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Functional Connectivity (FC) matrices measure the regional interactions in the brain and have been widely used in neurological brain disease classification. However, a FC matrix is neither a natural image which contains shape and texture information, nor a vector of independent features, which renders the extracting of efficient features from matrices as a challenging problem. A brain network, also named as connectome, could forma a graph structure naturally, the nodes of which are brain regions and the edges are interregional connectivity. Thus, in this study, we proposed novel graph convolutional networks (GCNs) to extract efficient disease-related features from FC matrices. Considering the time-dependent nature of brain activity, we computed dynamic FC matrices with sliding-windows and implemented a graph convolution based LSTM (long short term memory) layer to process dynamic graphs. Moreover, the demographics of patients were also used to guide the classification. However, unlike in conventional methods where personal information, i.e., gender and age were added as extra inputs, we argue that this kind of approach may not actually improve the classification performance, for such personal information given in dataset was usually balanced distributed. In this paper, we proposed to utilize the demographic information as extra outputs and to share parameters among three networks predicting subject status, gender and age, which serve as assistant tasks. We tested the performance of the proposed architecture in ADNI II dataset to classify Alzheimer's disease patients from normal controls. The classification accuracy, sensitivity and specificity reach 0.90, 0.92 and 0.89 on ADNI II dataset.