
John Urschel's Favorite Theorem - mgdo
https://blogs.scientificamerican.com/roots-of-unity/john-urschels-favorite-theorem/
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gjm11
John Urschel is a mathematician who recently retired from being an NFL
American football player, at least in part because of the dangers of brain
damage from repeated concussions. There have been several HN posts about him,
but the only one I can find with a lot of comments is this one:
[https://news.ycombinator.com/item?id=14866573](https://news.ycombinator.com/item?id=14866573)

His favourite theorem is described here
[https://arxiv.org/abs/0808.0163](https://arxiv.org/abs/0808.0163). It's a
theorem in graph theory, and it says that if you have any graph, you can find
a subgraph with "not too many edges" that is "like" the original graph, in a
specific sense.

"Not too many" means that for some value d which you get to choose, the number
of edges is no more than about d times the number of vertices. (In general,
the number of edges in a graph can scale like the number of vertices
_squared_.)

"Like the original graph" means that a thing called the "graph Laplacian" of
the cut-down graph is a good approximation to the graph Laplacian of the
original. Here's one way to think about what the graph Laplacian is. Imagine a
sort of "heat flow" process on the graph, where at any given time each vertex
has a temperature, and at each time step you move each vertex's temperature
towards the average of its neighbours' temperatures. All the possible
temperature assignments constitute a finite-dimensional vector space; call it
V; the graph Laplacian is a linear map from V to itself that maps a
temperature assignment to the resulting _change_ produced by one step of the
heat-flow process. It's a sort of discrete version of the _second derivative_
operator. And it turns out that lots of interesting properties of a graph can
be computed just from knowing the graph Laplacian, or even just from knowing
its eigenvalues.

"Good approximation" means that the relative error is on the order of
1/sqrt(d) where d is that value you got to choose earlier. So the more edges
you're allowed to keep, the better the approximation can be made.

(All of the above is a handwavy approximation. If you understood everything I
said and want the _correct_ version, you can find it in the paper itself.)

