Online Factorization and Partition of Complex Networks From Random Walks

TL;DR

This paper introduces an online, scalable method for factorizing and partitioning large-scale complex networks using random walk data, enabling dynamic analysis without explicit network observation.

Contribution

It proposes a novel stochastic generalized Hebbian algorithm for online network factorization and partitioning based on dependent random walk observations.

Findings

Algorithm converges globally to principal components of the Markov chain.

Achieves nearly optimal sample complexity.

Successfully recovers network partitions when the Markov process is lumpable.

Abstract

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.