Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications

TL;DR

This paper investigates the fundamental limits and practical algorithms for reconstructing graphs from linear measurements, providing theoretical bounds and demonstrating effectiveness in applications like electric grid modeling.

Contribution

It introduces a sparsity-based characterization for random graph distributions, derives measurement bounds, and develops a polynomial-time recovery algorithm with empirical validation.

Findings

Derived necessary and sufficient measurement conditions for graph recovery.

Established tight bounds for specific graph classes like trees and Erdos-Renyi graphs.

Demonstrated the algorithm's effectiveness in electric grid applications.

Abstract

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdos-Renyi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction.