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ABSTRACT: We introduce a stochastic model of binary opinion dynamics in one dimension.
The binary opinions $\pm 1$ are analogous to up and down Ising spins and in the
equivalent spin system, only the spins at the domain boundary can flip. The
probability that a spin at the boundary is up is taken as $P_{up} = \frac
{s_{up}} {s_{up} + \delta s_{down}}$ where $s_{up} (s_{down})$ denotes the size
of the domain with up (down) spins neighbouring it. With $x$ fraction of up
spins initially, a phase transition is observed in terms of the exit
probability and the phase boundary is obtained in the $\delta -x$ plane. In
addition, we investigate the coarsening behaviour starting from a completely
random state; conventional scaling is observed only at the phase transition
point $\delta = 1$. The scaling behaviour is compared to other dynamical
phenomena; the model apparently belongs to a new dynamical universility class
as far as persistence is concerned although the dynamical exponent, equal to
one, is identical to a similar model with no stochasticity.
05/2012;
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ABSTRACT: Randomness is known to affect the dynamical behavior of many systems to a large extent. In this paper we investigate how the nature of randomness affects the dynamics in a zero-temperature quench of the Ising model on two types of random networks. In both networks, which are embedded in a one-dimensional space, the first-neighbor connections exist and the average degree is 4 per node. In random model A the second-neighbor connections are rewired with a probability p, while in random model B additional connections between neighbors at a Euclidean distance l(l > 1) are introduced with a probability P(l) proportionally l(-α). We find that for both models, the dynamics leads to freezing such that the system gets locked in a disordered state. The point at which the disorder of the nonequilibrium steady state is maximum is located. The behavior of dynamical quantities such as residual energy, order parameter, and persistence are discussed and compared. Overall, the behavior of physical quantities are similar, although subtle differences are observed due to the difference in the nature of randomness.
Physical Review E 12/2011; 84(6 Pt 2):066107. · 2.26 Impact Factor
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ABSTRACT: A model for opinion dynamics (Model I) has been recently introduced in which
the binary opinions of the individuals are determined according to the size of
their neighboring domains (population having the same opinion). The coarsening
dynamics of the equivalent Ising model shows power law behavior and has been
found to belong to a new universality class with the dynamic exponent $z=1.0
\pm 0.01$ and persistence exponent $\theta \simeq 0.235$ in one dimension. The
critical behavior has been found to be robust for a large variety of annealed
disorder that has been studied. Further, by mapping Model I to a system of
random walkers in one dimension with a tendency to walk towards their nearest
neighbour with probability $\epsilon$, we find that for any $\epsilon > 0.5$,
the Model I dynamical behaviour is prevalent at long times.
04/2011;
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ABSTRACT: In two different classes of network models, namely, the Watts Strogatz type
and the Euclidean type, subtle changes have been introduced in the randomness.
In the Watts Strogatz type network, rewiring has been done in different ways
and although the qualitative results remain same, finite differences in the
exponents are observed. In the Euclidean type networks, where at least one
finite phase transition occurs, two models differing in a similar way have been
considered. The results show a possible shift in one of the phase transition
points but no change in the values of the exponents. The WS and Euclidean type
models are equivalent for extreme values of the parameters; we compare their
behaviour for intermediate values.
10/2010;
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ABSTRACT: The idea that the dynamics of a spin is determined by the size of its
neighbouring domains was recently introduced (S. Biswas and P. Sen, Phys. Rev.
E {\bf 80}, 027101 (2009)) in a Ising spin model
(henceforth, referred to as model I).
A parameter $p$ is now defined to modify the dynamics such that a spin can
sense domain sizes up to $R = pL/2$ in a one dimensional system of size $L$.
