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The Asymmetric BA model extends the Barab\'asi-Albert scale-free network model by introducing a parameter $\omega$. As $\omega$ varies, the model transitions through different network structures: an extended lattice at $\omega = -1$, a random graph at $\omega = 0$, and the original scale-free network at $\omega = 1$. We derive the exact degree distribution for $\omega = -r/(r+k)$, where $k \in \{0,1,\cdots\}$, and develop a perturbative expansion around these values of $\omega$. Additionally, we show that for $\omega = -1 + \varepsilon$, the clustering coefficient scales as $\ln t / \sqrt{\varepsilon} t$ and approaches zero as $t \to \infty$, confirming the absence of small-world properties., Comment: 20 pages,9 figures

Ant colony optimization (ACO) leverages the parameter $\alpha$ to modulate the decision function's sensitivity to pheromone levels, balancing the exploration of diverse solutions with the exploitation of promising areas. Identifying the optimal value for $\alpha$ and establishing an effective annealing schedule remain significant challenges, particularly in complex optimization scenarios. This study investigates the $\alpha$-annealing process of the linear Ant System within the infinite-range Ising model to address these challenges. Here, "linear" refers to the decision function employed by the ants. By systematically increasing $\alpha$, we explore its impact on enhancing the search for the ground state. We derive the Fokker-Planck equation for the pheromone ratios and obtain the joint probability density function (PDF) in stationary states. As $\alpha$ increases, the joint PDF transitions from a mono-modal to a multi-modal state. In the homogeneous fully connected Ising model, $\alpha$-annealing facilitates the transition from a trivial solution at $\alpha=0$ to the ground state. The parameter $\alpha$ in the annealing process plays a role analogous to the transverse field in quantum annealing. Our findings demonstrate the potential of $\alpha$-annealing in navigating complex optimization problems, suggesting its broader application beyond the infinite-range Ising model., Comment: 24 pages, 4 figures

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., Comment: 37 pages 13 figures

We study the continuous time limit of a self-exciting negative binomial process and discuss the critical properties of its intensity distribution. In this limit, the process transforms into a marked Hawkes process. The probability mass function of the marks has a parameter $\omega$, and the process reduces to a "pure" Hawkes process in the limit $\omega\to 0$. We investigate the Lagrange--Charpit equations for the master equations of the marked Hawkes process in the Laplace representation close to its critical point and extend the previous findings on the power-law scaling of the probability density function (PDF) of intensities in the intermediate asymptotic regime to the case where the memory kernel is the superposition of an arbitrary finite number of exponentials. We develop an efficient sampling method for the marked Hawkes process based on the time-rescaling theorem and verify the power-law exponents., Comment: 15 pages, 5 figures. arXiv admin note: text overlap with arXiv:2001.01197 by other authors

In this study, we apply a multidimensional self-exciting negative binomial distribution (SE-NBD) process to default portfolios with 13 sectors. The SE-NBD process is a Poisson process with a gamma-distributed intensity function. We extend the SE-NBD process to a multidimensional process. Using the multidimensional SE-NBD process (MD-SE-NBD), we can estimate interactions between these 13 sectors as a network. By applying impact analysis, we can classify upstream and downstream sectors. The upstream sectors are real-estate and financial institution (FI) sectors. From these upstream sectors, shock spreads to the downstream sectors. This is an amplifier of the shock. This is consistent with the analysis of bubble bursts. We compare these results to the multidimensional Hawkes process (MD-Hawkes) that has a zero-variance intensity function., Comment: 26 pages, 7 figures

This study considers a new multi-term urn process that has a correlation in the same term and temporal correlation. The objective is to clarify the relationship between the urn model and the Hawkes process. Correlation in the same term is represented by the P\'{o}lya urn model and the temporal correlation is incorporated by introducing the conditional initial condition. In the double-scaling limit of this urn process, the self-exciting negative binomial distribution (SE-NBD) process, which is a marked Hawkes process, is obtained. In the standard continuous limit, this process becomes the Hawkes process, which has no correlation in the same term. The difference is the variance of the intensity function in that the phase transition from the steady to the non-steady state can be observed. The critical point, at which the power law distribution is obtained, is the same for the Hawkes and the urn processes. These two processes are used to analyze empirical data of financial default to estimate the parameters of the model. For the default portfolio, the results produced by the urn process are superior to those obtained with the Hawkes process and confirm self-excitation., Comment: 22 pages, 3 figures, 3 tables

