A scientific and reasonable assessment of port competitiveness will facilitate a comprehensive understanding of the current development of Chinese container ports along the Maritime Silk Road. Considering the network attributes of ports and the repetitive information among indicators, this paper proposes a novel port competitiveness evaluation and grading model based on entropy-improved catastrophe progression method. On the basis of a directed and weighted shipping network, the “dominance” indicator is innovatively introduced. The evaluation system including land-oriented competitiveness, sea-oriented competitiveness, hinterland support capacity and production and operation capacity provides a comprehensive assessment of port competitiveness. It is found that: land-oriented and sea-oriented competitiveness have the greatest influence on port competitiveness. A pattern of “One core, two wings, and four ports’ coordination” has been developed among coastal ports in China. The competition among ports in the Bohai Rim region is fierce, and production and operation ability of ports in the southeast coast needs to be strengthened. Fuzhou port and Rizhao port have high hinterland support and high development potential. The results of this paper can provide decision support for the development and upgrading of China’s coastal ports under the Maritime Silk Road initiative.