Josep Dı́az

Varsha Dani, Josep Díaz, Thomas P. Hayes et al.

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or 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 the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if $r=n^α$ for any $α> 0$, with high probability reconstructs the vertex positions with a maximum error of $O(n^β)$ where $β=1/2-(4/3)α$, until $α\ge 3/8$ where $β=0$ and the error becomes $O(\sqrt{\log n})$. 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 extend our results to the surface of the sphere in $\R^3$ and to hypercubes in any constant fixed dimension. Additionally we examine the extent to which reconstruction is still possible when the original adjacency lists have had a subset of the edges independently deleted at random.

Josep Diaz, Colin McDiarmid, Dieter Mitsche

Suppose that there is a family of $n$ random points $X_v$ for $v \in V$, independently and uniformly distributed in the square $\left[-\sqrt{n}/2,\sqrt{n}/2\right]^2$ of area $n$. 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 $r$. If the threshold distance $r$ satisfies $n^{3/14} \ll r \ll n^{1/2}$, then 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 $O(\sqrt{\log n})$.