Weighted Graph Clustering with Non-Uniform Uncertainties

We study the graph clustering problem where each observation (edge or no-edge between a pair of nodes) may have a different level of confidence/uncertainty. We propose a clustering algorithm that is based on optimizing an appropriate weighted objective, where larger weights are given to observations with lower uncertainty. Our approach leads to a convex optimization problem that is efficiently solvable. We analyze our approach under a natural generative model, and establish theoretical guarantees for recovering the underlying clusters. Our main result is a general theorem that applies to any given weight and distribution for the uncertainty. By optimizing over the weights, we derive a provably optimal weighting scheme, which matches the information theoretic lower bound up to logarithmic factors and leads to strong performance bounds in several specific settings. By optimizing over the uncertainty distribution, we show that nonuniform uncertainties can actually help. In particular, if the graph is built by spending a limited amount of resource to take measurement on each node pair, then it is beneficial to allocate the resource in a non-uniform fashion to obtain accurate measurements on a few pairs of nodes, rather than obtaining inaccurate measurements on many pairs. We provide simulation results that validate our theoretical findings.