13,144 research outputs found
Macroscopic Noisy Bounded Confidence Models with Distributed Radical Opinions
In this article, we study the nonlinear Fokker-Planck (FP) equation that
arises as a mean-field (macroscopic) approximation of bounded confidence
opinion dynamics, where opinions are influenced by environmental noises and
opinions of radicals (stubborn individuals). The distribution of radical
opinions serves as an infinite-dimensional exogenous input to the FP equation,
visibly influencing the steady opinion profile. We establish mathematical
properties of the FP equation. In particular, we (i) show the well-posedness of
the dynamic equation, (ii) provide existence result accompanied by a
quantitative global estimate for the corresponding stationary solution, and
(iii) establish an explicit lower bound on the noise level that guarantees
exponential convergence of the dynamics to stationary state. Combining the
results in (ii) and (iii) readily yields the input-output stability of the
system for sufficiently large noises. Next, using Fourier analysis, the
structure of opinion clusters under the uniform initial distribution is
examined. Specifically, two numerical schemes for identification of
order-disorder transition and characterization of initial clustering behavior
are provided. The results of analysis are validated through several numerical
simulations of the continuum-agent model (partial differential equation) and
the corresponding discrete-agent model (interacting stochastic differential
equations) for a particular distribution of radicals
Opinion diversity and community formation in adaptive networks
It is interesting and of significant importance to investigate how network
structures co-evolve with opinions. The existing models of such co-evolution
typically lead to the final states where network nodes either reach a global
consensus or break into separated communities, each of which holding its own
community consensus. Such results, however, can hardly explain the richness of
real-life observations that opinions are always diversified with no global or
even community consensus, and people seldom, if not never, totally cut off
themselves from dissenters. In this article, we show that, a simple model
integrating consensus formation, link rewiring and opinion change allows
complex system dynamics to emerge, driving the system into a dynamic
equilibrium with co-existence of diversified opinions. Specifically, similar
opinion holders may form into communities yet with no strict community
consensus; and rather than being separated into disconnected communities,
different communities remain to be interconnected by non-trivial proportion of
inter-community links. More importantly, we show that the complex dynamics may
lead to different numbers of communities at steady state with a given tolerance
between different opinion holders. We construct a framework for theoretically
analyzing the co-evolution process. Theoretical analysis and extensive
simulation results reveal some useful insights into the complex co-evolution
process, including the formation of dynamic equilibrium, the phase transition
between different steady states with different numbers of communities, and the
dynamics between opinion distribution and network modularity, etc.Comment: 12 pages, 8 figures, Journa
Homogeneous symmetrical threshold model with nonconformity: independence vs. anticonformity
We study two variants of the modified Watts threshold model with a noise
(with nonconformity, in the terminology of social psychology) on a complete
graph. Within the first version, a noise is introduced via so-called
independence, whereas in the second version anticonformity plays the role of a
noise, which destroys the order. The modified Watts threshold model, studied
here, is homogeneous and posses an up-down symmetry, which makes it similar to
other binary opinion models with a single-flip dynamics, such as the
majority-vote and the q-voter models. Because within the majority-vote model
with independence only continuous phase transitions are observed, whereas
within the q-voter model with independence also discontinuous phase transitions
are possible, we ask the question about the factor, which could be responsible
for discontinuity of the order parameter. We investigate the model via the
mean-field approach, which gives the exact result in the case of a complete
graph, as well as via Monte Carlo simulations. Additionally, we provide a
heuristic reasoning, which explains observed phenomena. We show that indeed, if
the threshold r = 0.5, which corresponds to the majority-vote model, an
order-disorder transition is continuous. Moreover, results obtained for both
versions of the model (one with independence and the second one with
anticonformity) give the same results, only rescaled by the factor of 2.
However, for r > 0.5 the jump of the order parameter and the hysteresis is
observed for the model with independence, and both versions of the model give
qualitatively different results.Comment: 12 pages, 4 figures, accepted to Complexit
Game Theoretical Interactions of Moving Agents
Game theory has been one of the most successful quantitative concepts to
describe social interactions, their strategical aspects, and outcomes. Among
the payoff matrix quantifying the result of a social interaction, the
interaction conditions have been varied, such as the number of repeated
interactions, the number of interaction partners, the possibility to punish
defective behavior etc. While an extension to spatial interactions has been
considered early on such as in the "game of life", recent studies have focussed
on effects of the structure of social interaction networks.
However, the possibility of individuals to move and, thereby, evade areas
with a high level of defection, and to seek areas with a high level of
cooperation, has not been fully explored so far. This contribution presents a
model combining game theoretical interactions with success-driven motion in
space, and studies the consequences that this may have for the degree of
cooperation and the spatio-temporal dynamics in the population. It is
demonstrated that the combination of game theoretical interactions with motion
gives rise to many self-organized behavioral patterns on an aggregate level,
which can explain a variety of empirically observed social behaviors
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