I love the data type graphs and find it amazing how expressive it is and how naturally it can be used for modelling just about anything. I am currently writing my thesis on using Neighborhood Embedding algorithms (basically an algorithm operating on the distance matrix between datapoints) for visualization and I find the whole topic fascinating.
That being said, by virtue of being so flexible, they can, generally speaking, be computationally expensive to use, as without any restrictions just creating a graph would cost O(n^2) space and time. In many application domains it is possible to exploit the structure (usually sparsity) in order to push this down, however.
All in all, I think graphs are a powerful data structure and can become more important in the future, especially if theoretical progress continues to be made.
There seems to be several “rest of the fucking owl” leaps in this text. Maybe I’m just too dumb.