# Learning random points from geometric graphs or orderings

Suppose that there is a family of n random points X_v for v ∈ V, independently and uniformly distributed in the square [-√(n)/2,√(n)/2]^2. We do not see these points, but learn about them in one of the following two ways. Suppose first that we are given the corresponding random geometric graph G, where distinct vertices u and v are adjacent when the Euclidean distance d_E(X_u,X_v) is at most r. Assume that the threshold distance r satisfies n^3/14≪ r ≪ n^1/2. We shall see that the following holds with high probability. Given the graph G (without any geometric information), in polynomial time we can approximately reconstruct the hidden embedding, in the sense that, `up to symmetries', for each vertex v we find a point within distance about r of X_v; that is, we find an embedding with `displacement' at most about r. Now suppose that, instead of being given the graph G, we are given, for each vertex v, the ordering of the other vertices by increasing Euclidean distance from v. Then, with high probability, in polynomial time we can find an embedding with the much smaller displacement error O(√( n)).

• 7 publications
• 2 publications
• 6 publications
research
04/12/2018

### Connectivity in Random Annulus Graphs and the Geometric Block Model

Random geometric graphs are the simplest, and perhaps the earliest possi...
research
07/29/2021

### Improved Reconstruction of Random Geometric Graphs

Embedding graphs in a geographical or latent space, i.e., inferring loca...
research
08/03/2023

### Reconstruction of colours

Given a graph G, when is it possible to reconstruct with high probabilit...
research
07/13/2020

### Reconstruction of Line-Embeddings of Graphons

Consider a random graph process with n vertices corresponding to points ...
research
04/23/2018

### How to Realize a Graph on Random Points

We are given an integer d, a graph G=(V,E), and a uniformly random embed...
research
12/08/2021

### Consistency of Spectral Seriation

Consider a random graph G of size N constructed according to a graphon w...
research
03/28/2021

### Phase transition in noisy high-dimensional random geometric graphs

We study the problem of detecting latent geometric structure in random g...