Approaches relying on nonlinear dynamics and statistical mechanics have provided compelling models and crucial insights to understand interdisciplinary problems and emergent phenomena in complex systems. One paradigmatic example in the realm of social dynamics is the "voter model", where individuals in a population can be in one of two opinion states. The voter model is also closely related to evolutionary games used to model social and cooperation dilemmas. In this class of models, an individual is selected at random and adopts (with some probability) the state of its randomly-chosen neighbour; this update step is applied repeatedly. In this project, we propose to develop equally simple and paradigmatic individual-based models to investigate social behaviours like the emergence of cooperation, polarization and radicalization. For this, the dynamics will be implemented on various types of graphs and we will study a series of nonlinear (deterministic and stochastic) problems using a well-rounded combination of mathematical methods, notably the theory of dynamical systems and differential equations, stochastic processes and tools borrowed from statistical mechanics. Examples of problems and models which will be considered are the following:
(i) Dynamics of polarization: In many democracies, like in the UK, there are some major parties that trade governing roles every election and a number of other parties that access governing roles only very rarely. We shall devise voter-like models to describe to understand how polarization and marginalization emerge. We will typically study the probability for minority parties to reach the majority and the average time for such events to occur, and aim at characterizing the composition of the polarized states.
(ii) Consensus, cooperation and fanaticism: we will study the formation of consensus and the evolution of cooperation in voter-like models and in models of evolutionary game theory when the population is heterogeneous. As an example of population heterogeneity, we will consider the influence of fanatical individuals (zealots) favouring a specific "opinion" on the system's fate for various types of dynamics (update rules).
(iii) Emergence of radicalization: We shall consider a heterogeneous population of "compromisers", who move opinions closer to that of their interaction partner, and "rebels", who move their opinions further away. For this class of models, the goal will be to determine the conditions by which compromise or dissension wins at the societal level.