  Community detection is an important information mining task to uncover
modular structures in large networks. For increasingly common large network
data sets, global community detection is prohibitively expensive, and attention
has shifted to methods that mine local communities, i.e. identifying all latent
members of a particular community from a few labeled seed members. To address
such semi-supervised mining task, we systematically develop a local spectral
subspace-based community detection method, called LOSP. We define a family of
local spectral subspaces based on Krylov subspaces, and seek a sparse indicator
for the target community via an $\ell_1$ norm minimization over the Krylov
subspace. Variants of LOSP depend on type of random walks with different
diffusion speeds, type of random walks, dimension of the local spectral
subspace and step of diffusions. The effectiveness of the proposed LOSP
approach is theoretically analyzed based on Rayleigh quotients, and it is
experimentally verified on a wide variety of real-world networks across social,
production and biological domains, as well as on an extensive set of synthetic
LFR benchmark datasets.
