Based on the characteristics of time-varying for the topology of real networks, an online social network was constructed by Twitter data set. Some results were found that the activity distribution of users was virtually independent of the time scale and degree distribution was heterogeneous as well as the side length distribution through analysis. Combined with the characteristics of the network, a spatial activity network model was proposed. The network topology was affected by the activity of nodes and the preferential attachment probability, the accuracy of the mechanism was proved through the analysis of the statistical characteristics. In order to study the dynamic process in time-varying network, a random walk process was carried out in the spatial activity network, the conclusion was obtained that the Mean First-Passage Time (MFPT) was shorter when the node was more active. Finally, the relationship between the preferential attachment power exponent and the average search time was discussed, and it was found that the power exponent with the highest search efficiency was 2 in spatial activity network. The proposed activity network model can be applied to time-varying network.