  We study the problem of community recovery from coarse measurements of a
graph. In contrast to the problem of community recovery of a fully observed
graph, one often encounters situations when measurements of a graph are made at
low-resolution, each measurement integrating across multiple graph nodes. Such
low-resolution measurements effectively induce a coarse graph with its own
communities. Our objective is to develop conditions on the graph structure, the
quantity, and properties of measurements, under which we can recover the
community organization in this coarse graph. In this paper, we build on the
stochastic block model by mathematically formalizing the coarsening process,
and characterizing its impact on the community members and connections. Through
this novel setup and modeling, we characterize an error bound for community
recovery. The error bound yields simple and closed-form asymptotic conditions
to achieve the perfect recovery of the coarse graph communities.
