Mean-field approximation for the Sznajd model in complex networks

We will present a work in which we revisited mean fied approximations to 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 hybrid approach. Based in a mean-field approximation, we were able to include some features of the underling structure of the network though the estimate of some network parameters. 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. The main point in this work is the presentation of a hybrid mean-field approach, that includes interactions between second nearest neighbors, that are 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 Barab?si-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. The work has been published in PRE 91, 022813 (2015).