Clustering Partially Observed Graphs via Convex Optimization

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.

💡 Research Summary

The paper addresses the problem of clustering an undirected, unweighted graph when only a subset of node‑pair relationships is observed. Some pairs are known to be connected, some are known to be disconnected, and the rest are completely unobserved. The goal is to partition the nodes into disjoint clusters that exhibit dense internal connectivity and sparse external connectivity, while minimizing the number of “disagreements” – the sum of observed missing edges inside clusters and observed present edges across clusters.

The authors propose a novel reduction of this combinatorial problem to a matrix decomposition task. In an ideal clustering, after adding the identity matrix to the adjacency matrix, the result is block‑diagonal with all‑ones blocks; such a matrix has rank equal to the number of clusters. Any real graph can therefore be expressed as the sum of a low‑rank matrix (the ideal block‑diagonal component) and a sparse matrix (the disagreements). When only a subset of entries is observed, the task becomes to recover the low‑rank and sparse components from their partially observed sum.

To achieve this, they formulate a convex optimization problem (a semidefinite program):

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