  Finding "densely connected clusters" in a graph is in general an important
and well studied problem in the literature \cite{Schaeffer}. It has various
applications in pattern recognition, social networking and data mining
\cite{Duda,Mishra}. Recently, Ames and Vavasis have suggested a novel method
for finding cliques in a graph by using convex optimization over the adjacency
matrix of the graph \cite{Ames, Ames2}. Also, there has been recent advances in
decomposing a given matrix into its "low rank" and "sparse" components
\cite{Candes, Chandra}. In this paper, inspired by these results, we view
"densely connected clusters" as imperfect cliques, where imperfections
correspond missing edges, which are relatively sparse. We analyze the problem
in a probabilistic setting and aim to detect disjointly planted clusters. Our
main result basically suggests that, one can find \emph{dense} clusters in a
graph, as long as the clusters are sufficiently large. We conclude by
discussing possible extensions and future research directions.
