Our empirical results suggest a significantly more refined picture of community structure than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small size scales, and at larger size scales, the best possible communities gradually ``blend in'' with the rest of the network and thus become less ``community-like.'' This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are well-embeddable in a low-dimensional structure, and from small social networks that have served as testbeds of community detection algorithms. Possible mechanisms for reproducing our empirical observations will be discussed, as will implications of these findings for clustering, classification, and more general data analysis in modern large social and information networks.