Towards A Theoretical Framework for Analysis and Intervention of Random Drift on General Networks

It is well known that evolution is a fundamental phenomenon driving nature. And random drift is a basic force for population evolution. This technical note aims at constructing a unified theoretical framework for analyzing and intervening in random drift of binary states on general networks. In detail, a general methodology is developed for calculating the fixation probability with different dynamics, including the Wright-Fisher (WF), birth-death (BD), and death-birth (DB) processes. In particular, we prove that the fixation probability of a mutant at node k corresponds to the k-th element of stationary distribution of a stochastic matrix deduced from the weight matrix of the networks. Intuitively, it provides an effective way to discover the invasion hubs of structured population and further to intervene in random drift on networks. Finally, a typical example is then given to validate the above theoretical results.