# Optimal Dimension of Graph(Vertex) Embedding [closed]

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.

To conclude, finding the infimum can be done post-hoc, in the sense that you can evaluate the performance on the function and see which value of $d$ does not significantly decrease the performance and choose it this way. However, at a post-hoc level, the only way I can see of doing this is by evaluating the embedding space beforehand, I have not read too much literature about this.