Friedkin-Johnsen model

Another model I have seen recently featured a lot in the literature is the Friedkin-Johnsen model. This model is a generalization of the DeGroot model we have discussed earlier, but it adds a novel twist which allowed it to become a ground-breaking model on its own [1]. Namely, it has added stubbornness into the DeGroot model [2].

The model

So the agents still have real-valued opinions and there exits a trust network between them. The opinions are still averaged over the trusted agents, but the agents also remember their initial opinion and are pulled towards it. The updated formula is then given by:

\begin{equation} o_i(t+1) = (1 - s_i) \sum_j T_{i,j} o_j(t) + s_i o_i(0). \end{equation}

In the above all of the parameters are identical to the DeGroot model, with one new parameter - \( s_i \). The new parameter stands for the stubbornness (willingness to retain the initial opinion) of the \( i \)-th agent.

Interactive app

As should be expected, stubborn crowds become less wise. Wisdom of the crowd effect (shrinkage of the opinion distribution) is still present, but now the width of the final opinion distribution depends on the value of \( s_i \), which in this app we have kept the same for all agents. It is assumed to be equal to just \( s \).

References

• H. Noorazar. Recent advances in opinion propagation dynamics: a 2020 survey. The European Physical Journal Plus 135: 521 (2020). doi: 10.1140/epjp/s13360-020-00541-2.
• N. E. Friedkin, E. C. Johnsen. Social influence and opinions. The Journal of Mathematical Sociology 15: 193-206 (1990). doi: 10.1080/0022250X.1990.9990069.