The Internet is by all accounts an incredible success, but in spite or maybe because of this success, its deficiencies have come under increasing scrutiny and triggered calls for new architectures to succeed it. Those architectures will, however, face a formidable incumbent in the Internet, and their ability to ultimately replace it is likely to depend equally on technical superiority as on economic factors. The goal of this paper is to start developing models that can help provide a quantitative understanding of a competition between the Internet and a new system, and show what factors affect it most strongly. A model for the adoption of competing network technologies by individual users is formulated and solved. It accounts for both the intrinsic value of each technology and the positive externalities derived from their respective numbers of adopters. Using this model, different configurations are explored and possible outcomes characterized. More importantly, configurations are identified where small differences in the attributes of either technology can lead to vastly different results. The paper provides initial results that can help identify parameters that significantly affect the likelihood of success of new network technologies.