Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the topological structure of the data. In this paper, we explore several aspects that affect the quality of a vertex embedding of graph-structured data. To this effect, we first present a family of flexible distance functions that faithfully capture the topological distance between different vertices. Secondly, we analyze vertex embeddings as resulting from a fitted transformation of the distance matrix rather than as a direct result of optimization. Finally, we evaluate the effectiveness of our proposed embedding constructions by performing community detection on a host of benchmark datasets. The reported results are competitive with classical algorithms that operate on the entire graph while benefitting from a substantially reduced computational complexity due to the reduced dimensionality of the representations.
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