Search

Ant colony optimization (ACO) is a stochastic optimization algorithm inspired by the foraging behavior of ants. We investigate a simplified computational model of ACO, wherein ants sequentially engage in binary decision-making tasks, leaving pheromone trails contingent upon their choices. The quantity of pheromone left is the number of correct answers. We scrutinize the impact of a salient parameter in the ACO algorithm, specifically, the exponent α , which governs the pheromone levels in the stochastic choice function. In the absence of pheromone evaporation, the system is accurately modeled as a multivariate nonlinear Pólya urn, undergoing phase transition as α varies. The probability of selecting the correct answer for each question asymptotically approaches the stable fixed point of the nonlinear Pólya urn. The system exhibits dual stable fixed points for α ≥ α c and a singular stable fixed point for α < α c where α c is the critical value. When pheromone evaporates over a time scale τ , the phase transition does not occur and leads to a bimodal stationary distribution of probabilities for α ≥ α c and a monomodal distribution for α < α c . [ABSTRACT FROM AUTHOR]

In this paper, we discuss a voting model by considering three different kinds of networks: a random graph, the Barabási–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. [ABSTRACT FROM AUTHOR]

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 Hisakado and Mori (2011), 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. [ABSTRACT FROM AUTHOR]

Observational learning is an important information aggregation mechanism. However, it occasionally leads to a state in which an entire population chooses a suboptimal option. When this occurs and whether it is a phase transition remain unanswered. To address these questions we perform a voting experiment in which subjects answer a two-choice quiz sequentially with and without information about the prior subjects' choices. The subjects who could copy others are called herders. We obtain a microscopic rule regarding how herders copy others. Varying the ratio of herders leads to qualitative changes in the macroscopic behavior of about 50 subjects in the experiment. If the ratio is small, the sequence of choices rapidly converges to the correct one. As the ratio approaches 100%, convergence becomes extremely slow and information aggregation almost terminates. A simulation study of a stochastic model for 10* subjects based on the herder's microscopic rule shows a phase transition to the two-peak phase, where the convergence completely terminates as the ratio exceeds some critical value. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 a beta-binomial distribution. A non-equilibrium phase transition with an order parameter occurs when the temporal correlation decays by a power law. In this study, 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 the power law. When the power index is less than one, the PD estimator converges slowly. Thus, it is difficult to estimate the 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 the PD can have a long memory and that it is difficult to estimate parameters due to slow convergence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]