My understanding is that points (vertices) are defined only in terms of their connections (edges).
Apparent multi-dimensional space arises by taking paths (geodesics) through the graph.
e.g. Given {1 -> 2, 1 -> 3, 1 -> 4}
from point 1 you can get to point 2, 3, or 4, thus having local dimensionality 3. Similarly {1 -> 2, 1 -> 3}
has local dimensionality 2 at point 1.
Note that point 2 has local dimensionality 1, and that therefore the graph as a whole has fractional dimensionality.