  Random walk based node embedding algorithms learn vector representations of
nodes by optimizing an objective function of node embedding vectors and
skip-bigram statistics computed from random walks on the network. They have
been applied to many supervised learning problems such as link prediction and
node classification and have demonstrated state-of-the-art performance. Yet,
their properties remain poorly understood. This paper studies properties of
random walk based node embeddings in the unsupervised setting of discovering
hidden block structure in the network, i.e., learning node representations
whose cluster structure in Euclidean space reflects their adjacency structure
within the network. We characterize the ergodic limits of the embedding
objective, its generalization, and related convex relaxations to derive
corresponding non-randomized versions of the node embedding objectives. We also
characterize the optimal node embedding Grammians of the non-randomized
objectives for the expected graph of a two-community Stochastic Block Model
(SBM). We prove that the solution Grammian has rank $1$ for a suitable nuclear
norm relaxation of the non-randomized objective. Comprehensive experimental
results on SBM random networks reveal that our non-randomized ergodic
objectives yield node embeddings whose distribution is Gaussian-like, centered
at the node embeddings of the expected network within each community, and
concentrate in the linear degree-scaling regime as the number of nodes
increases.
