  In the modeling, monitoring, and control of complex networks, a fundamental
problem concerns the comprehensive determination of the state of the system
from limited measurements. Using power grids as example networks, we show that
this problem leads to a new type of percolation transition, here termed a {\it
network observability transition}, which we solve analytically for the
configuration model. We also demonstrate a dual role of the network's community
structure, which both facilitates optimal measurement placement and renders the
networks substantially more sensitive to `observability attacks'. Aside from
their immediate implications for the development of smart grids, these results
provide insights into decentralized biological, social, and technological
networks.
