Community detection has attracted increasing attention during the past decade, and many algorithms have been proposed to find the underlying community structure in a given network. Many of these algorithms are based on modularity maximization, and these methods suffer from the resolution limit. In order to detect the underlying cluster structure, we propose a new convex formulation to decompose a partially observed adjacency matrix of a network into low-rank and sparse components. In such decomposition, the low-rank component encodes the cluster structure under certain assumptions. We also devise an alternating direction method of multipliers with increasing penalty sequence to solve this problem; and compare it with Louvain method, which maximizes the modularity, on some synthetic randomly generated networks. Numerical results show that our method outperforms Louvain method on the randomly generated networks when variance among cluster sizes increases. Moreover, empirical results also demonstrate that our formulation is indeed tighter than the robust PCA formulation, and is able to find the true clustering when the robust PCA formulation fails.