Folk art education is an important way to inherit and develop traditional culture. In this paper, the cascade propagation of typical decentralized Internet-social networks is modeled as a propagation dynamic graph model, and an enhanced graph-aware neural network is proposed through the analysis of the learning process of neural graph networks. A further recurrent graph-aware neural network is proposed for the characteristics of information dissemination in the decentralized Internet, and the transmission and development of traditional culture in folk art education are analyzed based on this network model. In folk art education, the most common type of traditional culture dissemination is ink painting, accounting for 20.32%, which is 6.91%, 12.35%, and 14.86% higher than other types, respectively. From 2014 to 2021, the percentage of Internet-based communication media increased from 12.47% to 24.78%, an increase of 12.31 percentage points. The analysis based on the decentralized Internet can accurately extract the characteristics of traditional culture integrated into folk art education, which helps to inherit further and promote the excellent traditional culture.