Information cascade on networks and phase transitions

Herein, we consider a voting model for information cascades on several types of networks -- a random graph, the Barab\'{a}si-Albert(BA) model, and lattice networks -- by using one parameter $\omega$; $\omega=1,0, -1$ respectively correspond to these networks. $\omega$ is related to the size of hubs. We discuss the differences between the phases in which the networks depend. In $\omega\ne -1$, without, the following two types of phase transitions can be observed: information cascade transition and super-normal transition. The first is the transition between a state where most voters make correct choices and a state where most of them are wrong. This is an absorption transition that belongs to the non-equilibrium transition. In the symmetric case, the phase transition is continuous and the universality class is the same as nonlinear P\'{o}lya model. In contrast, in the asymmetric case, there is a discontinuous phase transition, where the gap depends on the network. The super-normal transition is the transition of the convergence speed, and the critical point of the convergence speed transition depends on $\omega$. At $\omega=1$, in the BA model, this transition disappears. Both phase transitions disappear at $\omega=-1$ in the lattice case. In conclusion, as the performance near the lattice case, $\omega\sim-1$ exhibits the best performance of the voting in all networks. As the hub size decreases, the performance improves.