Representation learning is one of the foundations of Deep Learning and allowed important improvements on several Machine Learning tasks, such as Neural Machine Translation, Question Answering and Speech Recognition. Recent works have proposed new methods for learning representations for nodes and edges in graphs. Several of these methods are based on the SkipGram algorithm, and they usually process a large number of multi-hop neighbors in order to produce the context from which node representations are learned. In this paper, we propose an effective and also efficient method for generating node embeddings in graphs that employs a restricted number of permutations over the immediate neighborhood of a node as context to generate its representation, thus ego-centric representations. We present a thorough evaluation showing that our method outperforms state-of-the-art methods in six different datasets related to the problems of link prediction and node classification, being one to three orders of magnitude faster than baselines when generating node embeddings for very large graphs.
The paper includes the terms first-order proximity ("the concept that connected nodes in a graph should have similar properties") and second-order proximity ("the concept that nodes with similar neighborhoods should have common characteristics"). These are called homophily in social network analysis. It is also known as assortativity in network science literature. The paper states on Page 4: "A trade-off between first and second order proximity can be achieved by changing the parameter k, which simultaneously controls both the sizes of sentences generated and the size of the wind used in the SkipGram algorithm." It is not readily clear why this statement should hold. Also the paper does not include a discussion on how the amount of homophily in the graph affects the results. There are various ways of measuring the level of homophily in a graph. There is simple local consistency, which is % of edges connecting nodes that have the same characteristics at each endpoint. Neville & Jensen's JMLR 2007 paper describes relational auto-correlation, which is Pearson contingency coefficient on the characteristics of endpoints of edges. Park & Barabasi's PNAS 2007 paper describes dyadicity and heterophilicity, which measures connections of nodes with the same characteristics compared to a random model and the connections of nodes with different characteristics compared to a random model. 
k ("which simultaneously controls both the sizes of sentences generated and the size of the wind used in the SkipGram algorithm") is a free-parameter in the proposed algorithm. The paper needs an in-depth discussion of the role of k in the results. Currently, no discussion is provided on k except that it was set to 5 for the experiments.  From a network science perspective, it makes sense to have k vary per node.
It is also not clear why d = 128 was chosen as the size of the embedding.
From the description of the experimental setup for link prediction, it is not clear if a stratified sample of the entries of the adjacency matrix (i.e., both 0 and 1 entries) where selected.
For the node classification experiments, information on class distribution and homophily levels would be helpful. 
In Section 5.1, the paper states: "For highly connected graphs, larger numbers of permutations should be chosen (n in [10, 1000]) to better represent distributions, while for sparser graphs, smaller values can be used (n in [1, 10])." How high is highly connected graphs? How spare is a sparser graph? In general, the paper lacks an in-depth analysis of when the approach works and when it does not.  I recommend running experiments on synthetic graphs (such as Barabasi-Albert, Watts-Strogatz, Forest Fire, Kronecker, and/or BTER graphs), systematically changing various characteristics of the graph, and reporting the results.
The faster runtime is interesting but not surprising given the ego-centric nature of the approach.