Reconstructing undirected graphs from eigenspaces
This addresses graph recovery challenges in applications like stationary signals or Markov chains, offering a novel method with theoretical guarantees but is incremental in graph reconstruction techniques.
The paper tackles the problem of reconstructing undirected weighted graphs from perturbed eigenspaces of their adjacency matrices, showing that identifiability follows a sharp phase transition with a threshold of N log N/2 edges in the Erdős-Rényi model, and provides support selection procedures with test statistics scaling as O(1/n) for overestimation.
In this paper, we aim at recovering an undirected weighted graph of $N$ vertices from the knowledge of a perturbed version of the eigenspaces of its adjacency matrix $W$. For instance, this situation arises for stationary signals on graphs or for Markov chains observed at random times. Our approach is based on minimizing a cost function given by the Frobenius norm of the commutator $\mathsf{A} \mathsf{B}-\mathsf{B} \mathsf{A}$ between symmetric matrices $\mathsf{A}$ and $\mathsf{B}$. In the Erdős-Rényi model with no self-loops, we show that identifiability (i.e., the ability to reconstruct $W$ from the knowledge of its eigenspaces) follows a sharp phase transition on the expected number of edges with threshold function $N\log N/2$. Given an estimation of the eigenspaces based on a $n$-sample, we provide support selection procedures from theoretical and practical point of views. In particular, when deleting an edge from the active support, our study unveils that our test statistic is the order of $\mathcal O(1/n)$ when we overestimate the true support and lower bounded by a positive constant when the estimated support is smaller than the true support. This feature leads to a powerful practical support estimation procedure. Simulated and real life numerical experiments assert our new methodology.