  Community detection in network analysis aims at partitioning nodes in a
network into $K$ disjoint communities. Most currently available algorithms
assume that $K$ is known, but choosing a correct $K$ is generally very
difficult for real networks. In addition, many real networks contain outlier
nodes not belonging to any community, but currently very few algorithm can
handle networks with outliers. In this paper, we propose a novel model free
tightness criterion and an efficient algorithm to maximize this criterion for
community detection. This tightness criterion is closely related with the graph
Laplacian with $L_0$ penalty. Unlike most community detection methods, our
method does not require a known $K$ and can properly detect communities in
networks with outliers.
  Both theoretical and numerical properties of the method are analyzed. The
theoretical result guarantees that, under the degree corrected stochastic block
model, even for networks with outliers, the maximizer of the tightness
criterion can extract communities with small misclassification rates even when
the number of communities grows to infinity as the network size grows.
Simulation study shows that the proposed method can recover true communities
more accurately than other methods. Applications to a college football data and
a yeast protein-protein interaction data also reveal that the proposed method
performs significantly better.
