Recovering Small Communities in the Planted Partition Model

We analyze community recovery in the planted partition model (PPM) in regimes where the number of communities is arbitrarily large. We examine the three standard recovery regimes: exact recovery, almost exact recovery, and weak recovery. When communities vary in size, traditional accuracy- or alignment-based metrics become unsuitable for assessing the correctness of a predicted partition. To address this, we redefine these recovery regimes using the correlation coefficient, a more versatile metric for comparing partitions. We then demonstrate that $\textit{Diamond Percolation}$, an algorithm based on common-neighbors, successfully recovers communities under mild assumptions on edge probabilities, with minimal restrictions on the number and sizes of communities. As a key application, we consider the case where community sizes follow a power-law distribution, a characteristic frequently found in real-world networks. To the best of our knowledge, we provide the first recovery results for such unbalanced partitions.