53 research outputs found
Mean field models for large data-clustering problems
We consider mean-field models for data--clustering problems starting from a
generalization of the bounded confidence model for opinion dynamics. The
microscopic model includes information on the position as well as on additional
features of the particles in order to develop specific clustering effects. The
corresponding mean--field limit is derived and properties of the model are
investigated analytically. In particular, the mean--field formulation allows
the use of a random subsets algorithm for efficient computations of the
clusters. Applications to shape detection and image segmentation on standard
test images are presented and discussed
How friends and non-determinism affect opinion dynamics
The Hegselmann-Krause system (HK system for short) is one of the most popular
models for the dynamics of opinion formation in multiagent systems. Agents are
modeled as points in opinion space, and at every time step, each agent moves to
the mass center of all the agents within unit distance. The rate of convergence
of HK systems has been the subject of several recent works. In this work, we
investigate two natural variations of the HK system and their effect on the
dynamics. In the first variation, we only allow pairs of agents who are friends
in an underlying social network to communicate with each other. In the second
variation, agents may not move exactly to the mass center but somewhere close
to it. The dynamics of both variants are qualitatively very different from that
of the classical HK system. Nevertheless, we prove that both these systems
converge in polynomial number of non-trivial steps, regardless of the social
network in the first variant and noise patterns in the second variant.Comment: 14 pages, 3 figure
Opinion dynamics: models, extensions and external effects
Recently, social phenomena have received a lot of attention not only from
social scientists, but also from physicists, mathematicians and computer
scientists, in the emerging interdisciplinary field of complex system science.
Opinion dynamics is one of the processes studied, since opinions are the
drivers of human behaviour, and play a crucial role in many global challenges
that our complex world and societies are facing: global financial crises,
global pandemics, growth of cities, urbanisation and migration patterns, and
last but not least important, climate change and environmental sustainability
and protection. Opinion formation is a complex process affected by the
interplay of different elements, including the individual predisposition, the
influence of positive and negative peer interaction (social networks playing a
crucial role in this respect), the information each individual is exposed to,
and many others. Several models inspired from those in use in physics have been
developed to encompass many of these elements, and to allow for the
identification of the mechanisms involved in the opinion formation process and
the understanding of their role, with the practical aim of simulating opinion
formation and spreading under various conditions. These modelling schemes range
from binary simple models such as the voter model, to multi-dimensional
continuous approaches. Here, we provide a review of recent methods, focusing on
models employing both peer interaction and external information, and
emphasising the role that less studied mechanisms, such as disagreement, has in
driving the opinion dynamics. [...]Comment: 42 pages, 6 figure
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
Modelling opinion dynamics under the impact of influencer and media strategies
Digital communication has made the public discourse considerably more complex, and new actors and strategies have emerged as a result of this seismic shift. Aside from the often-studied interactions among individuals during opinion formation, which have been facilitated on a large scale by social media platforms, the changing role of traditional media and the emerging role of “influencers” are not well understood, and the implications of their engagement strategies arising from the incentive structure of the attention economy even less so. Here we propose a novel framework for opinion dynamics that can accommodate various versions of opinion dynamics as well as account for different roles, namely that of individuals, media and influencers, who change their own opinion positions on different time scales. Numerical simulations of instances of this framework show the importance of their relative influence in creating qualitatively different opinion formation dynamics: with influencers, fragmented but short-lived clusters emerge, which are then counteracted by more stable media positions. The framework allows for mean-field approximations by partial differential equations, which reproduce those dynamics and allow for efficient large-scale simulations when the number of individuals is large. Based on the mean-field approximations, we can study how strategies of influencers to gain more followers can influence the overall opinion distribution. We show that moving towards extreme positions can be a beneficial strategy for influencers to gain followers. Finally, our framework allows us to demonstrate that optimal control strategies allow other influencers or media to counteract such attempts and prevent further fragmentation of the opinion landscape. Our modelling framework contributes to a more flexible modelling approach in opinion dynamics and a better understanding of the different roles and strategies in the increasingly complex information ecosystem
Asymptotic Consensus Without Self-Confidence
This paper studies asymptotic consensus in systems in which agents do not
necessarily have self-confidence, i.e., may disregard their own value during
execution of the update rule. We show that the prevalent hypothesis of
self-confidence in many convergence results can be replaced by the existence of
aperiodic cores. These are stable aperiodic subgraphs, which allow to virtually
store information about an agent's value distributedly in the network. Our
results are applicable to systems with message delays and memory loss.Comment: 13 page
Clustering in a network of non-identical and mutually interacting agents
Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behaviour. It is observed in fields ranging from the exact sciences to social and life sciences; consider, for example, swarm behaviour of animals or social insects, the dynamics of opinion formation or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modelling brain or heart cells. We consider a clustering model with a general network structure and saturating interaction functions. We derive both necessary and sufficient conditions for clustering behaviour of the model and we
investigate the cluster structure for varying coupling strength. Generically, each cluster asymptotically reaches a (relative) equilibrium state. We discuss the relationship of the model to swarming, and we explain how the model equations naturally arise in a system of interconnected water basins. We also indicate how the model applies to opinion formation dynamics
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