The automated analysis of digital human communication data often focuses on specific aspects such as content or network structure in isolation. This can provide limited perspectives while making cross-methodological analyses in domains like investigative journalism difficult. Communication research in psychology and the digital humanities instead stresses the importance of a holistic approach to overcome these limiting factors. In this work, we conduct an extensive survey on the properties of over forty semi-automated communication analysis systems and investigate how they cover concepts described in theoretical communication research. From these investigations, we derive a design space and contribute a conceptual framework based on communication research, technical considerations, and the surveyed approaches. The framework describes the systems' properties, capabilities, and composition through a wide range of criteria organized in the dimensions (1) Data, (2) Processing and Models, (3) Visual Interface, and (4) Knowledge Generation. These criteria enable a formalization of digital communication analysis through visual analytics, which, we argue, is uniquely suited for this task by tackling automation complexity while leveraging domain knowledge. With our framework, we identify shortcomings and research challenges, such as group communication dynamics, trust and privacy considerations, and holistic approaches. Simultaneously, our framework supports the evaluation of systems and promotes the mutual exchange between researchers through a structured, common language, laying the foundations for future research on communication analysis.