  Optimal operation of distribution grid resources relies on accurate
estimation of its state and topology. Practical estimation of such quantities
is complicated by the limited presence of real-time meters. This paper
discusses a theoretical framework to jointly estimate the operational topology
and statistics of injections in radial distribution grids under limited
availability of nodal voltage measurements. In particular we show that our
proposed algorithms are able to provably learn the exact grid topology and
injection statistics at all unobserved nodes as long as they are not adjacent.
The algorithm design is based on novel ordered trends in voltage magnitude
fluctuations at node groups, that are independently of interest for radial
physical flow networks. The complexity of the designed algorithms is
theoretically analyzed and their performance validated using both linearized
and non-linear AC power flow samples in test distribution grids.
