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

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We study fundamental trade-offs between the number of measurements (sample complexity), the complexity of the graph class, and the probability of error by first deriving a necessary condition (fundamental limit) on the number of measurements. Then, by considering a two-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. 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. Applying the Kirchhoff's matrix tree theorem, our results are extended to the scenario when part of the topology information is known a priori. In addition, we design and implement a polynomial-time (in n) algorithm based on the two-stage recovery scheme. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness of the proposed algorithm for accurate topology and parameter recovery.