Spectral Clustering with Side Information

In the graph clustering problem with a planted solution, the input is a graph on $n$ vertices partitioned into $k$ clusters, and the task is to infer the clusters from graph structure. A standard assumption is that clusters induce well-connected subgraphs (i.e. $\Omega(1)$-expanders), and form $\epsilon$-sparse cuts. Such a graph defines the clustering uniquely up to $\approx \epsilon$ misclassification rate, and efficient algorithms for achieving this rate are known. While this vanilla version of graph clustering is well studied, in practice, vertices of the graph are typically equipped with labels that provide additional information on cluster ids of the vertices. For example, each vertex could have a cluster label that is corrupted independently with probability $\delta$. Using only one of the two sources of information leads to misclassification rate $\min\{\epsilon, \delta\}$, but can they be combined to achieve a rate of $\approx \epsilon \delta$? In this paper, we give an affirmative answer to this question and present a sublinear-time algorithm in the number of vertices $n$. Our key algorithmic insight is a new observation on ``spectrally ambiguous'' vertices in a well-clusterable graph. While our sublinear-time classifier achieves the nearly optimal $\approx \widetilde O(\epsilon \delta)$ misclassification rate, the approximate clusters that it outputs do not necessarily induce expanders in the graph $G$. In our second result, we give a polynomial-time algorithm that reweights edges of the original $(k, \epsilon, \Omega(1))$-clusterable graph to transform it into a $(k, \widetilde O(\epsilon \delta), \Omega(1))$-clusterable one (for constant $k$), improving sparsity of cuts nearly optimally and preserving expansion properties of the communities - an algorithm for refining community structure of the input graph.