The phenomenon of information dissemination in social networks is widespread, and Social Network Information Cascade Analysis (SNICA) aims to acquire valuable knowledge in the process of information dissemination in social networks. As the number, volume, and resolution of social network data increase rapidly, traditional social network data analysis methods, especially the analysis method of social network graph (SNG) data becoming overwhelmed in SNICA. Recently, deep learning models have changed this situation, and it has achieved success in SNICA with its powerful implicit feature extraction capabilities. In this paper, we provide a comprehensive survey of recent progress in applying deep learning techniques for SNICA. We first introduce related concepts and summarize the advantages of deep learning technology in SNICA. Then, we propose a general framework for deep learning technology, which applies to SNICA. Then, different SNICA application scenarios in the framework are classified and discussed according to user behavior analysis, information cascade analysis, rumor detection, and social network event analysis. Finally, we discuss the limitations of current work and suggest future directions.