Link analysis for web search spawned a surge of interest in mathematical techniques for improving the user experience in information retrieval. Although developed initially to combat the ”spamming” of text-based search engines, link analysis has over time become susceptible to innovative forms of ”link spam”; the battle continues between web search engines seeking to preserve editorial integrity and their adversaries with strong commercial interests. Such link analysis is but one visible example of the broader area of social network analysis, which originated with the classic experiment from the 60’s due to Stanley Milgram, leading to the popular folklore that any two humans have at most ”six degrees of separation”. Over the last ten years, these ideas from social network analysis have progressed beyond networks of links between people. The divestiture of the telephone monopoly in the United States (and subsequently in other countries) led to the study of networks of phone calls. In the network of emails between users, dense regions are know to form around participants with common topical interests. A new breed of examples comes from so-called recommendation systems deployed at eCommerce sites on the web: by analyzing the links between people and the products they purchase or rate, the system recommends to users what products they might be interested in, based on their and other users’ past behavior. The simplest incarnation is a recommendation of the form ”people who purchased this item also purchased ...” While these examples are promising, much work remains to be done in this nascent area. For instance, what are the models and criteria by which to optimize such systems? How should they be evaluated? How (if at all) should users be compensated for offering their opinions on items, given that the eCommerce site is profiting from these opinions? We describe analytical approaches to these questions drawing from linear algebra and game theory. Recent work has begun exploring so-called webs of trust-networks in which a link from one user to another carries a real number denoting how much the first user trusts (or distrusts) the second. Such networks are a critical ingredient at websites such as eBay and epinions. Studies in economics suggest that there is a direct relationship between the trust assigned to a seller and the price that he can charge for a given item. Many computational questions arise in such webs of trust. If A distrusts B and B distrusts C, what does this imply about A’s trust for C? If C likes a book, is A likely to? Questions such as these must first be resolved at a philosophical level before being approached mathematically. We describe some recent results in this area that raise a number of questions for further work.