  Community detection, which aims to cluster $N$ nodes in a given graph into
$r$ distinct groups based on the observed undirected edges, is an important
problem in network data analysis. In this paper, the popular stochastic block
model (SBM) is extended to the generalized stochastic block model (GSBM) that
allows for adversarial outlier nodes, which are connected with the other nodes
in the graph in an arbitrary way. Under this model, we introduce a procedure
using convex optimization followed by $k$-means algorithm with $k=r$. Both
theoretical and numerical properties of the method are analyzed. A theoretical
guarantee is given for the procedure to accurately detect the communities with
small misclassification rate under the setting where the number of clusters can
grow with $N$. This theoretical result admits to the best-known result in the
literature of computationally feasible community detection in SBM without
outliers. Numerical results show that our method is both computationally fast
and robust to different kinds of outliers, while some popular computationally
fast community detection algorithms, such as spectral clustering applied to
adjacency matrices or graph Laplacians, may fail to retrieve the major clusters
due to a small portion of outliers. We apply a slight modification of our
method to a political blogs data set, showing that our method is competent in
practice and comparable to existing computationally feasible methods in the
literature. To the best of the authors' knowledge, our result is the first in
the literature in terms of clustering communities with fast growing numbers
under the GSBM where a portion of arbitrary outlier nodes exist.
