The notion of local intrinsic dimensionality (LID) is an important
advancement in data dimensionality analysis, with applications in data mining,
machine learning and similarity search problems. Existing distance-based LID
estimators were designed for tabular datasets encompassing data points
represented as vectors in a Euclidean space. After discussing their limitations
for graph-structured data considering graph embeddings and graph distances, we
propose NC-LID, a novel LID-related measure for quantifying the discriminatory
power of the shortest-path distance with respect to natural communities of
nodes as their intrinsic localities. It is shown how this measure can be used
to design LID-aware graph embedding algorithms by formulating two LID-elastic
variants of node2vec with personalized hyperparameters that are adjusted
according to NC-LID values. Our empirical analysis of NC-LID on a large number
of real-world graphs shows that this measure is able to point to nodes with
high link reconstruction errors in node2vec embeddings better than node
centrality metrics. The experimental evaluation also shows that the proposed
LID-elastic node2vec extensions improve node2vec by better preserving graph
structure in generated embeddings