Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?)

I don't know what is a good definition of a ``good" embedding is! May be my rough intuition is that there should be minimum distortion of the shortest path metric on the graph. I am open to knowing if there are other definitions which are also considered in literature.

Two related MO discussions (for which I don't see hyperbolic counterparts) ,