Anisotropic opinion dynamics

We consider the process of opinion formation in a society of interacting agents, where there is a set $B$ of socially accepted rules. In this scenario, we observed that agents, represented by simple feed-forward, adaptive neural networks, may have a conservative attitude (mostly in agreement with $B$) or liberal attitude (mostly in agreement with neighboring agents) depending on how much their opinions are influenced by their peers. The topology of the network representing the interaction of the society's members is determined by a graph, where the agents' properties are defined over the vertexes and the interagent interactions are defined over the bonds. The adaptability of the agents allows us to model the formation of opinions as an online learning process, where agents learn continuously as new information becomes available to the whole society (online learning). Through the application of statistical mechanics techniques we deduced a set of differential equations describing the dynamics of the system. We observed that by slowly varying the average peer influence in such a way that the agents attitude changes from conservative to liberal and back, the average social opinion develops a hysteresis cycle. Such hysteretic behavior disappears when the variance of the social influence distribution is large enough. In all the cases studied, the change from conservative to liberal behavior is characterized by the emergence of conservative clusters, i.e., a closed knitted set of society members that follow a leader who agrees with the social status quo when the rule $B$ is challenged.

Figure 1

Pictorial representation of the synaptic vectors B, ${\mathbf{J}}_a$, and ${\mathbf{J}}_b$ and the angles between them.

Figure 2

Hysteresis cycle for a system with disorder parameter $\mathrm{\Delta }=1.5\times {10}^{-3}$. The square symbols represent the ${\mu }_{\mathrm{cp}}$ curve, and the circles represent the ${\mu }_{\mathrm{pc}}$ curve.

Figure 3

Right shift in the curve ${\mu }_{\mathrm{cp}}, |{\eta }_{\mathrm{cp}}-2/n|$ and the left shift in the curve ${\mu }_{\mathrm{pc}}, |{\eta }_{\mathrm{pc}}-2/n|$, as a function of the disorder $\mathrm{\Delta }.$ The label in the vertical axis is $|{\eta }^{☆}-2/n|$ where ${\eta }^{☆}$ stands for either ${\eta }_{\mathrm{cp}}$ or ${\eta }_{\mathrm{pc}}.$ The straight line is the best fit for the right shift of ${\mu }_{\mathrm{cp}}$, which is a scaling law proportional to ${\mathrm{\Delta }}^{0.8}$. The left shift in ${\mu }_{\mathrm{pc}}$ asymptotically approaches this behavior for values of $\mathrm{\Delta }$ larger than $1.8\times {10}^{-3}$ (indicated by the dotted line).

Figure 4

Data collapse of ${\mu }_{\mathrm{cp}}$ as a function of $x\equiv [\eta -{\eta }_{\mathrm{cp}}\left(\mathrm{\Delta }\right)]/{\mathrm{\Delta }}^{0.3}$ and ${\mu }_{\mathrm{pc}}$ as a function of $x\equiv [\eta -{\eta }_{\mathrm{pc}}\left(\mathrm{\Delta }\right)]/{\mathrm{\Delta }}^{0.3}$ for several values of $\mathrm{\Delta }<1.8\times {10}^{-3}.$ In the inset we have the correspondent snapshots of the arrays $\mathbf{R}$ at $x=0$. In all the cases the number of conservative clusters supported by the system is mostly 1.

Figure 5

Data collapse of ${\mu }_{\mathrm{cp}}$ as a function of $x\equiv [\eta -{\eta }_{\mathrm{cp}}\left(\mathrm{\Delta }\right)]/\mathrm{\Delta }$ and ${\mu }_{\mathrm{pc}}$ as a function of $x\equiv [\eta -{\eta }_{\mathrm{pc}}\left(\mathrm{\Delta }\right)]/\mathrm{\Delta }$ for several values of $\mathrm{\Delta }>1.8\times {10}^{-3}.$ In the inset we have the correspondent snapshots of the arrays $\mathbf{R}$ at $x=0$. In all the cases the number of conservative clusters supported by the system is larger than one and the larger the disorder parameter $\mathrm{\Delta }$ the smaller the size of the clusters observed.