Community Detection with and without Prior Information

We study the problem of graph partitioning, or clustering, in sparse networks with prior information about the clusters. Specifically, we assume that for a fraction ρ of the nodes the true cluster assignments are known in advance. This can be understood as a semi–supervised version of clustering, in contrast to unsupervised clustering where the only available information is the graph structure. In the unsupervised case, it is known that there is a threshold of the inter–cluster connectivity beyond which clusters cannot be detected. Here we study the impact of the prior information on the detection threshold, and show that even minute [but generic] values of ρ > 0 shift the threshold downwards to its lowest possible value. For weighted graphs we show that a small semi-supervising can be used for a non-trivial definition of communities.