P\'{o}lya urn is a stochastic process in which balls are randomly drawn from an urn of red and blue balls, and balls of the same color as the drawn balls are added. The probability of a ball of a certain color being drawn is equal to the percentage of balls of that color in the urn. We introduce arbitrary memory kernels to modify this probability. If the memory kernel decays exponentially, it is a stationary process and is mean-reverting. If the memory kernel decays by a power-law, a phase transition occurs and the asymptotic behavior of the autocorrelation function changes. An auxiliary field variable is introduced to transform the process Markovian and the field obeys a multivariate Ornstein-Uhlenbeck process. The exponents of the power law are estimated for the decay of the leading and subleading terms of the autocorrelation function. It is shown that the power law exponents changes discontinuously at the critical point., Comment: 22 pages, 1 figure

In this article, we discuss the random graph, Barab\'asi-Albert (BA) model, and lattice networks from a unified view point, with the parameter $\omega$ with values $1,0,-1$ characterizing these networks, respectively. The parameter is related to the preferential attachment of nodes in the networks and has different weights for the incoming and outgoing links. In addition, we discuss the correspondence between quantum statistics and the networks. Positive and negative $\omega$ correspond to Bose and Fermi-like statistics, respectively, and we obtain the distribution that connects the two. When $\omega$ is positive, it is related to the threshold of Bose-Einstein condensation (BEC). As $\omega$ decreases, the area of the BEC phase is narrowed, and disappears in the limit $\omega=0$. When $\omega$ is negative, nodes have limits in the number of attachments for newly added nodes (outgoing links), which corresponds to Fermi statistics. We also observe the Fermi degeneracy of the network. When $\omega=-1$, a standard Fermion-like network is observed. Fermion networks are realized in the cryptocurrency network "Tangle.", Comment: 19 pages, 6 figures

We study the phase transition and the critical properties of a nonlinear P\'{o}lya urn, which is a simple binary stochastic process $X(t)\in \{0,1\},t=1,\cdots$ with a feedback mechanism. Let $f$ be a continuous function from the unit interval to itself, and $z(t)$ be the proportion of the first $t$ variables $X(1),\cdots,X(t)$ that take the value 1. $X(t+1)$ takes the value 1 with probability $f(z(t))$. When the number of stable fixed points of $f(z)$ changes, the system undergoes a non-equilibrium phase transition and the order parameter is the limit value of the autocorrelation function. When the system is $Z_{2}$ symmetric, that is, $f(z)=1-f(1-z)$, a continuous phase transition occurs, and the autocorrelation function behaves asymptotically as $\ln(t+1)^{-1/2}g(\ln(t+1)/\xi)$, with a suitable definition of the correlation length $\xi$ and the universal function $g(x)$. We derive $g(x)$ analytically using stochastic differential equations and the expansion about the strength of stochastic noise. $g(x)$ determines the asymptotic behavior of the autocorrelation function near the critical point and the universality class of the phase transition., Comment: 19 pages, 4 figures

We discuss the parameter estimation of the probability of default (PD), the correlation between the obligors, and a phase transition. In our previous work, we studied the problem using the beta-binomial distribution. A non-equilibrium phase transition with an order parameter occurs when the temporal correlation decays by power law. In this article, we adopt the Merton model, which uses an asset correlation as the default correlation, and find that a phase transition occurs when the temporal correlation decays by power law. When the power index is less than one, the PD estimator converges slowly. Thus, it is difficult to estimate PD with limited historical data. Conversely, when the power index is greater than one, the convergence speed is inversely proportional to the number of samples. We investigate the empirical default data history of several rating agencies. The estimated power index is in the slow convergence range when we use long history data. This suggests that PD could have a long memory and that it is difficult to estimate parameters due to slow convergence., Comment: 19 pages, 5 figures

Social learning is learning through the observation of or interaction with other individuals; it is critical in the understanding of the collective behaviors of humans in social physics. We study the learning process of agents in a restless multiarmed bandit (rMAB). The binary payoff of each arm changes randomly and agents maximize their payoffs by exploiting an arm with payoff 1, searching the arm at random (individual learning), or copying an arm exploited by other agents (social learning). The system has Pareto and Nash equilibria in the mixed strategy space of social and individual learning. We study several models in which agents maximize their expected payoffs in the strategy space, and demonstrate analytically and numerically that the system converges to the equilibria. We also conducted an experiment and investigated whether human participants adopt the optimal strategy. In this experiment, three participants play the game. If the reward of each group is proportional to the sum of the payoffs, the median of the social learning rate almost coincides with that of the Pareto equilibrium., Comment: 16 pages, 9 figures

