Papers › Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science
Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science
George C. Linderman, Gal Mishne, Yuval Kluger, Stefan Steinerberger
If we pick n random points uniformly in [0,1]ᵈ and connect each point to its $k-$nearest neighbors, then it is well known that there exists a giant connected component with high probability. We prove that in $[0,1]^d$ it suffices to connect every point to c_(d,1) loglogn points chosen randomly among its $ c_{d,2} \log{n}-$nearest neighbors to ensure a giant component of size n - o(n) with high probability. This construction yields a much sparser random graph with ∼n loglogn instead of ∼n logn edges that has comparable connectivity properties. This result has nontrivial implications for problems in data science where an affinity matrix is constructed: instead of picking the $k-$nearest neighbors, one can often pick k′ ≪k random points out of the $k-$nearest neighbors without sacrificing efficiency. This can massively simplify and accelerate computation, we illustrate this with several numerical examples.
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