Phase transitions for detecting latent geometry in random graphs

Testing thresholds for high-dimensional sparse random geometric graphs

“…Our result settles the conjecture of [BDER16] up to logarithmic factors, an exponential improvement over the previous bound of [BBN20], which required d ≫ n 3/2 . We remark that we have not made an effort to optimize the logarithmic factors; it is possible that our current proofs in combination with chaining-style arguments will yield log 3 n, matching their conjecture.…”

“…This improves by polynomial factors (in p and n) on the previous bound of [BBN20], which required d ≫ min{pn 3 log 1 p , p 2 n 7/2 poly log n} and d ≫ npolylog n. However, this result is not tight (at least for small p) since in particular it does not recover Theorem 1.1. Given that we have come close to establishing the conjecture of [BDER16] in the sparse case, it is tempting to interpolate between the upper and lower bounds of [BDER16] in the p = Θ(1) regime and their conjecture for the p = Θ( 1 n ) regime and speculate that for all p 1 2 , the testing threshold occurs at d ≍ (nH(p)) 3 = O(n 3 p 3 log 3 1 p ), for H(p) the binary entropy function.…”

“…. ., v n−2 not spanning the entirety of R d , and the authors of [BBN20] explicitly state that improving their bound to any d < n requires a different approach.…”

“…The current best bound, due to Brennan, Bresler, and Nagaraj [BBN20], asserts that in the regime p = Θ 1 n , d TV (Geo d (n, p), G(n, p)) → 0 so long as d ≫ n 3/2 . In essence, their bound relies on the fact that independent random vectors v i , v j ∼ S d−1 have |〈v i , v j 〉| = O( 1 d ) with high probability; when d ≫ n 3/2 , these inner products are small enough relative to n that v i has negligible projection (of order ≈ j =i 〈v i , v j 〉 2 = O( n/d)) into span{v j } j =i , which is enough to guarantee approximate independence of edges.…”

“…In essence, their bound relies on the fact that independent random vectors v i , v j ∼ S d−1 have |〈v i , v j 〉| = O( 1 d ) with high probability; when d ≫ n 3/2 , these inner products are small enough relative to n that v i has negligible projection (of order ≈ j =i 〈v i , v j 〉 2 = O( n/d)) into span{v j } j =i , which is enough to guarantee approximate independence of edges. This argument is carried out with technical sophistication in [BBN20], but clearly, this technique cannot be extended to d < n, much less to d = poly log n, as the vectors in {v j } j =i then span all of R d .…”