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 their 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.