Near-Perfect Recovery in the One-Dimensional Latent Space Model
Suppose a graph $G$ is stochastically created by uniformly sampling vertices along a line segment and connecting each pair of vertices with a probability that is a known decreasing function of their distance. We ask if it is possible to reconstruct the actual positions of the vertices in $G$ by only observing the generated unlabeled graph. We study this question for two natural edge probability functions -- one where the probability of an edge decays exponentially with the distance and another where this probability decays only linearly. We initiate our study with the weaker goal of recovering only the order in which vertices appear on the line segment. For a segment of length $n$ and a precision parameter $\delta$, we show that for both exponential and linear decay edge probability functions, there is an efficient algorithm that correctly recovers (up to reflection symmetry) the order of all vertices that are at least $\delta$ apart, using only $\tilde{O}(\frac{n}{\delta ^ 2})$ samples (vertices). Building on this result, we then show that $O(\frac{n^2 \log n}{\delta ^2})$ vertices (samples) are sufficient to additionally recover the location of each vertex on the line to within a precision of $\delta$. We complement this result with an $\Omega (\frac{n^{1.5}}{\delta})$ lower bound on samples needed for reconstructing positions (even by a computationally unbounded algorithm), showing that the task of recovering positions is information-theoretically harder than recovering the order. We give experimental results showing that our algorithm recovers the positions of almost all points with high accuracy.