Mean-field approximation for the Sznajd model in complex networks

This paper studies the Sznajd model for opinion formation in a population connected through a general network. A master equation describing the time evolution of opinions is presented and solved in a mean-field approximation. Although quite simple, this approximation allows us to capture the most important features regarding the steady states of the model. When spontaneous opinion changes are included, a discontinuous transition from consensus to polarization can be found as the rate of spontaneous change is increased. In this case we show that a hybrid mean-field approach including interactions between second nearest neighbors is necessary to estimate correctly the critical point of the transition. The analytical prediction of the critical point is also compared with numerical simulations in a wide variety of networks, in particular Barabási-Albert networks, finding reasonable agreement despite the strong approximations involved. The same hybrid approach that made it possible to deal with second-order neighbors could just as well be adapted to treat other problems such as epidemic spreading or predator-prey systems.

Figure 1

Node $k$ may influence the opinion of $i$ only together with $j$ through the formation of the convincing couple $(j,k)$. If $k$ chooses the neighbor $l$, for instance, the convincing couple is unable to change the opinion of $i$, which happens to be out of range.

Figure 2

Bifurcation diagram for the CSM. Mean-field results are shown by solid (stable) or dashed (unstable) lines, depending upon their stability, whereas numerical results taken in a fully connected network are represented by symbols. The error bars in the numerical results are smaller than the symbols.

Figure 3

Order parameter for the CSM. Mean-field results are shown by solid lines whereas numerical results taken in a BA network are shown by symbols. The error bars in the numerical results are smaller than the symbols.

Figure 4

Typical time series for the model in a BA network near its transition point $\omega \approx {\omega }_t$. In this case the system can alternate between polarization and consensus, even interchanging the majority opinion.

Figure 5

Phase diagram for the CSM in a BA network. Typical time series for each phase are shown in the bottom row of insets: the ordered phase on the left and the disordered phase on the right. The error bars in the numerical results are smaller than the symbols.