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vgr's profile
Venkatesh Rao
Venkatesh Rao
Venkatesh Rao
@vgr

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Venkatesh Rao

@vgr

Conversational account. For work follow @ribbonfarm, @breaking_smart, @artofgig. Tweets are 90% vacuous views, apathetically held. Mediocritopian. IKEA builder.

Los Angeles, CA
venkateshrao.com
Joined August 2007

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    Venkatesh Rao‏ @vgr 2 Nov 2019
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    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 🤔

    9:59 AM - 2 Nov 2019
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    • have a holly jolly era of vice and indulgence Orthos Mark (👨‍💻's meditation instructions (See 🔗s👇)) ebr Bushra Farooqui 🎄🌉
    6 replies 0 retweets 5 likes
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      2. Olaf‏ @QiaochuYuan 2 Nov 2019
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        Replying to @vgr

        can take e.g. the average degree / 2, although i don't know how useful this is. you can think of the average degree / 2 localized in a region of a graph as being a "local dimension" which can then vary across regions of the graph

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      3. Olaf‏ @QiaochuYuan 2 Nov 2019
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        Replying to @QiaochuYuan @vgr

        ah no, this doesn't really capture enough structure. better: every graph is a metric space with metric given by the length of the shortest path and then there's a notion of local dimension given by how fast balls wrt this metric grow

        1 reply 0 retweets 4 likes
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      1. David Longman‏ @DavidJLongman 2 Nov 2019
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        Replying to @vgr

        errrr ????

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      1. Stanislav Fort‏ @stanislavfort 2 Nov 2019
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        Replying to @vgr

        Start at a vertex, look at the number of first neighbors, second, ... . Repeat and average for all vertices. Try fitting r^{d-1},where r is the order of the neighbor. This should be *a* valid dimension, though not sure if it matches the Hausdorff dimension.

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      2.  ☃️A Thousand Merry F**kin Plateaus 🎄‏ @interpretantion 2 Nov 2019
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        Replying to @vgr

        I'm pretty sure a graph simply has a hausdorff dimension of zero since the nodes are points and there's only one link between any two points. I suppose you could think of a graph with an uncountable number of nodes by constructing some set but that sounds a bit iffy

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      3. Venkatesh Rao‏ @vgr 2 Nov 2019
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        Replying to @interpretantion

        You're technically correct, but that's why I said something *like* a hausdorff dimension that captures the sense of spatial density of an embedded graph

        0 replies 0 retweets 2 likes
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      2. Mark ( 👨‍💻's meditation instructions (See  🔗s 👇))‏ @meditationstuff 2 Nov 2019
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        Replying to @vgr

        Maybe average or median fan-out per node for a directed graph? Or average number of choices (also fan-out) at each node for a DAG? Which flavor or quality are trying to capture, transferred over from the original scalar?

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      3. Venkatesh Rao‏ @vgr 2 Nov 2019
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        Replying to @meditationstuff

        If you squint at a (biological) tree or cauliflower it looks like a sankey diagram of an aggregate flow. The coarse topology of that flow given a resolution level. Like a recursive approximation of min-cut-max-flow view. DAGs are the easiest case.

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      1. Biswajit Banerjee‏ @parresianz 2 Nov 2019
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        Replying to @vgr

        Some ideas here: https://arxiv.org/pdf/1808.01079.pdf …

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