Improved Reconstruction of Random Geometric Graphs

Embedding graphs in a geographical or latent space, i.e., inferring locations for vertices in Euclidean space or on a smooth submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to symmetry, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r=n^α for α > 0, with high probability reconstructs the vertex positions with a maximum error of O(n^β) where β=1/2-(4/3)α, until α≥ 3/8 where β=0 and the error becomes O(√(log n)). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We sketch proofs that our results also apply on the surface of a sphere, and (with somewhat different exponents) in any fixed dimension.

• 4 publications
• 7 publications
• 8 publications
• 24 publications
research
09/26/2018

Learning random points from geometric graphs or orderings

Suppose that there is a family of n random points X_v for v ∈ V, indepen...
research
06/08/2020

Near-Perfect Recovery in the One-Dimensional Latent Space Model

Suppose a graph G is stochastically created by uniformly sampling vertic...
research
04/27/2018

On the Estimation of Latent Distances Using Graph Distances

We are given the adjacency matrix of a geometric graph and the task of r...
research
10/26/2020

Random Geometric Graphs on Euclidean Balls

We consider a latent space model for random graphs where a node i is ass...
research
09/15/2019

Latent Distance Estimation for Random Geometric Graphs

Random geometric graphs are a popular choice for a latent points generat...
research
06/10/2019

Latent Channel Networks

Latent Euclidean embedding models a given network by representing each n...
research
05/21/2013

Out-of-sample Extension for Latent Position Graphs

We consider the problem of vertex classification for graphs constructed ...