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

Graph Community Detection from Coarse Measurements: Recovery Conditions for the Coarsened Weighted Stochastic Block Model