  Physical Flow Networks are different infrastructure networks that allow the
flow of physical commodities through edges between its constituent nodes. These
include power grid, natural gas transmission network, water pipelines etc. In
such networks, the flow on each edge is characterized by a function of the
nodal potentials on either side of the edge. Further the net flow in and out of
each node is conserved. Learning the structure and state of physical networks
is necessary for optimal control as well as to quantify its privacy needs. We
consider radial flow networks and study the problem of learning the operational
network from a loopy graph of candidate edges using statistics of nodal
potentials. Based on the monotonic properties of the flow functions, the key
result in this paper shows that if variance of the difference of nodal
potentials is used to weight candidate edges, the operational edges form the
minimum spanning tree in the loopy graph. Under realistic conditions on the
statistics of nodal injection (consumption or production), we provide a greedy
structure learning algorithm with quasilinear computational complexity in the
number of candidate edges in the network. Our learning framework is very
general due to two significant attributes. First it is independent of the
specific marginal distributions of nodal potentials and only uses order
properties in their second moments. Second, the learning algorithm is agnostic
to exact flow functions that relate edge flows to corresponding potential
differences and is applicable for a broad class of networks with monotonic flow
functions. We demonstrate the efficacy of our work through realistic
simulations on diverse physical flow networks and discuss possible extensions
of our work to other regimes.
