An online fountain code is defined as a fountain code for which an optimal encoding strategy can be found efficiently given any instantaneous decoding state. This property is important for data distribution in practical networks. In this paper, we formalize the problem of online fountain code construction, and propose new online fountain codes that outperform known ones in having factor 3-5 lower redundancy overhead. The bounding of the code overhead is carried out using the analysis of the dynamics of random-graph processes.