We propose a dense insight network framework to encode the relationships between automatically generated insights from a complex dashboard based on their shared characteristics. Our insight network framework includes five high-level categories of relationships (e.g., type, topic, value, metadata, and compound scores). The goal of this insight network framework is to provide a foundation for implementing new insight interpretation and exploration strategies, including both user-driven and automated approaches. To illustrate the complexity and flexibility of our framework, we first describe a visualization playground to directly visualize key network characteristics; this playground also demonstrates potential interactive capabilities for decomposing the dense insight network. Then, we discuss a case study application for ranking insights based on the underlying network characteristics captured by our framework, before prompting a large language model to generate a concise, natural language summary. Finally, we reflect on next steps for leveraging our insight network framework to design and evaluate new systems.