## Clustering Partially Observed Graphs via Convex Optimization

*Yudong Chen, Ali Jalali, Sujay Sanghavi, Huan Xu*; 15(Jun):2213−2238, 2014.

### Abstract

This paper considers the problem of clustering a partially
observed unweighted graph---i.e., one where for some node pairs
we know there is an edge between them, for some others we know
there is no edge, and for the remaining we do not know whether
or not there is an edge. We want to organize the nodes into
disjoint clusters so that there is relatively dense (observed)
connectivity within clusters, and sparse across clusters. We
take a novel yet natural approach to this problem, by focusing
on finding the clustering that minimizes the number of
“disagreements”---i.e., the sum of the number of (observed)
missing edges within clusters, and (observed) present edges
across clusters. Our algorithm uses convex optimization; its
basis is a reduction of disagreement minimization to the problem
of recovering an (unknown) low-rank matrix and an (unknown)
sparse matrix from their partially observed sum. We evaluate the
performance of our algorithm on the classical Planted
Partition/Stochastic Block Model. Our main theorem provides
sufficient conditions for the success of our algorithm as a
function of the minimum cluster size, edge density and
observation probability; in particular, the results characterize
the tradeoff between the observation probability and the edge
density gap. When there are a constant number of clusters of
equal size, our results are optimal up to logarithmic factors.

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