The probability of default (PD) estimation is an important process for financial institutions. The difficulty of the estimation depends on the correlations between borrowers. In this paper, we introduce a hierarchical Bayesian estimation method using the beta binomial distribution and consider a multi-year case with a temporal correlation. A phase transition occurs when the temporal correlation decays by power decay. When the power index is less than one, the PD estimator does not converge. It is difficult to estimate the PD with limited historical data. Conversely, when the power index is greater than one, the convergence is the same as that of the binomial distribution. We provide a condition for the estimation of the PD and discuss the universality class of the phase transition. We investigate the empirical default data history of rating agencies and their Fourier transformations to confirm the form of the correlation decay. The power spectrum of the decay history seems to be 1/f, which corresponds to a long memory. But the estimated power index is much greater than one. If we collect adequate historical data,the parameters can be estimated correctly., Comment: 24 pages, 9 figures

In elections, the vote shares or turnout rates show a strong spatial correlation. The logarithmic decay with distance suggests that a 2D noisy diffusive equation describes the system. Based on the study of U.S. presidential elections data, it was determined that the fluctuations of vote shares also exhibit a strong and long-range spatial correlation. Previously, it was considered difficult to induce strong and long-range spatial correlation of the vote shares without breaking the empirically observed narrow distribution. We demonstrate that a voter model on networks shows such a behavior. In the model, there are many voters in a node who are affected by the agents in the node and by the agents in the linked nodes. A multivariate Wright-Fisher diffusion equation for the joint probability density of the vote shares is derived. The stationary distribution is a multivariate generalization of the beta distribution. In addition, we also estimate the equilibrium values and the covariance matrix of the vote shares and obtain a correspondence with a multivariate normal distribution. This approach largely simplifies the calibration of the parameters in the modeling of elections., Comment: 16 pages, 6 figures

This study discusses choice behavior using a voting model in which voters can obtain information from a finite number of previous $r$ voters. Voters vote for a candidate with a probability proportional to the previous vote ratio, which is visible to the voters. We obtain the Pitman sampling formula as the equilibrium distribution of $r$ votes. We present the model as a process of posting on a bulletin board system, 2ch.net, where users can choose one of many threads to create a post. We explore how this choice depends on the last $r$ posts and the distribution of these last $r$ posts across threads. We conclude that the posting process is described by our voting model with analog herders for a small $r$, which might correspond to the time horizon of users' responses., Comment: 29 pages,11 figures

In this study, we propose a mechanical model of a plurality election based on a mean field voter model. We assume that there are three candidates in each electoral district, i.e., one from the ruling party, one from the main opposition party, and one from other political parties. The voters are classified as fixed supporters and herding (floating) voters with ratios of $1-p$ and $p$, respectively. Fixed supporters make decisions based on their information and herding voters make the same choice as another randomly selected voter. The equilibrium vote-share probability density of herding voters follows a Dirichlet distribution. We estimate the composition of fixed supporters in each electoral district and $p$ using data from elections to the House of Representatives in Japan (43rd to 47th). The spatial inhomogeneity of fixed supporters explains the long-range spatial and temporal correlations. The estimated values of $p$ are close to the estimates obtained from a survey., Comment: 11 pages, 7 figures

An "Echo chamber" is the state of social learning agents whose performances are deteriorated by excessive observation of others. We understand this to be the collective behavior of agents in a restless multi-armed bandit. The bandit has $M$ good levers and bad levers. A good lever changes to a bad one randomly with probability $q_{C}$ and a new good lever appears. $N$ agents exploit ones' lever if they know that it is a good one. Otherwise, they search for a good one by (i) random search (success probability $q_{I}$) and (ii) observe a good lever that is known by other agents (success probability $q_{O}$) with probability $1-p$ and $p$, respectively. The distribution of agents in good levers obeys the Yule distribution with power law exponent $1+\gamma$ in the limit $N,M\to \infty$ and $\gamma=1+\frac{(1-p)q_{I}}{pq_{O}}$. The expected value of the number of the agents with a good lever $N_{1}$ increases with $p$. The system shows a phase transition at $p_{c}=\frac{q_{I}}{q_{I}+q_{o}}$. For $pp_{c})$, the variance of $N_{1}$ per agent $\mbox{Var}(N_{1})/N$ is finite (diverges as $\propto N^{2-\gamma}$ with $N$). There is a threshold value $N_{s}$ for the system size that scales as $\ln N_{s} \propto 1/(\gamma-1)$. For $p>p_{c}$ and $N

