Multivariate time series data suffer from the problem of missing values, which hinders the application of many analytical methods. To achieve the accurate imputation of these missing values, exploiting inter-correlation by employing the relationships between sequences (i.e., a network) is as important as the use of temporal dependency, since a sequence normally correlates with other sequences. Moreover, exploiting an adequate network depending on time is also necessary since the network varies over time. However, in real-world scenarios, we normally know neither the network structure nor when the network changes beforehand. Here, we propose a missing value imputation method for multivariate time series, namely MissNet, that is designed to exploit temporal dependency with a state-space model and inter-correlation by switching sparse networks. The network encodes conditional independence between features, which helps us understand the important relationships for imputation visually. Our algorithm, which scales linearly with reference to the length of the data, alternatively infers networks and fills in missing values using the networks while discovering the switching of the networks. Extensive experiments demonstrate that MissNet outperforms the state-of-the-art algorithms for multivariate time series imputation and provides interpretable results.