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.

Apps are emerging as an important form of on-line content, and they combine aspects of Web usage in interesting ways --- they exhibit a rich temporal structure of user adoption and long-term engagement, and they exist in a broader social ecosystem that helps drive these patterns of adoption and engagement. It has been difficult, however, to study apps in their natural setting since this requires a simultaneous analysis of a large set of popular apps and the underlying social network they inhabit. In this work we address this challenge through an analysis of the collection of apps on Facebook Login, developing a novel framework for analyzing both temporal and social properties. At the temporal level, we develop a retention model that represents a user's tendency to return to an app using a very small parameter set. At the social level, we organize the space of apps along two fundamental axes --- popularity and sociality --- and we show how a user's probability of adopting an app depends both on properties of the local network structure and on the match between the user's attributes, his or her friends' attributes, and the dominant attributes within the app's user population. We also devolop models that show the importance of different feature sets with strong performance in predicting app success.

In many applications we have a social network of people and would like to identify the members of an interesting but unlabeled group or community. We start with a small number of exemplar group members -- they may be followers of a political ideology or fans of a music genre -- and need to use those examples to discover the additional members. This problem gives rise to the seed expansion problem in community detection: given example community members, how can the social graph be used to predict the identities of remaining, hidden community members? In contrast with global community detection (graph partitioning or covering), seed expansion is best suited for identifying communities locally concentrated around nodes of interest. A growing body of work has used seed expansion as a scalable means of detecting overlapping communities. Yet despite growing interest in seed expansion, there are divergent approaches in the literature and there still isn't a systematic understanding of which approaches work best in different domains. Here we evaluate several variants and uncover subtle trade-offs between different approaches. We explore which properties of the seed set can improve performance, focusing on heuristics that one can control in practice. As a consequence of this systematic understanding we have found several opportunities for performance gains. We also consider an adaptive version in which requests are made for additional membership labels of particular nodes, such as one finds in field studies of social communities. This leads to interesting connections and contrasts with active learning and the trade-offs of exploration and exploitation. Finally, we explore topological properties of communities and seed sets that correlate with algorithm performance, and explain these empirical observations with theoretical ones. We evaluate our methods across multiple domains, using publicly available datasets with labeled, ground-truth communities.