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We study the misclassification error for community detection in generalheterogeneous stochastic block models (SBM) with noisy or partial labelinformation. We establish a connection between the misclassification rate andthe notion of minimum energy on the local neighborhood of the SBM. We developan optimally weighted message passing algorithm to reconstruct labels for SBMbased on the minimum energy flow and the eigenvectors of a certain Markovtransition matrix. The general SBM considered in this paper allows forunequal-size communities, degree heterogeneity, and different connectionprobabilities among blocks. We focus on how to optimally weigh the messagepassing to improve misclassification.

Weighted Message Passing and Minimum Energy Flow for Heterogeneous Stochastic Block Models with Side Information