454,175 research outputs found
On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence
This paper introduces a class of non-linear and continuous-time opinion
dynamics model with additive noise and state dependent interaction rates
between agents. The model features interaction rates which are proportional to
a negative power of opinion distances. We establish a non-local partial
differential equation for the distribution of opinion distances and use Mellin
transforms to provide an explicit formula for the stationary solution of the
latter, when it exists. Our approach leads to new qualitative and quantitative
results on this type of dynamics. To the best of our knowledge these Mellin
transform results are the first quantitative results on the equilibria of
opinion dynamics with distance-dependent interaction rates. The closed form
expressions for this class of dynamics are obtained for the two agent case.
However the results can be used in mean-field models featuring several agents
whose interaction rates depend on the empirical average of their opinions. The
technique also applies to linear dynamics, namely with a constant interaction
rate, on an interaction graph
Dynamics of Majority Rule
We introduce a 2-state opinion dynamics model where agents evolve by majority
rule. In each update, a group of agents is specified whose members then all
adopt the local majority state. In the mean-field limit, where a group consists
of randomly-selected agents, consensus is reached in a time that scales ln N,
where N is the number of agents. On finite-dimensional lattices, where a group
is a contiguous cluster, the consensus time fluctuates strongly between
realizations and grows as a dimension-dependent power of N. The upper critical
dimension appears to be larger than 4. The final opinion always equals that of
the initial majority except in one dimension.Comment: 4 pages, 3 figures, 2-column revtex4 format; annoying typo fixed in
Eq.(1); a similar typo fixed in Eq.(6) and some references update
Non-Markovian models of opinion dynamics on temporal networks
Traditional models of opinion dynamics, in which the nodes of a network
change their opinions based on their interactions with neighboring nodes,
consider how opinions evolve either on time-independent networks or on temporal
networks with edges that follow Poisson statistics. Most such models are
Markovian. However, in many real-life networks, interactions between
individuals (and hence the edges of a network) follow non-Poisson processes and
thus yield dynamics with memory-dependent effects. In this paper, we model
opinion dynamics in which the entities of a temporal network interact and
change their opinions via random social interactions. When the edges have
non-Poisson interevent statistics, the corresponding opinion models are have
non-Markovian dynamics. We derive an opinion model that is governed by an
arbitrary waiting-time distribution (WTD) and illustrate a variety of induced
opinion models from common WTDs (including Dirac delta distributions,
exponential distributions, and heavy-tailed distributions). We analyze the
convergence to consensus of these models and prove that homogeneous
memory-dependent models of opinion dynamics in our framework always converge to
the same steady state regardless of the WTD. We also conduct a numerical
investigation of the effects of waiting-time distributions on both transient
dynamics and steady states. We observe that models that are induced by
heavy-tailed WTDs converge to a steady state more slowly than those with light
tails (or with compact support) and that entities with larger waiting times
exert a larger influence on the mean opinion at steady state.Comment: 24 pages, 7 figure
Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems
Recently, significant attention has been dedicated to the models of opinion
dynamics in which opinions are described by real numbers, and agents update
their opinions synchronously by averaging their neighbors' opinions. The
neighbors of each agent can be defined as either (1) those agents whose
opinions are in its "confidence range," or (2) those agents whose "influence
range" contain the agent's opinion. The former definition is employed in
Hegselmann and Krause's bounded confidence model, and the latter is novel here.
As the confidence and influence ranges are distinct for each agent, the
heterogeneous state-dependent interconnection topology leads to a
poorly-understood complex dynamic behavior. In both models, we classify the
agents via their interconnection topology and, accordingly, compute the
equilibria of the system. Then, we define a positive invariant set centered at
each equilibrium opinion vector. We show that if a trajectory enters one such
set, then it converges to a steady state with constant interconnection
topology. This result gives us a novel sufficient condition for both models to
establish convergence, and is consistent with our conjecture that all
trajectories of the bounded confidence and influence models eventually converge
to a steady state under fixed topology.Comment: 22 pages, Submitted to SIAM Journal on Control and Optimization
(SICON
Opinion Dynamics With Cross-Coupling Topics: Modeling and Analysis
To model the cross couplings of multiple topics, we develop a set of rules for opinion updates of a group of agents. The rules are used to design or assign values to the elements of weighting matrices. The cooperative and anticooperative couplings are modeled in both the inverse-proportional and proportional structures. The behaviors of opinion dynamics are analyzed using a nullspace property of the state-dependent matrix-weighted Laplacian matrices and a Lyapunov candidate. Various consensus properties of the state-dependent matrix-weighted Laplacian matrices are predicted according to the interagent network topology and interdependent topical coupling topologies
Signed bounded confidence models for opinion dynamics
The aim of this paper is to modify continuous-time bounded confidence opinion dynamics models so
that ‘‘changes of opinion’’ (intended as changes of the sign of the initial states) are never induced during
the evolution. Such sign invariance can be achieved by letting opinions of different sign localized near the
origin interact negatively, or neglect each other, or even repel each other. In all cases, it is possible to obtain
sign-preserving bounded confidence models with state-dependent connectivity and with a clustering
behavior similar to that of a standard bounded confidence model
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