Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations

Figure 1.  Evolution of the opinion of 10 agents (blue) from a homogeneous population of $ N = 1000 $ agents interacting according the interaction rule (16) studied in this paper (left) or the interaction rule (22) considered in [2,38] (right). The red dashed line indicates the theoretical limit opinion in both cases ($ m_\infty\approx -0.35 $ for interaction rule (16), (left), and $ \tilde m_\infty\approx 0.41 $ for interaction rule (22), (right)) The early evolution is shown in inset

Figure 2.  Evolution of $ \ln(\max\,w-\min\,w) $, where the $ \max $ and $ \min $ are taken on the support of the distribution of opinions, for interaction rule (16) studied in this paper (left), and interaction rule (22) considered in [2,38] (right)

Figure 3.  Evolution of the opinion of 10 agents (blue) from a population of $ N = 1000 $ agents with $ \alpha_0 = 2\% $ (left) and $ \alpha_0 = 60\% $ (right) stubborn agents. The red dashed line indicates the theoretical limit opinion $ m_\infty = 1/2 $, and the blue dotted lines the opinion of the stubborn agents (half with opinion $ 1/4 $ and the other half with opinion $ 3/4 $). The early evolution is shown in inset

Figure 4.  Evolution of the distribution of $ (w,q) $ among the non-stubborn population during one simulation ($ w $ in the horizontal axis and $ q $ in the vertical axis) with $ \lambda(w) = (w-0.5)^2 + \varepsilon $, $ \varepsilon = 0.01 $. From left to right and top to bottom, figures show the distribution of $ (w,q) $ at time $ 1,500,1000,1500,3000,5000,10000,15000,30000 $