  Based on the definition of local spectral subspace, we propose a novel
approach called LOSP for local overlapping community detection. Instead of
using the invariant subspace spanned by the dominant eigenvectors of the entire
network, we run the power method for a few steps to approximate the leading
eigenvectors that depict the embedding of the local neighborhood structure
around seeds of interest. We then seek a sparse approximate indicator vector in
the local spectral subspace spanned by these vectors such that the seeds are in
its support.
  We evaluate LOSP on five large real world networks across various domains
with labeled ground-truth communities and compare the results with the
state-of-the-art community detection approaches. LOSP identifies the members of
a target community with high accuracy from very few seed members, and
outperforms the local Heat Kernel or PageRank diffusions as well as the global
baselines.
  Two candidate definitions of the local spectral subspace are analyzed, and
different community scoring functions for determining the community boundary,
including two new metrics, are thoroughly evaluated. The structural properties
of different seed sets and the impact of the seed set size are discussed. We
observe low degree seeds behave better, and LOSP is robust even when started
from a single random seed.
  Using LOSP as a subroutine and starting from each ego connected component, we
try the harder yet significant task of identifying all communities a single
vertex is in. Experiments show that the proposed method achieves high F1
measures on the detected multiple local overlapping communities containing the
seed vertex.
