Consistency of community structure in complex networks

The most widely used techniques for community detection in networks, including methods based on modularity, statistical inference, and information theoretic arguments, all work by optimizing objective functions that measure the quality of network partitions. There is a good case to be made, however, that one should not look solely at the single optimal community structure under such an objective function, but rather at a selection of high-scoring structures. If one does this one typically finds that the resulting structures show considerable variation, and this has been taken as evidence that these community detection methods are unreliable, since they do not appear to give consistent answers. Here we argue that, upon closer inspection, the structures found are in fact consistent in a certain way. Specifically, we show that they can all be assembled from a set of underlying "building blocks", groups of network nodes that are usually found together in the same community. Different community structures correspond to different arrangements of blocks, but the blocks themselves are largely invariant. We propose an information theoretic method for discovering the building blocks in specific networks and demonstrate it with several example applications. We conclude that traditional community detection is not the failure some have suggested it is, and that in fact it gives a significant amount of insight into network structure, although perhaps not in exactly the way previously imagined.