The non-linear noisy voter model

Stylized facts of financial markets (fat tails, volatility clustering) can be considered as an emergent phenomenon of the interactions among traders. One of the simplest agent-based models capable of reproducing these statistical properties is the one proposed by Kirman based on herding and idiosincratic behavior. A fundamental aspect of the model is that agents can adopt an opinion based on the proportion of neighbor agents holding it. However, most of the times, crucial features such as the network of agent interactions and the functional form of the probability transition rates are disregarded.

In this talk we report results on the generalization of the model by including non-linear transition rates that minic a more general imitation process than the indiscriminate copy of the original model. The consideration of this new ingredient implies that the phase transition (from an unimodal to a bimodal distribution of the opinion index) is no longer a finite-size effect. Furthermore, we report a rich pattern of critical behavior characterized by an additional transition to a trimodal distribution.

We also address the effect of the network structure on the results of the model by using recent analytical tools known as heterogeneous mean field approximations. This approach suggests that the dynamics in an heterogeneous degree network is equivalent to the usual all-to-all approximation with an effective system size.