  Community detection is a fundamental and important problem in network
science, as community structures often reveal both topological and functional
relationships between different components of the complex system. In this
paper, we first propose a gradient descent framework of modularity optimization
called vector-label propagation algorithm (VLPA), where a node is associated
with a vector of continuous community labels instead of one label. Retaining
weak structural information in vector-label, VLPA outperforms some well-known
community detection methods, and particularly improves the performance in
networks with weak community structures. Further, we incorporate stochastic
gradient strategies into VLPA to avoid stuck in the local optima, leading to
the stochastic vector-label propagation algorithm (sVLPA). We show that sVLPA
performs better than Louvain Method, a widely used community detection
algorithm, on both artificial benchmarks and real-world networks. Our
theoretical scheme based on vector-label propagation can be directly applied to
high-dimensional networks where each node has multiple features, and can also
be used for optimizing other partition measures such as modularity with
resolution parameters.