We propose a method of detecting a phase transition in a generalized P\'olya urn in an information cascade experiment. The method is based on the asymptotic behavior of the correlation $C(t)$ between the first subject's choice and the $t+1$-th subject's choice, the limit value of which, $c\equiv \lim_{t\to \infty}C(t)$, is the order parameter of the phase transition. To verify the method, we perform a voting experiment using two-choice questions. An urn X is chosen at random from two urns A and B, which contain red and blue balls in different configurations. Subjects sequentially guess whether X is A or B using information about the prior subjects' choices and the color of a ball randomly drawn from X. The color tells the subject which is X with probability $q$. We set $q\in \{5/9,6/9,7/9,8/9\}$ by controlling the configurations of red and blue balls in A and B. The (average) lengths of the sequence of the subjects are 63, 63, 54.0, and 60.5 for $q\in \{5/9,6/9,7/9,8/9\}$, respectively. We describe the sequential voting process by a nonlinear P\'olya urn model. The model suggests the possibility of a phase transition when $q$ changes. We show that $c>0\,\,\,(=0)$ for $q=5/9,6/9\,\,\,(7/9,8/9 )$ and detect the phase transition using the proposed method., Comment: 26 pages, 12 figures

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., Comment: 10 pages, 4 figures

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., Comment: 31 pages, 7 figures. arXiv admin note: text overlap with arXiv:1203.3274

We obtain the conditions for the emergence of the swarm intelligence effect in an interactive game of restless multi-armed bandit (rMAB). A player competes with multiple agents. Each bandit has a payoff that changes with a probability $p_{c}$ per round. The agents and player choose one of three options: (1) Exploit (a good bandit), (2) Innovate (asocial learning for a good bandit among $n_{I}$ randomly chosen bandits), and (3) Observe (social learning for a good bandit). Each agent has two parameters $(c,p_{obs})$ to specify the decision: (i) $c$, the threshold value for Exploit, and (ii) $p_{obs}$, the probability for Observe in learning. The parameters $(c,p_{obs})$ are uniformly distributed. We determine the optimal strategies for the player using complete knowledge about the rMAB. We show whether or not social or asocial learning is more optimal in the $(p_{c},n_{I})$ space and define the swarm intelligence effect. We conduct a laboratory experiment (67 subjects) and observe the swarm intelligence effect only if $(p_{c},n_{I})$ are chosen so that social learning is far more optimal than asocial learning., Comment: 18 pages, 4 figures

We describe a universality class of the transitions of a generalized P\'{o}lya urn by studying the asymptotic behavior of the normalized correlation function $C(t)$ using finite-size scaling analysis. $X(1),X(2),\cdots$ are the successive additions of a red (blue) ball [$X(t)=1\,(0)$] at stage $t$ and $C(t)\equiv \mbox{Cov}(X(1),X(t+1))/\mbox{Var}(X(1))$. Furthermore, $z(t)=\sum_{s=1}^{t}X(s)/t$ represents the successive proportions of red balls in an urn to which, at the $t+1$-th stage, a red ball is added, [$X(t+1)=1$], with probability $q(z(t))=(\tanh [J(2z(t)-1)+h]+1)/2,J\ge 0$, and a blue ball is added, [$X(t)=0$], with probability $1-q(z(t))$. A boundary $(J_{c}(h),h)$ exists in the $(J,h)$ plane between a region with one fixed point and another region with two stable fixed points for $q(z)$. $C(t) \sim c+a\cdot t^{l-1}$ with $c=0\,(>0)$ for $JJ_{c})$, and $l$ is the (larger) value of the slope(s) of $q(z)$ at the stable fixed point(s). On the boundary $J=J_{c}(h)$, $C(t)\simeq c+a\cdot \log(t)^{-\alpha'}$ and $c=0\,(c>0), \alpha'=0.5\,(1.0)$ for $h=0\,(h\neq 0)$. The system shows a continuous phase transition for $h=0$ and $C(t)$ behaves as $C(t)\simeq t^{-\alpha'}g((1-l)\log t)$ with an universal function $g(x)$ and a length scale $1/(1-l)$ with respect to $\log t$. $\beta=\nu_{||}\cdot \alpha'$ holds with critical exponent $\beta=1/2$ and $\nu_{||}=1$., Comment: 26 pages, 8 figures

