Town hall-type debates are increasingly moving online, irrevocably transforming public discourse. Yet, we know relatively little about crucial social dynamics that determine which arguments are more likely to be successful. This study investigates the impact of one's position in the discussion network created via responses to others' arguments on one's persuasiveness in unfacilitated online debates. We propose a novel framework for measuring the impact of network position on persuasiveness, using a combination of social network analysis and machine learning. Complementing existing studies investigating the effect of linguistic aspects on persuasiveness, we show that the user's position in a discussion network influences their persuasiveness online. Moreover, the recognition of successful persuasion further increases this dominant network position. Our findings offer important insights into the complex social dynamics of online discourse and provide practical insights for organizations and individuals seeking to understand the interplay between influential positions in a discussion network and persuasive strategies in digital spaces.