  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.
