We study the evolution of the p53 core regulation network across the taxonomic span of humans to protozoans and nematodes. We introduce a new model for the core regulation network in mammalian cells, and conduct a formal analysis of the different network configurations that emerge in the evolutionary path to complexity. Solving the high dimensional equations associated with this model is typically challenging, and we develop a novel algorithm to overcome this problem. A key technical tool used is the representation of the distinct pathways in the core regulation networks as "modules", such that the behavior of the composite of two or more modules can be inferred from the characteristics of each of the individual modules. Apart from simplifying the complexity of the algorithm, this modular representation also allows us to qualitatively compare the distinct types of switching behaviors each network can exhibit. This then allows us to demonstrate how our model for the core regulation network in mammalian cells matches experimentally observed phenomena, and contrast this with the plausible behaviors admitted by the network configurations in putative primordial organisms. We show that the complexity of the p53 core regulation network in vertebrates permits a range of behaviors that can bring about distinct cell fate decisions not possible in the putative primordial organisms.