  The geometric block model is a recently proposed generative model for random
graphs that is able to capture the inherent geometric properties of many
community detection problems, providing more accurate characterizations of
practical community structures compared with the popular stochastic block
model. Galhotra et al. recently proposed a motif-counting algorithm for
unsupervised community detection in the geometric block model that is proved to
be near-optimal. They also characterized the regimes of the model parameters
for which the proposed algorithm can achieve exact recovery. In this work, we
initiate the study of active learning in the geometric block model. That is, we
are interested in the problem of exactly recovering the community structure of
random graphs following the geometric block model under arbitrary model
parameters, by possibly querying the labels of a limited number of chosen
nodes. We propose two active learning algorithms that combine the idea of
motif-counting with two different label query policies. Our main contribution
is to show that sampling the labels of a vanishingly small fraction of nodes
(sub-linear in the total number of nodes) is sufficient to achieve exact
recovery in the regimes under which the state-of-the-art unsupervised method
fails. We validate the superior performance of our algorithms via numerical
simulations on both real and synthetic datasets.
