Differential Analysis for Networks Obeying Conservation Laws

TL;DR

This paper introduces a convex method for differential network analysis in systems obeying conservation laws, enabling the estimation of structural changes from node potential data, with proven uniqueness conditions and efficient computation.

Contribution

It formulates a novel differential analysis problem for conservation law networks and proposes a convex estimator with theoretical guarantees and practical algorithms.

Findings

Successful estimation of network changes on synthetic data.

Effective application to benchmark power network data.

Theoretical conditions for unique solutions in high-dimensional settings.

Abstract

Networked systems that occur in various domains, such as the power grid, the brain, and opinion networks, are known to obey conservation laws. For instance, electric networks obey Kirchoff's laws, and social networks display opinion consensus. Such conservation laws are often modeled as balance equations that relate appropriate injected flows and potentials at the nodes of the networks. A recent line of work considers the problem of estimating the unknown structure of such networked systems from observations of node potentials (and only the knowledge of the statistics of injected flows). Given the dynamic nature of the systems under consideration, an equally important task is estimating the change in the structure of the network from data -- the so called differential network analysis problem. That is, given two sets of node potential observations, the goal is to estimate the structural differences between the underlying networks. We formulate this novel differential network analysis problem for systems obeying conservation laws and devise a convex estimator to learn the edge changes directly from node potentials. We derive conditions under which the estimate is unique in the high-dimensional regime and devise an efficient ADMM-based approach to perform the estimation. Finally, we demonstrate the performance of our approach on synthetic and benchmark power network data.