Analysis of a voter model with an evolving number of opinion states

In traditional voter models, opinion dynamics are driven by interactions between individuals, where an individual adopts the opinion of a randomly chosen neighbor. However, these models often fail to capture the emergence of entirely new opinions, which can arise spontaneously in real-world scenarios. Our study introduces a novel element to the classic voter model: the concept of innovation, where individuals have a certain probability of generating new opinions independently of their neighbors' states. By incorporating this mechanism of spontaneous innovation, our model provides insights into how new ideas can propagate and influence the overall opinion dynamics.

The dynamical rules of the model with evolving sets of opinions is similar to that of the conventional voter model. The key difference is the possibility for an individual to introduce a new opinion state into the population. Initially, there are two opinions present in the population. Each individual is randomly assigned one of these states, each with probability 1/2, and with no correlations between different individuals. At each time step, a node, referred to as node i, is randomly selected. Node i adopts the state of one of its neighbor, say node j, which is chosen at random. In dependent to the adoption process, a node that is chosen at random again conceives an idea and transitions to a new state at an innovation rate of α. Here, a "new" opinion means an opinion that has not previously appeared in a whole system. We repeat the procedures.

This model allows us to examine how the stability of consensus and the existence of diverse opinions change depending on the frequency of new ideas emerging. We observed that the average number of distinct opinions present in the system changes as the parameter α varies. When α exceeds a certain threshold, the probability of reaching consensus becomes significantly lower. This contrasts with the traditional voter model, where opinion dynamics occur within a fixed pool of opinions and consensus is always achieved.