Non-consensus Opinion Models on Complex Networks

Social dynamic opinion models have been widely studied to understand how interactions among individuals cause opinions to evolve. Most opinion models that utilize spin interaction models usually produce a consensus steady state in which only one opinion exists. Because in reality different opinions usually coexist, we focus on non-consensus opinion models in which above a certain threshold two opinions coexist in a stable relationship. We revisit and extend the non-consensus opinion (NCO) model introduced by Shao et al. (Phys. Rev. Lett. 103:01870, 2009). The NCO model in random networks displays a second order phase transition that belongs to regular mean field percolation and is characterized by the appearance (above a certain threshold) of a large spanning cluster of the minority opinion. We generalize the NCO model by adding a weight factor W to each individual’s original opinion when determining their future opinion (NCOW model). We find that as W increases the minority opinion holders tend to form stable clusters with a smaller initial minority fraction than in the NCO model. We also revisit another non-consensus opinion model based on the NCO model, the inflexible contrarian opinion (ICO) model (Li et al. in Phys. Rev. E 84:066101, 2011), which introduces inflexible contrarians to model the competition between two opinions in a steady state. Inflexible contrarians are individuals that never change their original opinion but may influence the opinions of others. To place the inflexible contrarians in the ICO model we use two different strategies, random placement and one in which high-degree nodes are targeted. The inflexible contrarians effectively decrease the size of the largest rival-opinion cluster in both strategies, but the effect is more pronounced under the targeted method. All of the above models have previously been explored in terms of a single network, but human communities are usually interconnected, not isolated. Because opinions propagate not only within single networks but also between networks, and because the rules of opinion formation within a network may differ from those between networks, we study here the opinion dynamics in coupled networks. Each network represents a social group or community and the interdependent links joining individuals from different networks may be social ties that are unusually strong, e.g., married couples. We apply the non-consensus opinion (NCO) rule on each individual network and the global majority rule on interdependent pairs such that two interdependent agents with different opinions will, due to the influence of mass media, follow the majority opinion of the entire population. The opinion interactions within each network and the interdependent links across networks interlace periodically until a steady state is reached. We find that the interdependent links effectively force the system from a second order phase transition, which is characteristic of the NCO model on a single network, to a hybrid phase transition, i.e., a mix of second-order and abrupt jump-like transitions that ultimately becomes, as we increase the percentage of interdependent agents, a pure abrupt transition. We conclude that for the NCO model on coupled networks, interactions through interdependent links could push the non-consensus opinion model to a consensus opinion model, which mimics the reality that increased mass communication causes people to hold opinions that are increasingly similar. We also find that the effect of interdependent links is more pronounced in interdependent scale free networks than in interdependent Erd?s Rényi networks.     

Non-Equilibrium Behaviour in Unidirectionally Coupled Map Lattices

A spatially one dimensional coupled map lattice with a local and unidirectional coupling is introduced. This model is studied analytically by a perturbation theory that is valid for small coupling strength. In parameter space three phases with different ergodic behaviour are observed. Via coarse graining the deterministic model is mapped to a stochastic spin model that can be described by a master equation. Because of the anisotropic coupling non-equilibrium behaviour is found on the coarse grained level. However, the stationary statistical properties of the spin dynamics can still be described with a nearest neighbour Ising model whereby the ordering is predominantly antiferromagnetic.     

Estimating parameters in stochastic systems: A variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.     

Combined update scheme in the Sznajd model

We analyze the Sznajd opinion formation model, where a pair of neighboring individuals sharing the same opinion on a square lattice convinces its six neighbors to adopt their opinions, when a fraction of the individuals is updated according to the usual random sequential updating rule (asynchronous updating), and the other fraction, the simultaneous updating (synchronous updating). This combined updating scheme provides that the bigger the synchronous frequency becomes, the more difficult the system reaches a consensus. Moreover, in the thermodynamic limit, the system needs only a small fraction of individuals following a different kind of updating rules to present a non-consensus state as a final state.     

