How can we reduce the size of a graph without significantly altering its basic properties? We approach the graph reduction problem from the perspective of restricted similarity, a modification of a well-known measure for graph approximation. Our choice is motivated by the observation that restricted similarity implies strong spectral guarantees and can be used to prove statements about certain unsupervised learning problems. The paper then focuses on coarsening, a popular type of graph reduction. We derive sufficient conditions for a small graph to approximate a larger one in the sense of restricted similarity. Our theoretical findings give rise to a novel quasi-linear algorithm. Compared to both standard and advanced graph reduction methods, the proposed algorithm finds coarse graphs of improved quality -often by a large margin- without sacrificing speed.
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