Finding experts in question answering platforms has important applications, such as question routing or identification of best answers. Addressing the problem of ranking users with respect to their expertise, we propose Competition-Based Expertise Networks (CBEN), a novel community expertise network structure based on the principle of competition among the answerers of a question. We evaluate our approach on a very large dataset from Yahoo Answers using a variety of centrality measures. We show that it outperforms state-of-the-art network structures and, unlike previous methods, is able to consistently outperform simple metrics like best answer count. We also analyse question answering forums in Yahoo Answers, and show that they can be characterised by factual or subjective information seeking behavior, social discussions and the conducting of polls or surveys. We find that the ability to identify experts greatly depends on the type of forum, which is directly reflected in the structural properties of the expertise networks.