How would you calculate something like a Hausdorff dimension of a graph given there’s no unique embedding in Euclidean space, but we still think of (for eg) bushy vs sparse branching etc? A long skinny DAG with few branches should be lower dim than dense random graph for eg 
You have to embed them to get a meaningful notion of spatial dimension. Pure graph-theoretic metrics lack a notion of dimension. Or even proper norms. You can only get quasi-spatial metrics like “diameter” which yield an O(n) notation type analog to dimension (log, linear etc).
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Sure; you can get norms if not metrics if you add weights to edges (like edit distance between commits). But I'm just surprised there isn't something that's made it onto wikipedia that has some relevant flavor of fractal dimension, thickness, bushiness, self-similarity for graphs
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The moment you add weights you’re bringing in a whole new information set. I’d like a “natural” embedding that minimizes the amount of extra information added. Also weights are halfway to an embedding so you might as well go all the way.
End of conversation
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