In this paper, we discuss a voting model by considering three different kinds of networks: a random graph, the Barab\'{a}si-Albert(BA) model, and a fitness model. A voting model represents the way in which public perceptions are conveyed to voters. Our voting model is constructed by using two types of voters--herders and independents--and two candidates. Independents conduct voting based on their fundamental values; on the other hand, herders base their voting on the number of previous votes. Hence, herders vote for the majority candidates and obtain information relating to previous votes from their networks. We discuss the difference between the phases on which the networks depend. Two kinds of phase transitions, an information cascade transition and a super-normal transition, were identified. The first of these is a transition between a state in which most voters make the correct choices and a state in which most of them are wrong. The second is a transition of convergence speed. The information cascade transition prevails when herder effects are stronger than the super-normal transition. In the BA and fitness models, the critical point of the information cascade transition is the same as that of the random network model. However, the critical point of the super-normal transition disappears when these two models are used. In conclusion, the influence of networks is shown to only affect the convergence speed and not the information cascade transition. We are therefore able to conclude that the influence of hubs on voters' perceptions is limited.

We propose a method of detecting non-self-correcting information cascades in experiments in which subjects choose an option sequentially by observing the choices of previous subjects. The method uses the correlation function $C(t)$ between the first and the $t+1$-th subject's choices. $C(t)$ measures the strength of the domino effect, and the limit value $c\equiv \lim_{t\to \infty}C(t)$determines whether the domino effect lasts forever$(c>0)$ or not $(c=0)$. The condition $c>0$ is an adequate condition for a non-self-correcting system, and the probability that the majority's choice remains wrong in the limit $t\to \infty$ is positive. We apply the method to data from two experiments in which $T$ subjects answered two-choice questions: (i) general knowledge questions ($T_{avg}=60$) and (ii) urn-choice questions ($T=63$). We find $c>0$ for difficult questions in (i) and all cases in (ii), and the systems are not self-correcting.

We propose a finite-size scaling analysis of binary stochastic processes $X(t)\in \{0,1\}$ based on the second moment correlation length $\xi$ for the autocorrelation function $C(t)$. The purpose is to clarify the critical properties and provide a new data analysis method for information cascades. As a simple model to represent the different behaviors of subjects in information cascade experiments, we assume that $X(t)$ is a mixture of an independent random variable that takes 1 with probability $q$ and a random variable that depends on the ratio $z$ of the variables taking 1 among recent $r$ variables. We consider two types of the probability $f(z)$ that the latter takes 1: (i) analog [$f(z)=z$] and (ii) digital [$f(z)=\theta(z-1/2)$]. We study the universal functions of scaling for $\xi$ and the integrated correlation time $\tau$. For finite $r$, $C(t)$ decays exponentially as a function of $t$, and there is only one stable renormalization group (RG) fixed point. In the limit $r\to \infty$, where $X(t)$ depends on all the previous variables, $C(t)$ in model (i) obeys a power law, and the system becomes scale invariant. In model (ii) with $q\neq 1/2$, there are two stable RG fixed points, which correspond to the ordered and disordered phases of the information cascade phase transition with critical exponents $\beta=1$ and $\nu_{||}=2$.