The effective visual exploration of dynamic networks has been one of the toughest challenges and an unsolved problem; however, it is very important to understand network evolution. Although many developments have been achieved in modeling evolutionary networks, the closely related task of visualizing continues to remain a major concern. Therefore, in this study, quantitative analysis is used to assign node attributes in the network topology, and then, the evolutionary process of networks is analyzed. By fixing the position of nodes, the possibility of a stationary shape of the network is suggested, and a more intuitive and comprehensive explanation of the enumeration of the types for the evolutionary process is provided. Further, a large amount of information is presented in this study in an extremely economical and accessible way by incorporating a circular layout and evolution laws, which offers a new approach for the estimation and evaluation of network evolution. Finally, this three-pronged approach—network analysis, quantitative method, and topological modeling—is expected to provide a revelatory insight into the principle of network evolution.