Monte Carlo study of low/high spin transitions

We study the finite temperature property of a model on two dimensional square lattices with two Ising spins at each lattice site by Monte Carlo simulations. When those Ising spins at a lattice site are parallel the site is said to be in the high-spin state (HS), while when they are antiparallel the site is said to be in the low-spin state (LS). Throughout the study, the energy of HS is presumed to be higher than that of LS. Two Ising spins at each site are added to form a total spin, which interacts with its nearest neighbour total spins via spin-spin couplings. The spin-phonon coupling also is introduced via harmonic springs between nearest neighbour sites with spring constants and equilibrium distances depending on the spin states of the sites involved. In this system, we investigate the feature of transitions between LS and HS (to be called low/high spin transition (LHST)) by varying the temperature. As for the ferromagnetic interaction between total spins, the second order phase transition: pure HSmixed state of HS and LS is possible to occur in a pure spin system, as is expected from mean field calculations. The role of lattice distortions by the change of lattice spacings is shown to be essential for LHST: pure LS(pure)HS. In the model investigated, there appears an indication of the strong first order phase transition which reveals a conspicuous hysteresis.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

A deterministic sandpile automaton revisited

The Bak-Tang-Wiesenfeld (BTW) sandpile model is a cellular automaton which has been intensively studied during the last years as a paradigm for self-organized criticality. In this paper, we reconsider a deterministic version of the BTW model introduced by Wiesenfeld, Theiler and McNamara, where sand grains are added always to one fixed site on the square lattice. Using the Abelian sandpile formalism we discuss the static properties of the system. We present numerical evidence that the deterministic model is only in the BTW universality class if the initial conditions and the geometric form of the boundaries do not respect the full symmetry of the square lattice. Received 19 August 1999     

Dynamics of stochastic and nearly stochastic two-party competitions

We introduce a stochastic model to study the problem of time evolution and outcome of simple two-party competitions or battles on a lattice where each party randomly deploys its constituents or elements to the lattice. The elements have assigned strength levels that determine how competitive or effective they are against the opponent. In our models, the elements neutralize one another when they are at the same site with a combination of strength levels and numbers determining which one gains control of the site. The competitions last until complete dominance has been established by one side by eliminating the opponents or a draw is achieved (unless the time evolution is terminated via some ad hoc condition). A Markov chain approach is used to describe the time-dependent dynamics of such competitions. The advantage of the approach is that it allows us to develop a theoretical framework for describing competitive systems where a combination of random and correlated events decide the outcome. We use the approach here for studies of highly contentious stochastic battles and for that of a battle with correlated events along with stochastic events. We present the method, a simple illustrative example on how the method works and close by considering two non-trivial cases including one with a combination of stochasticity and correlations.     

Ferromagnetism in the Hubbard Model with a Gapless Nearly-Flat Band

We present a version of the Hubbard model with a gapless nearly-flat lowest band which exhibits ferromagnetism in two or more dimensions. The model is defined on a lattice obtained by placing a site on each edge of the hypercubic lattice, and electron hopping is assumed to be only between nearest and next nearest neighbor sites. The lattice, where all the sites are identical, is simple, and the corresponding single-electron band structure, where two cosine-type bands touch without an energy gap, is also simple. We prove that the ground state of the model is unique and ferromagnetic at half-filling of the lower band, if the lower band is nearly flat and the strength of on-site repulsion is larger than a certain value which is independent of the lattice size. This is the first example of ferromagnetism in three dimensional non-singular models with a gapless band structure.     

On stochastic parameter estimation using data assimilation

Data assimilation-based parameter estimation can be used to deterministically tune forecast models. This work demonstrates that it can also be used to provide parameter distributions for use by stochastic parameterization schemes. While parameter estimation is (theoretically) straightforward to perform, it is not clear how one should physically interpret the parameter values obtained. Structural model inadequacy implies that one should not search for a deterministic “best” set of parameter values, but rather allow the parameter values to change as a function of state; different parameter values will be needed to compensate for the state-dependent variations of realistic model inadequacy. Over time, a distribution of parameter values will be generated and this distribution can be sampled during forecasts. The current work addresses the ability of ensemble-based parameter estimation techniques utilizing a deterministic model to estimate the moments of stochastic parameters. It is shown that when the system of interest is stochastic the expected variability of a stochastic parameter is biased when a deterministic model is employed for parameter estimation. However, this bias is ameliorated through application of the Central Limit Theorem, and good estimates of both the first and second moments of the stochastic parameter can be obtained. It is also shown that the biased variability information can be utilized to construct a hybrid stochastic/deterministic integration scheme that is able to accurately approximate the evolution of the true stochastic system.     

