Let's define a embedding of a graph structure G = (V,E) where $\mid V\mid=v, \mid E \mid=e$
Now define an embedding $f: V \to R^d$ where $d\in \Bbb N$, an optimal dimension of embedding which contains every edge information of $G$.
(G is a directed graph and there exist no weight, thus it's not a network.)
I'd like to find an infimum formula of $d$ represented with $v$ and $e$.
[Backgroud of this problem]
I am trying to construct a neural network which can discern whether the given explanation of word is about the word "be" or the word "exist".
For example "having a real existence" is "exist".
To do this, first I need to find the most smallest dimension of each words' corresponding embedding of vertex for training of my network.