Stochastic optimization algorithms are widely used for large-scale data analysis due to their low per-iteration costs, but they often suffer from slow asymptotic convergence caused by inherent variance. Variance-reduced techniques have been therefore used to address this issue in structured sparse models utilizing sparsity-inducing norms or $\ell_0$-norms. However, these techniques are not directly applicable to complex (non-convex) graph sparsity models, which are essential in applications like disease outbreak monitoring and social network analysis. In this paper, we introduce two stochastic variance-reduced gradient-based methods to solve graph sparsity optimization: GraphSVRG-IHT and GraphSCSG-IHT. We provide a general framework for theoretical analysis, demonstrating that our methods enjoy a linear convergence speed. Extensive experiments validate