Testing thresholds for high-dimensional sparse random geometric graphs
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In the random geometric graph model 𝖦𝖾𝗈_d(n,p), we identify each of our n vertices with an independently and uniformly sampled vector from the d-dimensional unit sphere, and we connect pairs of vertices whose vectors are “sufficiently close”, such that the marginal probability of an edge is p. We investigate the problem of testing for this latent geometry, or in other words, distinguishing an Erdős-Rényi graph 𝖦(n, p) from a random geometric graph 𝖦𝖾𝗈_d(n, p). It is not too difficult to show that if d→∞ while n is held fixed, the two distributions become indistinguishable; we wish to understand how fast d must grow as a function of n for indistinguishability to occur. When p = α/n for constant α, we prove that if d ≥polylog n, the total variation distance between the two distributions is close to 0; this improves upon the best previous bound of Brennan, Bresler, and Nagaraj (2020), which required d ≫ n^3/2, and further our result is nearly tight, resolving a conjecture of Bubeck, Ding, Eldan, & Rácz (2016) up to logarithmic factors. We also obtain improved upper bounds on the statistical indistinguishability thresholds in d for the full range of p satisfying 1/n≤ p≤1/2, improving upon the previous bounds by polynomial factors. Our analysis uses the Belief Propagation algorithm to characterize the distributions of (subsets of) the random vectors conditioned on producing a particular graph. In this sense, our analysis is connected to the “cavity method” from statistical physics. To analyze this process, we rely on novel sharp estimates for the area of the intersection of a random sphere cap with an arbitrary subset of the sphere, which we prove using optimal transport maps and entropy-transport inequalities on the unit sphere.
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