  We introduce a new model of competition on growing networks. This extends the
preferential attachment model, with the key property that node choices evolve
simultaneously with the network. When a new node joins the network, it chooses
neighbours by preferential attachment, and selects its type based on the number
of initial neighbours of each type. The model is analysed in detail, and in
particular, we determine the possible proportions of the various types in the
limit of large networks. An important qualitative feature we find is that, in
contrast to many current theoretical models, often several competitors will
coexist. This matches empirical observations in many real-world networks.
