Reconstructing the Geometry of Random Geometric Graphs
This work addresses a problem in machine learning and network analysis by providing a method to infer manifold structure from graph data, complementing existing manifold learning approaches.
The paper tackles the problem of reconstructing the geometry of random geometric graphs under the manifold assumption, showing how to efficiently recover the underlying low-dimensional manifold from the sampled graph.
Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work, we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the manifold assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in $\mathbb{R}^N$. Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.