arXiv subjects

Kazuaki Nakayama

Publications and source records attributed to Kazuaki Nakayama.

8 recordsLinked to original sources

arXiv2024

Structural Properties of the Asymmetric Barabási-Albert Model in the Lattice Limit

The Asymmetric BA model extends the Barabási-Albert scale-free network model by introducing a parameter $ω$. As $ω$ varies, the model transitions through different network structures: an extended lattice at $ω= -1$, a random graph at $ω= 0$, and the original scale-free network at $ω= 1$. We derive the exact degree distribution for $ω= -r/(r+k)$, where $k \in \{0,1,\cdots\}$, and develop a perturbative expansion around these values of $ω$. Additionally, we show that for $ω= -1 + \varepsilon$, the clustering coefficient scales as $\ln t / \sqrt{\varepsilon} t$ and approaches zero as $t \to \infty$, confirming the absence of small-world properties.

cond-mat.stat-mech↗

arXiv2024

$α$ Annealing of Ant Colony Optimization in the infinite-range Ising model

Ant colony optimization (ACO) leverages the parameter $α$ 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 $α$ and establishing an effective annealing schedule remain significant challenges, particularly in complex optimization scenarios. This study investigates the $α$-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 $α$, 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 $α$ increases, the joint PDF transitions from a mono-modal to a multi-modal state. In the homogeneous fully connected Ising model, $α$-annealing facilitates the transition from a trivial solution at $α=0$ to the ground state. The parameter $α$ in the annealing process plays a role analogous to the transverse field in quantum annealing. Our findings demonstrate the potential of $α$-annealing in navigating complex optimization problems, suggesting its broader application beyond the infinite-range Ising model.

cond-mat.stat-mech↗

arXiv2024

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. $ω$ is related to the size of hubs. We discuss the differences between the phases in which the networks depend. In $ω\ne -1$, 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 Pólya 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 $ω$. At $ω=1$, in the BA model, this transition disappears. Both phase transitions disappear at $ω=-1$ in the lattice case. In conclusion, as the performance near the lattice case, $ω\sim-1$ exhibits the best performance of the voting in all networks. As the hub size decreases, the performance improves.

physics.soc-ph↗

arXiv2021

Pólya urn with memory kernel and asymptotic behaviours of autocorrelation function

Pólya 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.

cond-mat.stat-mech↗

arXiv2020

Universal function of the non-equilibrium phase transition of nonlinear Pólya urn

We study the phase transition and the critical properties of a nonlinear Pólya 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 $\ln(t+1)^{-1/2}g(\ln(t+1)/ξ)$, with a suitable definition of the correlation length $ξ$ 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.

cond-mat.stat-mech↗

arXiv2020

Optimal Learning Dynamics of Multi Agents in Restless Multiarmed Bandit Game

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.

physics.soc-ph↗

arXiv2019

A voter model on networks and multivariate beta distribution

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.

physics.soc-ph↗

arXiv2016

Phase transition of social learning collectives and "Echo chamber"

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 $1+γ$ in the limit $N,M\to \infty$ and $γ=1+\frac{(1-p)q_{I}}{pq_{O}}$. 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 $p p_{c})$, the variance of $N_{1}$ per agent $\mbox{Var}(N_{1})/N$ is finite (diverges as $\propto N^{2-γ}$ with $N$). There is a threshold value $N_{s}$ for the system size that scales as $\ln N_{s} \propto 1/(γ-1)$. For $p>p_{c}$ and $N<N_{s}$, all agents tend to share only one good lever. $\mbox{E}(N_{1})$ decreases to zero as $p\to 1$, which is referred to as the "Echo chamber".

physics.soc-ph↗