Small-world bifurcations in an opinion model.

We study a cellular automaton opinion formation model of Ising type, with antiferromagnetic pair interactions modeling anticonformism, and fer- romagnetic plaquette terms modeling the social norm constraints. For a sufficiently large connectivity, the mean-field equation for the average mag- netization (opinion density) is chaotic. This “chaoticity” would imply ir- regular coherent oscillations of the whole society, that may eventually lead to a sudden jump into an absorbing state, if present. However, simula- tions on regular one-dimensional lattices show a different scenario: local patches may oscillate following the mean-field description, but these oscil- lations are not correlated spatially, so the average magnetization fluctuates around zero (average opinion near one half). The system is chaotic, but in a microscopic sense where local fluctuations tend to compensate each other. By varying the long-range rewiring of links, we trigger a small-world effect. We observe a bifurcation diagram for the magnetization, with pe- riod doubling cascades ending in a chaotic phase. As far as we know, this is the first observation of a small-world induced bifurcation diagram. The social implications of this transition are also interesting. In the presence of strong “anticonformistic” (or “antinorm”) behavior, efforts for promoting social homogenization may trigger violent oscillations.