Nonlinear Voter Models: The Transition from Invasion to Coexistence

In nonlinear voter models the transitions between two states depend in a nonlinear manner on the frequen ies of these states in the neighborhood. We investigate the role of these nonlinearities on the global out ome of the dynami s for a homogeneous network where ea h node is onne ted to m = 4 neighbors. The paper unfolds in two dire tions. We rst develop a general sto hasti framework for frequen y dependent pro esses from whi h we derive the ma ros opi dynami s for key variables, su h as global frequen ies and orrelations. Expli it expressions for both the meaneld limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter spa e that distinguishes between di erent dynami regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) omplete invasion, (ii) random oexisten e, and most interestingly (iii) orrelated oexisten e. These ndings are ontrasted with predi tions from the meaneld phase diagram and are on rmed by extensive omputer simulations of the mi ros opi dynami s. PACS : 87.23.C Population dynami s and e ologi al pattern formation, 87.23.Ge Dynami s of so ial systems