Bifurcation Analysis of an Opinion Formation Model in a Small-World Network

In this opinion formation model (Bagnoli, 2009), an individual can change its opinion by influence of their neighborhood. The opinions are represented by spins σi ∈ 0, 1. The interactions between the individuals follow a Watts-Strogatz small-world (SW) model, with a fraction p of random edges. The opinion formation model is of Ising type, with antiferromagnetic pair interactions modeling anticonformism, and ferromagnetic plaquette terms modeling the social norm constraints. Dynamics The influence of the neighbors is given by hi = ij aijσj ki where aij comes from the adjancency matrix and ki is the number of neighbors of the i th individual. The individuals change its opinion according the following rules: τ (1|h) = if h < q (1 + exp(−2J(2h − 1)))−1 if q ≤ h ≤ 1 − q 1 − if h> 1 − q The quantity of interest is c = σi, the density of active sites. Special Limits I For = 0 (infinite plaquette terms) we have the absorbing states s = 0 and s = 1 if the social pressure is above (below) the threshold 1 − q (q), respectively. I J> 0 or ferromagnetic interactions model a conformistic society while for J < 0 (anti-ferro) model an anti-conformistic society. Whereas in the ferromagnetic case leads to two absorbing states, the anti-ferro leads to an active phase where the former absorbing states are unstable. I q = 0 and q = 1 represents the kinetic Ising model