People are the most important factors of production and the primary carriers of social culture. In addition to the impacts on the origin and destination, the volume of cross-border migrants can reflect the relationship between two countries. In fact, the migration relationship of countries is complex and multilateral, and network theories could provide a better description and more clearly show the structure and statistical characteristics. Based on the estimated bilateral migration data and disparity filter algorithm, we extract the global migration networks describing the multilateral migration relationships between 200 countries over fifty years. The results show that the global migration network during 1960-2015 exhibits obvious clustering and a disassortative property and that it is experiencing globalized and multipolarized changes during these years. Moreover, the global migration network has a typical "core-periphery" structure, and we certify its hyperbolic geometry and embed the network into a Poincar disk, which could present the hierarchical structure of the global migration network and describe the status, role and evolution trends of different countries. Finally, we analyze the correlation and evolution of communities. The results show the stability of most communities, but there are still structural changes, such as the influence of the collapse of the Soviet Union on the Eurasian pattern, the intimacy between France and Africa, Canada's relative affinity between America and the British Commonwealth, and the combination of the countries around the Indian Ocean.