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 $\nu_{||}=2$., Comment: 32 pages, 9 figures

In this paper, we discuss a voting model in which voters can obtain information from a finite number of previous voters. There exist three groups of voters: (i) digital herders and independent voters, (ii) analog herders and independent voters, and (iii) $\tanh$-type herders. In our previous paper [1], we used the mean field approximation for case (i). In that study, if the reference number $r$ is above three, phase transition occurs and the solution converges to one of the equilibria. However, the conclusion is different from mean field approximation. In this paper, we show that the solution oscillates between the two states. A good (bad) equilibrium is where a majority of $r$ select the correct (wrong) candidate. In this paper, we show that there is no phase transition when $r$ is finite. If the annealing schedule is adequately slow from finite $r$ to infinite $r$, the voting rate converges only to the good equilibrium. In case (ii), the state of reference votes is equivalent to that of Kirman's ant colony model, and it follows beta binomial distribution. In case (iii), we show that the model is equivalent to the finite-size kinetic Ising model. If the voters are rational, a simple herding experiment of information cascade is conducted. Information cascade results from the quenching of the kinetic Ising model.As case (i) is the limit of case (iii) when $\tanh$ function becomes a step function, the phase transition can be observed in infinite size limit. We can confirm that there is no phase transition when the reference number $r$ is finite., Comment: 27 pages, 4 figures

We consider a situation where one has to choose an option with multiplier m. The multiplier is inversely proportional to the number of people who have chosen the option and is proportional to the return if it is correct. If one does not know the correct option, we call him a herder, and then there is a zero-sum game between the herder and other people who have set the multiplier. The max-min strategy where one divides one's choice inversely proportional to m is optimal from the viewpoint of the maximization of expected return. We call the optimal herder an analog herder. The system of analog herders takes the probability of correct choice to one for any value of the ratio of herders, p<1, in the thermodynamic limit if the accuracy of the choice of informed person q is one. We study how herders choose by a voting experiment in which 50 to 60 subjects sequentially answer a two-choice quiz. We show that the probability of selecting a choice by the herders is inversely proportional to m for 4/3 < m < 4 and they collectively adopt the max-min strategy in that range., Comment: 25 pages,9 figures

In this paper, we discuss a voting model with two candidates, C_0 and C_1. We consider two types of voters--herders and independents. The voting of independents is based on their fundamental values; on the other hand, the voting of herders is based on the number of previous votes. We can identify two kinds of phase transitions. One is an information cascade transition similar to a phase transition seen in Ising model. The other is a transition of super and normal diffusions. These phase transitions coexist. We compared our results to the conclusions of experiments and identified the phase transitions in the upper limit of the time t by using analysis of human behavior obtained from experiments., Comment: 24 pages, 6 figures

Observational learning is an important information aggregation mechanism. However, it occasionally leads to a state in which an entire population chooses a sub-optimal option. When it occurs and whether it is a phase transition remain unanswered. To address these questions, we performed a voting experiment in which subjects answered a two-choice quiz sequentially with and without information about the prior subjects' choices. The subjects who could copy others are called herders. We obtained a microscopic rule regarding how herders copy others. Varying the ratio of herders led to qualitative changes in the macroscopic behavior in the experiment of about 50 subjects. If the ratio is small, the sequence of choices rapidly converges to the true one. As the ratio approaches 100%, convergence becomes extremely slow and information aggregation almost terminates. A simulation study of a stochastic model for 10^{6} subjects based on the herder's microscopic rule showed a phase transition to the two-peaks phase, where the convergence completely terminates, as the ratio exceeds some critical value., Comment: 11 pages, 9 figures, 3 tables

In this paper, we discuss a voting model with two candidates, C_1 and C_2. We set two types of voters--herders and independents. The voting of independent voters is based on their fundamental values; on the other hand, the voting of herders is based on the number of votes. Herders always select the majority of the previous $r$ votes, which is visible to them. We call them digital herders. We can accurately calculate the distribution of votes for special cases. When r>=3, we find that a phase transition occurs at the upper limit of t, where t is the discrete time (or number of votes). As the fraction of herders increases, the model features a phase transition beyond which a state where most voters make the correct choice coexists with one where most of them are wrong. On the other hand, when r<3, there is no phase transition. In this case, the herders' performance is the same as that of the independent voters. Finally, we recognize the behavior of human beings by conducting simple experiments., Comment: 26 pages, 10 figures