Dynamic partition of complex networks

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 propose an efficient and scalable nonconvex stochastic gradient algorithm. It is able to process dependent 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 nonconvex stochastic algorithm achieves 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. The proposed approach is experimented on Manhattan taxi data.