  Finding the reduced-dimensional structure is critical to understanding
complex networks. Existing approaches such as spectral clustering are
applicable only when the full network is explicitly observed. In this paper, we
focus on the online factorization and partition of implicit large-scale
networks based on observations from an associated random walk. We formulate
this into a nonconvex stochastic factorization problem and propose an efficient
and scalable stochastic generalized Hebbian algorithm. The algorithm is able to
process dependent state-transition data dynamically generated by the underlying
network and learn a low-dimensional representation for each vertex. By applying
a diffusion approximation analysis, we show that the continuous-time limiting
process of the stochastic algorithm converges globally to the "principal
components" of the Markov chain and achieves a nearly optimal sample
complexity. Once given the learned low-dimensional representations, we further
apply clustering techniques to recover the network partition. We show that when
the associated Markov process is lumpable, one can recover the partition
exactly with high probability. We apply the proposed approach to model the
traffic flow of Manhattan as city-wide random walks. By using our algorithm to
analyze the taxi trip data, we discover a latent partition of the Manhattan
city that closely matches the traffic dynamics.
