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 Barabási–Albert(BA) model, and lattice networks – by using one parameter ω; ω=1,0,−1 respectively correspond to these networks. Our objective is to study the relation between the phase transitions and networks using the parameter, ω which is related to the size of hubs. We discuss the differences between the phases in which the networks depend. In ω≠−1, without a lattice, 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 Pólya model. In contrast, in the asymmetric case, there is a discontinuous phase transition, where the gap depends on the network. As ω increases, the size of the hub and the gap increase. Therefore, a network that has hubs has a greater effect through this phase transition. The critical point of information cascade transition does not depend on ω. The super-normal transition is the transition of the convergence speed, and the critical point of the convergence speed transition depends on ω. At ω=1, in the BA model, this transition disappears, and the range where we can observe the phase transition is the same as that in the percolation model on the network. Both phase transitions disappear at ω=−1 in the lattice case. In conclusion, as the performance near the lattice case, ω∼−1 exhibits the best performance of the voting in all networks. As the hub size decreases, the performance improves. Finally, we show the relation between the voting model and the elephant walk model.