We consider the extraction of “coarse” descriptions of networks strictly from observing data supported on their nodes. Taking a graph signal processing perspective, we model the observed data as the output of a graph filter applied to white noise. We then consider two tasks. First, we infer the eigenvector centrality ranking of the underlying graph, drawing connections between network diffusion processes, graph filters, and the power method for eigenvector computation. In doing so, we derive statistical guarantees for correctly comparing pairs of nodes in terms of their eigenvector centrality ranking from a simple PCA-type procedure. Second, we extract the community structure of a collection of planted partition graphs over a shared set of nodes from nodal data. In this case, we show that the optimal conditions for centrality ranking are suboptimal for community detection. We also derive statistical guarantees for estimating the number of communities in the underlying planted partition model as well as exactly inferring the community membership structure.