  Given the widespread popularity of spectral clustering (SC) for partitioning
graph data, we study a version of constrained SC in which we try to incorporate
the fairness notion proposed by Chierichetti et al. (2017). According to this
notion, a clustering is fair if every demographic group is approximately
proportionally represented in each cluster. To this end, we develop variants of
both normalized and unnormalized constrained SC and show that they help find
fairer clusterings on both synthetic and real data. We also provide a rigorous
theoretical analysis of our algorithms on a natural variant of the stochastic
block model, where $h$ groups have strong inter-group connectivity, but also
exhibit a "natural" clustering structure which is fair. We prove that our
algorithms can recover this fair clustering with high probability.
