Internet Collaboration on Extremely Difficult Problems: Research versus Olympiad Questions on the Polymath Site

Despite the existence of highly successful Internet collaborations on complex projects, including open-source software, little is known about how Internet collaborations work for solving "extremely" difficult problems, such as open-ended research questions. We quantitatively investigate a series of efforts known as the Polymath projects, which tackle mathematical research problems through open online discussion. A key analytical insight is that we can contrast the polymath projects with mini-polymaths -- spinoffs that were conducted in the same manner as the polymaths but aimed at addressing math Olympiad questions, which, while quite difficult, are known to be feasible. Our comparative analysis shifts between three elements of the projects: the roles and relationships of the authors, the temporal dynamics of how the projects evolved, and the linguistic properties of the discussions themselves. We find interesting differences between the two domains through each of these analyses, and present these analyses as a template to facilitate comparison between Polymath and other domains for collaboration and communication. We also develop models that have strong performance in distinguishing research-level comments based on any of our groups of features. Finally, we examine whether comments representing research breakthroughs can be recognized more effectively based on their intrinsic features, or by the (re-)actions of others, and find good predictive power in linguistic features.