For the cutoff factor $p$ \to 0$, the dynamics is Ising like and the domains
grow with time $t$ diffusively as $ t^{1/z}$ with $z=2$, while for $p=1$, the
original model I showed ballistic dynamics with $z \simeq 1$. For intermediate
values of $p$, the domain growth, magnetisation and persistence show model I
like behaviour up to a macroscopic crossover time $ t_1 \sim pL/2$. Beyond
$t_1$, characteristic power law variations of the dynamic quantities are no
longer observed. The total time to reach equilibrium is found to be $t = apL +
b(1-p)^3L^2$, from which we conclude that the later time behaviour is
diffusive. We also consider the case when a random but quenched value of $p$ is
used for each spin for which ballistic behaviour is once again obtained.
04/2010;
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ABSTRACT: We propose a model of binary opinion in which the opinion of the individuals changes according to the state of their neighboring domains. If the neighboring domains have opposite opinions then the opinion of the domain with the larger size is followed. Starting from a random configuration, the system evolves to a homogeneous state. The dynamical evolution shows a scaling behavior with the persistence exponent theta approximately 0.235 and dynamic exponent z approximately 1.02 + or - 0.02. Introducing disorder through a parameter called rigidity coefficient rho (probability that people are completely rigid and never change their opinion), the transition to a heterogeneous society at rho=0(+) is obtained. Close to rho=0, the equilibrium values of the dynamic variables show power-law scaling behavior with rho. We also discuss the effect of having both quenched and annealed disorder in the system.
Physical Review E 08/2009; 80(2 Pt 2):027101. · 2.26 Impact Factor
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ABSTRACT: We propose a model of binary opinion in which the opinion of the individuals change according to the state of their neighbouring domains. If the neighbouring domains have opposite opinions, then the opinion of the domain with the larger size is followed. Starting from a random configuration, the system evolves to a homogeneous state. The dynamical evolution show novel scaling behaviour with the persistence exponent $\theta \simeq 0.235$ and dynamic exponent $z \simeq1.02 \pm 0.02$. Introducing disorder through a parameter called rigidity coefficient $\rho$ (probability that people are completely rigid and never change their opinion), the transition to a heterogeneous society at $\rho = 0^{+}$ is obtained. Close to $\rho =0$, the equilibrium values of the dynamic variables show power law scaling behaviour with $\rho$. We also discuss the effect of having both quenched and annealed disorder in the system. Comment: 4 pages, 6 figures, Final version of PRE
04/2009;
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ABSTRACT: We investigate the dynamics of a two dimensional axial next nearest neighbour Ising (ANNNI) model following a quench to zero temperature. The Hamiltonian is given by $H = -J_0\sum_{i,j=1}^L S_{i,j}S_{i+1,j} - J_1\sum_{i,j=1} [S_{i,j} S_{i,j+1} -\kappa S_{i,j} S_{i,j+2}]$. For $\kappa <1$, the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For $\kappa > 1$, the system shows a behaviour similar to the two dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent $\theta = 0.235 \pm 0.001$ while the dynamical exponent of growth $z=2.08\pm 0.01$. For $\kappa =1$, the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domains walls. We also compare the dynamical behaviour to that of a Ising model in which both the nearest and next nearest neighbour interactions are ferromagnetic. Comment: 9 pages, revtex4, to appear in PRE
10/2008;
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ABSTRACT: We investigate the dynamics of a two-dimensional axial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H= -J_(0) summation operator(L)_(i,j=1)S_(i,j)S_(i+1,j)-J_(1)summation operator_(i,j=1)(S_{i,j}S_{i,j+1}-kappaS_{i,j}S_{i,j+2}) . For kappa<1 , the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For kappa>1 , the system shows a behavior similar to that of the two-dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent theta=0.235+/-0.001 while the dynamical exponent of growth z=2.08+/-0.01 . For kappa=1 , the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domain walls. We also compare the dynamical behavior to that of a Ising model in which both the nearest and next-nearest-neighbor interactions are ferromagnetic.
Physical Review E 10/2008; 78(4 Pt 1):041119. · 2.26 Impact Factor
