Growing literature explores toxicity and polarization in discourse, with comparatively less work on characterizing what makes dialogue prosocial and constructive. We explore conversational discourse and investigate a method for characterizing its quality built upon the notion of responsivity''---whether one person's conversational turn is responding to a preceding turn. We develop and evaluate methods for quantifying responsivity---first through semantic similarity of speaker turns, and second by leveraging state-of-the-art large language models (LLMs) to identify the relation between two speaker turns. We evaluate both methods against a ground truth set of human-annotated conversations. Furthermore, selecting the better performing LLM-based approach, we characterize the nature of the response---whether it responded to that preceding turn in a substantive way or not. We view these responsivity links as a fundamental aspect of dialogue but note that conversations can exhibit significantly different responsivity structures. Accordingly, we then develop conversation-level derived metrics to address various aspects of conversational discourse. We use these derived metrics to explore other conversations and show that they support meaningful characterizations and differentiations across a diverse collection of conversations.