Time series classification is an important topic in data mining. Time series classification methods have been studied by many researchers. A feature-weighted classification method is proposed based on complex network. There are four stages in the proposed classification algorithm. First, we obtain visible points and edge weights by applying an improved weighted visibility graph algorithm to transform the original time series into weighted complex networks. Next, weighted features are calculated based on visible points and weights, and unable-weighted features are obtained by utilizing NetworkX. After that, we combine two types of features to obtain a total features matrix. We then reconsider the weight of each independent feature by utilizing the importance function of random forest to measure the relative importance of features and construct a importance weighted features matrix. Finally, we utilize a traditional classifier called random forest to cluster the weighted features matrix and generate classification results. Simultaneously, a novel feature weight calculation method is applied during the classification process. We compare the proposed method to other classification methods and the results indicate that the proposed method can improve classification accuracy for time series datasets.