-art scalable and unsupervised node embedding alternatives. The
proposed method enjoys superior performance in many cases, while also yielding
interpretable information on the underlying structure of the graph.
