In this paper, new synchronization and state estimation problems are considered for an array of coupled discrete time-varying stochastic complex networks over a finite horizon. A novel concept of bounded H_∞ synchronization is proposed to handle the time-varying nature of the complex networks. Such a concept captures the transient behavior of the time-varying complex network over a finite horizon, where the degree of bounded synchronization is quantified in terms of the H_∞-norm. A general sector-like nonlinear function is employed to describe the nonlinearities existing in the network. By utilizing a timevarying real-valued function and the Kronecker product, criteria are established that ensure the bounded H_∞ synchronization in terms of a set of recursive linear matrix inequalities (RLMIs), where the RLMIs can be computed recursively by employing available MATLAB toolboxes. The bounded H_∞ state estimation problem is then studied for the same complex network, where the purpose is to design a state estimator to estimate the network states through available output measurements such that, over a finite horizon, the dynamics of the estimation error is guaranteed to be bounded with a given disturbance attenuation level. Again, an RLMI approach is developed for the state estimation problem. Finally, two simulation examples are exploited to show the effectiveness of the results derived in this paper.