Superfluid to Mott‐insulator transition in an anisotropic two‐dimensional optical lattice

We study the superfluid to Mott‐insulator transition of bosons in an optical anisotropic lattice by employing the Bose‐Hubbard model living on a two‐dimensional lattice with anisotropy parameter κ. The compressible superfluid state and incompressible Mott‐insulator (MI) lobes are efficiently described analytically, using the quantum U(1) rotor approach. The ground state phase diagram showing the evolution of the MI lobes is quantified for arbitrary values of κ, corresponding to various kind of lattices: from square, through rectangular to almost one‐dimensional.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension

Deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non-Markovian stochastic evolution equations after an average over realization using a theorem. Examples are given, show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model. Two explicit examples, the first containing updating rules that depend on two previous time steps and the second on three, are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations.     

Correlations and conductivity of interacting fermions in a disordered chain

A one dimensional system of interacting fermions in a random potential is investigated within a Hartree-Fock approximation. The model in consideration combines the onsite interaction of the Hubbard model and the stochastic site energies of the Anderson model. For a system of 120 sites and several values of the model parameters the equal time correlation functions for the charge and the spin density as well as the low frequency conductivity are calculated numerically. The gross features of the conductivity thus obtained agree with published results obtained in a Monte Carlo simulation, but there are also interesting differences.     

Time evolution of the probability distribution in stochastic and chaotic systems with enhanced diffusion

We perform a detailed study of the time evolution of the probability distribution for two processes displaying enhanced diffusion: a stochastic process named the Lévy walk and a deterministic chaotic process, the amplified climbing-sine map. The time evolution of the probability distribution differs in the two cases and carries information which is peculiar to the investigated process.     

噪声对远程态制备迹距的影响

用一纠缠对作为信道，考察噪声对单个量子比特远程制备的影响。我们采用迹距来描述终态和待制备的初态之间的接近程度，其研究包括两种情况：第一种情况，发送方和接收方采用两Bell态的混合态作为信道去实现远程态制备[RSP],计算发现迹距是两态混合比的函数；第二种情况，假定环境噪声是一种随机波动模式，通过求解Bloch方程，得到系统的密度矩阵，详细讨论在随机噪声模式下，噪声对远程态制备迹距的影响。     

Chemical diffusion in a one dimensional lattice with two sites inequivalent2

Making use of the exact form of the Nernst-Einstein equation and an exact expression for the d.c. ionic conductivity derived by Richards, I have derived an exact expression for the chemical diffusion coefficient in a one dimensional lattice with two sites inequivalent and self-blocking by the atoms. It was found that the chemical diffusion coefficient is independent of concentration. This particular result, however, can apply only to the one dimensional lattice.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations

The deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of stochastic evolution equations after taking an average over realizations using a theorem. Examples are given that show that deterministic quantum mechanical evolution equations, obtained initially by R.P. Feynman and subsequently studied by Boghosian and Taylor IV [B.M. Boghosian, W. Taylor IV, Phys. Rev. E 57 (1998) 54. See also arXiv:quant-ph/9904035] and Meyer [D.A. Meyer, Phys. Rev. E 55 (1997) 5261], among others, are derived from a set of stochastic evolution equations. In addition, a deterministic classical evolution equation for the diffusion of monomers, similar to the second Fick law, is also obtained.     

涨落作用下周期驱动的分数阶过阻尼棘轮模型的混沌输运现象

考虑涨落作用下周期驱动的过阻尼分数阶棘轮模型, 通过模型的数值求解, 研究确定性棘轮的混沌特性与噪声的作用对输运行为的影响, 进而讨论过阻尼分数阶分子马达反向输运的机理. 分析表明: 随着势垒高度、 势不对称性与模型记忆性的变化, 随机棘轮的反向输运并不必然地要求确定性棘轮也反向输运; 随着模型阶数的减小, 亦即分数阻尼介质记忆性的增强, 确定性棘轮在反向输运之前会经历一个周期倍化导致的混沌状态, 但在噪声作用下, 反向流的发生会提前, 即混沌状态的确定性棘轮在噪声的作用下即可进行反向输运. 也就是说, 噪声能定性地改变棘轮的输运状态: 从无噪声时的混沌运动到有噪声时的定向输运. 这是过阻尼随机棘轮反向输运的一种机理, 也是噪声在定向输运过程中发挥积极作用的一个体现.