  Spectral clustering is widely used in practice due to its flexibility,
computational efficiency, and well-understood theoretical performance
guarantees. Recently, spectral clustering has been studied to find balanced
clusters under population-level constraints. These constraints are specified by
additional information available in the form of auxiliary categorical node
attributes. In this paper, we consider a scenario where these attributes may
not be observable, but manifest as latent features of an auxiliary graph.
Motivated by this, we study constrained spectral clustering with the aim of
finding balanced clusters in a given \textit{similarity graph} $\mathcal{G}$,
such that each individual is adequately represented with respect to an
auxiliary graph $\mathcal{R}$ (we refer to this as representation graph). We
propose an individual-level balancing constraint that formalizes this idea. Our
work leads to an interesting stochastic block model that not only plants the
given partitions in $\mathcal{G}$ but also plants the auxiliary information
encoded in the representation graph $\mathcal{R}$. We develop unnormalized and
normalized variants of spectral clustering in this setting. These algorithms
use $\mathcal{R}$ to find clusters in $\mathcal{G}$ that approximately satisfy
the proposed constraint. We also establish the first statistical consistency
result for constrained spectral clustering under individual-level constraints
for graphs sampled from the above-mentioned variant of the stochastic block
model. Our experimental results corroborate our theoretical findings.
